JEE Advanced Problems in Mathematics 5th Edition - Book

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Fourth Edition :2017
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Preface
Authors
We are pleased to present this book before the readers. It is specially 
designed to help the students preparing for IIT-JEE and book is strictly based 
upon the current pattern being followed in IIT-JEE. Each chapter includes 
different types of problem, which are:
Here we have made an effort to provide a unique set of problems and its 
answer key experienced by us in our teaching career.
We are human being and errors are always probable. If you find any in this 
book that will be absolutely our responsibility. However before deciding upon a 
slip up, the readers should feel free to check with us by post. We would also 
welcome all suggestions from our readers for improving this edition of our book.
Finally, We convey our best wishes for you in your studies and success in 
your bright career.
(1) Only one choice is correct, (2) One or more than one is/are correct, (3) 
Comprehension, (4) Match Type problems, (5) Subjective problems.
Vikas Gupta
(Email: gvikas1378@yahoo.com)
Pankaj Joshi
(Email: humekant2@gmail.com)
mailto:gvikas1378@yahoo.com
mailto:humekant2@gmail.com
Authors
We would like to pay our sincere thanks to all our colleagues : 
Mr. Nitin Jain (NJ), Mr. Neel Kamal Sethia (NKS), Mr. M.S. 
Chouhan (MSC), Mr. Narendra Awasthi (NA) and Mr. Vimal 
Kumar Jaiswal (VKJ), our faculty members, Sanjay Kumar 
Singhal, Nishant Gupta, Gopal Patidar, Ravi Pratap Singh, 
Manoj Chouhan, Shubhanu Purohit, Sudershan Vairagi, 
Mandhir Sood, Aditya Bhardwaj, Sonu Prakash Agarwal, Jai 
Prakash Dixit, Arun Kumar Panwar, Pradeep Kumar, Nitesh 
Goyal, Shubham Agarwal, Shubham Gupta, Siripalli Nitin 
Sailesh, Shubham Jhanwar, Subrato Karmakar, Hanush Kumar, 
Vasu Jain, Devendra Sukheja & our students.
We would also like to convey my thanks to M/s Shri Balaji 
Publications for publishing this book in such a beautiful format.
Grow Groon
Save Notwro
CONTENTS
3-291. Function
30-442. Limit
3. Continuity, Differentiability and Differentiation 45-74
75-974. Application of Derivatives
98-1275. Indefinite and Definite Integration
128-1346. Area Under Curves
135-1447. Differential Equations
147-1768. Quadratic Equations
177-1979. Sequence and Series
198-20610. Determinants
207-21611. Complex Numbers
217-22412. Matrices
225-23313. Permutation and Combinations
234 - 24214. Binomial Theorem
243 - 25115. Probability
252 - 26416. Logarithms
EAECULUS-W
LGEBRA^~»
267 - 28017. Straight Lines
281 -29518. Circle
296 - 30219. Parabola
303 - 30720. Ellipse
308-31221. Hyperbola
315-33422. Compound Angles
335 - 34323. Trigonometric Equations
344 - 35924. Solution of Triangles
25. Inverse Trigonometric Functions 360 - 370
26. Vector & 3Dimensional Geometry 373 - 389
O-ORDINATE GEOMETRY*
TRIGONOMETRY—**
VECTOR & 3DIMENSIONALGEOMETRY-
BalaJi
Advanced Problems in
mat:: ematics
for
by :
Pankaj Joshi
Director
Vibrant Academy India(P) Ltd. 
KOTA (Rajasthan)
Vikas Gupta
Director 
Vibrant Academy India(P) Ltd. 
KOTA (Rajasthan)
SHRI BALAJI PUBLICATIONS
(EDUCATIONAL PUBLISHERS & DISTRIBUTORS) 
[AN ISO 9001-2008 CERTIFIED ORGANIZATION] 
Muzaffarnagar (U.P.) - 251001
JEE (MAIN & ADVANCED)
Calculus
1. Funtion
2. Limit
3. Continuity, Differentiability and Differentiation
4. Application of Derivatives
5. Indefinite and Definite Integration
6. Area under Curves
7. Differential Equations
-i
•V
Function
•zExercise-1: Single Choice Problems
X
(d) (-3h/4,3ti/4)
(b) 1 (d) 3
- a is satisfied for maximum number
7. The maximum value of sec is :
(c)
7n
12
(d)
3
 
k 2(x2 + 2) ,
(b) n
(c) 2 
. i^l
5. The complete range of values of ‘ a’ such that I -
f 5n (a).-
I- .-sin 
2
(a) (-7i/2,7i/2) (b) [-ti/2,71/2]-{0} (c) [-n/2,n/2]
4. Find the number of real ordered pair(s) (x,y) for which : 
16x2+> +16x+j'2 =1 
(a) 0
of values of x is :
(a) (-a),-l) (b) (-oo.oo) (c) (-1,1) (d) (-l,oo)
6. For a real number x, let [x] denotes the greatest integer less than or equal to x. Let f: R -> R be 
defined by /(x) = 2x + [x] + sin x cosx. Then f is :
(a) One-one but not onto (b) Onto but not one-one
(c) Both one-one and onto (d) Neither one-one nor onto
= x2
1. Range of the function/(x) = log2(2-log^2(16sin2x + l))is :
(a) [0,1] (b) (-oo,l] (c) [-1,1] (d) (-00,00)
2. The value of a and b for which | - a| = 2, has four distinct solutions, are :
(a) a e (-3,oo),b = 0 (b) a e (2,oo),b = 0 (c) a e (3,oo), beR (d) a e(2,oo),b =a
3. The range of the function :
/(x) = tan
Advanced Problems in Mathematics for JEE4
(b) reflexive (d) transitive
14. The true set of values of ‘K’ for which sin
(a) (b) [1,3] (d) [2,4](c)
valued
(d) None of these
17. Let/ :D -> R be defined as :/(x) =
1
1 + sin2 x
‘1 1'
.6’2.
satisfies
(c) not symmetric
= ~ may have a solution is :
(c) /(a) + /(a-x) (d) /(-x) 
og(x)n times. Then inverse of
(b) (x-l + 4n)4“n (c) (x + l + 4n)4
x2 + 2x + Q—------------where D and R denote the domain off and the
x + 4x + 3a
set of all real numbers respectively. If/is surjective mapping, then the complete range of a is:
(a) 0 B (b) A B, ffx) = 2x3 - 3x2 + 6 will have an inverse for 
the smallest real value of a, if:
(a) a=l,B=[5,oo) (b) a = 2,B =[10,oo) (c) a = 0,B=[6,oo) (d) a =-1,B = [1,oo)
11. Solution of the inequation {x} ({x} -l)({x} + 2) > 0
(where {•} denotes fractional part function) is :
(a) x e (-2,1) (b) x el (1 denote set of integers)
(c) x g[0,1) (d) xe[-2,0)
12. Let /(x),g(x) be two real valued functions then the function h(x) = 2max{/(x) -g(x), 0} is 
equal to :
(a) /(x) -g(x) -|g(x) -/(x) | (b) /(x) + g(x) —|g(x) —/(x) |
(c) /(x)-g(x)+|g(x)-/(x)| (d) /(x) + g(x)+|g(x)-/(x)|
13. LetR ={(1,3), (4,2), (2,4), (2, 3), (3,1)} be a relation on the set A ={1, 2, 3,4}. The relation 
R is :
(a) a function
1 1
.4’2.
15. A real valued function /(x) satisfies the functional equation 
/(x-y) = /(x)/(y)-/(a -x)/(a + y) where ‘a’ is a given constant and /(0) = 1, /(2a-x) is 
equal to : . • .
(a) -/(x) (b) /(x)
16. Let g :R -> R be given by g(x) = 3 + 4x if gn(x) = gogogo 
gn(x) is equal to :
(a) (x + l-4n)-4
5Function
(d)(b) ,-l(a) [-2,-1) ,-l
(b)(a)
(d)(c)
22. If f:R -*R,ftf =
(d) Infinite(c) 7
0
Then the graph ofg(x) =
greatest integer function) :
(a) 5 (b) 6
25. The graph of function /(x) is shown below :
3-73 + 2
5
3V3 + 4
5
n (n +1)
2
n(n + l) n2 + n + 2
2 ’ 2
18. If/:(-oo, 2]----> (—oo, 4], where /(x) = x(4-x), then / 1 (x) is given by :
(a) 2-/4-X (b) 2 + /4-X (c) -2 + V4-X (d) -2-V4-X
19. If [5 sin x] + [cosx] + 6 = 0, then range of /(x) = cosx + sin x corresponding to solution set of
the given equation is : (where [•]90. Let f (x) =
x->0
(d) 3
f'M
7
10
5
fM
1
2
3
10
x
3
6
9
, x * 0
, x = 0
1 + sin
1 - tan
x + l-{x} ; ll
1. If/(x) = tan 
be :
(a) 1
If f is continuous at x = 0 then correct statement(s) is/are :
(a) a + c = —1 (b) b + c = — 4
(c) a + b = -5 (d) c + d = an irrational number
6. If f(x) =||| x|- 2| + p| have more than 3 points of non-derivability then the value of p can be :
(a) 0 (b) _x
W ~2 (d) 2
(a) /(x) is non-differentiable at exactly three points
(b) /(x) is continuous in (- Mt
cx + dx3
y-
59Continuity, Differentiability and Differentiation
is discontinuous at exactly one point(d) /(x) =
(d) -2(c) 2
7. Identify the options having correct statement:
(a) /(x) = ^x2|x| — 1 —| x| is no where non-differentiable
(b) lim ((x + 5) tan-1 (x +1)) - ((x +1) tan-1 (x +1)) = 2rc 
X—>co
(c) /(x) = sin (In (x + v x 
4-x2
4x-x3
8. A twice differentiable function /(x) is defined for all real numbers and satisfies the following 
conditions :
/CO) = 2;
The function g(x) is defined by g(x) = e 
g'(0) + g"(0) = 0 then ‘a’ can be equal to : 
(a) 1 (b) -1
9. If /(x) =| x| sin x, then f is :
(a) differentiable everywhere
(c) not differentiable at x = 0
(b) not differentiable at x = nn,n el
(d) continuous at x = 0
10. Let [ ] denotes the greatest integer function and /(x) =[tan2 x], then
(a) lim /(x) does not exist (b) /(x) is continuous at x = 0
x-»0
(c) /(x) is not differentiable at x = 0 (d) /'CO) = 0
11. Let f be a differentiable function satisfying /'(x) = /'(-x) V x g R. Then
(a) If /(I) = /(2), then /(-l) = f (-2)
1 1 (1 A(b) -/(x) + -/(y) =/l -(x + y) J for all real values of x,y
(c) Let /(x) be an even function, then J(x) = 0 V x g R
(d) /(x) +/(-x) = 2/(0) Vx gR
12. Let f:R -> R be a function, such that |/(x) | 1
(b) lim /(x) does not exist 
x->l
(c) gof is continuous for all x
(d) fog is continuous for all x
2 when x is not an integer ([ ] is
2 +1)) is an odd function
f'(0) = -5 and /(O) = 3.
“ + fW V x g R, where ‘ a’ is any constant. If
1
2
Advanced Problems in Mathematics for JEE60
14. Let the function f be defined by /(x) =
(c) J(x) is not differentiable at x = 0
= 0
19. Let the function f be defined by /(x) =
-35
4
35 
8
./ 3
(b) /Cx) is not differentiable at x = 1
Cd) /'(-!)/'
17. If /Cx) be a differentiable function satisfying /Cy)/ — = /(x) V x,y e R, y * 0 and /(I) * 0,
Oj) lim
x->0
/'Cl) = 3, then :
(a) sgn C/Cx)) is non-differentiable at exactly one point 
x2(cosx-l)
7w
Cc) /Cx) = x has 3 solutions
(d) /(/Cx)) - f 3 (x) = 0 has infinitely many solutions
18. Let/(x) = (x2 - 3x + 2)(x2 + 3x + 2) and a,p, y satisfy a 2
Ca) /(x) is continuous in R if 3p + lOq = 4
(b) f (x) is differentiable in R if p = q =
Cc) If p = -2, q =1, then /(x) is continuous in R
Cd) f Cx) is differentiable in R if 2p + llq = 4
16. Let /(x) = max Cx, x2, x3) in -2 (c) /(-!) + /! |j =
’ x 2
p + qx + x2 
2px+ 3qx2
Ca) /(x) is continuous in R if 3p + lOq = 4
(b) fM is differentiable in R if p = q =
Cc) If p = -2, q = 1, then /(x) is continuous in R
Cd) /(x) is differentiable in R if 2p + llq = 4
15. Let/(x) =|2x-9| + |2x| + |2x + 9|. Which of the following are true ?
9 -9Ca) /(x) is not differentiable at x = — Cb) /(x) is not differentiable at x = —
2 2
-9 9Cd) /(x) is differentiable at x = —, 0, - 
2
-9
2
 9
2
Continuity, Differentiability and Differentiation 61
20. If y = exs,n(*3) + (tanx)x then — may be equal to :
3
(d) exsin(x3)[3x3 cos(x3) + sin(x3)] + (tanx) In tan x +
)x";(n >4)
r3x + b ; 0 100(c) n 2
(d) -^ = 3 
A»
(l-xn-1
1+x 0 [x3 cos(x3) + sin(x3)] + (tanx)x[lntanx + 2xcosec2x]
(c) exs,n(x )[x3 sin(x3) + cos(x3)] + (tanx)x[lntanx + 2cosec2x] 
xsec2 x
tanx
L 2cosnx +tan-1 x ;lat x = 0
(d) is discontinuous at x = 1
30. A function /(x) satisfies the relation :
f(x + y) = /(x) + f(y) + xy(x + y)Vx,y e R. If f'(0) = -1, then :
(a) /(x) is a polynomial function
(b) /(x) is an exponential function
(c) f (x) is twice differentiable for all x e R
(d) f'(3) = 8
Then which of the following is/are correct ?
(a) a + b + c = 0 (b) b=a + c (c) c = l + b (d)b2+c2=l
28. /(x)is differentiable function satisfying the relationship/2(x) + /2(y) + 2(xy -1) = /2(x + y)
V x,y eR
Also /(x) > 0 V x e R and /(V2) = 2. Then which of the following statement(s) is/are correct 
about /(x) ?
(a) [/(3)] = 3 ([•] denotes greatest integer function)
(b) /(V7) = 3
(c) /(x) is even
(d) f'(0) = 0
29. The function /(x) = yl--Jl-x
2—|x| , x2
If h(x) = /(x) + g(x) is discontinuous at exactly one point, then which of the following are 
correct ?
(a) a = -3,b = 0 (b) a = -3,b = -l (c) a = 2,b=l
27. Let /(x) be a continuous function in [-1, 1] such that 
ln(qx2 + to + c) 
x2
I*!"3 > *l’gW
; 02
(c) /(x) is continuous at x = 2
71
2
(')3
v'4>" 
(')2
(d) 7t
f -v 71
W 6
; x3
(a) g"(e) = 
(1 + e)
3x-x2
2 
[x-1] 
x2
x^
2
2x2 - 3x + — Kx ')3
, x el 
, xel
(a) go/ is continuous for all x
(b) gof is not continuous for all x
(c) fog is continuous everywhere
(d) fog is not continuous everywhere
35. Let f :R+ -> R defined as /(x) = ex + Inx and g = /-1 then correct statement(s) is/are :
e -1 1(c) g'(e) = e + l (d)g'(e) = —
(1 + e)3 e + 1
64 Advanced Problems in Mathematics forJEE
Answers
(c, d) (a, c, d) Cb, c)2. (a, b, c) (a, b, d)3. (a, b, c, d) 6.4. 5.
(a, b, c)7. 8. (a, d) (a, d) (b.d)(b, d) (a, d)9. 10. 11. 12.
(a, b, c) (a, b, c)13. (a, b, c) (a, b, c) (a, c)14. 15. 16. (a, b, c, d)17. 18.
(a, b, c)19. (a, d) (b.c) (a, b, c, d) (a, b)20. Cb, c)21. 22. 23. 24.
(a,c,d)(a, c) (a, b, c, d)25. 26. fed) (a, b, c, d)27. (a, b, d)28. 29. 30.
(b,c) (a, b, d) Cb, d) (a, c, d) '31. (a) (a, d)32. 33. 36.34. 35. J
I
Continuity, Differentiability and Differentiation 65
Exercise-3 : Comprehension Type Problems
Paragraph for Question Nos. 1 to 2
and g(x) = min (f(x), {x})
(where {•} denotes fractional part function)
(d) Does not exist
(d) 3
(d) 3(c) 2
4. The number of integers in the domain of the function F(x) =
(c) 2(b) 1(a) 0
h (x) =
(d) V2-1
Paragraph for Question Nos. 3 to 4
Let /(x) and g(x) be two differentiable functions, defined as : 
/(x) =x2+ xg'(l) + g"(2) and g(x) =/(l)x2 + x/'(x) +/''(x).
5. /(x) is non-differentiable at
(a) 3,6 (b) 2,3
6. h(x) is discontinuous at x = 
(a) 1 + V2
3. The value of /(I) + g(-1) is :
(a) 0 (b) 1
points and the sum of corresponding xvalue(s) is 
(c) 2, 4 (d) 2, 5
1. Left hand derivative of 4>(x) = e',ZJ'~xl at x = 0 is :
(a) 0 (b) 1 (c) -1
2. Number of points in x e [-1,2] at which g(x) is discontinuous :
(a) 2 (b) 1 (c) 0
lil *tan In sec —
I \ n
g(x)
(d) Infinite
(b) tan —
8
(c) tan —
8
Let/(x) = lim n2
Paragraph for Question Nos. 5 to 6
Define : /(x) =|x2 -4x + 3|lnx + 2(x-2)1/3,x > 0 
x-1 , xeQ
x2-x-2 , xgQ
66 Advanced Problems in Mathematicsfor J El
(d) 5
10.
(b) 1 (c) -1 (d) 2
Let /(x) = where [x] = greatest integer less than or equal to x, then :
(d) 6
Paragraph for Question Nos. 14 to 16
Let f:R -> R be a continuous and differentiable function such that /(x + y) = /(x)- /(y) 
V x, y, f (x) * 0 and /(0) = 1 and /'(O) = 2.
LetgOy) =g(x)-g(y) Vx,y and g'(l) = 2;g(l) * 0
Paragraph for Question Nos. 11 to 13 
0729
(d) g(3) = 6
(d) 3
Paragraph for Question Nos. 17 to 18
andg(x) = Xtanx + (1 - X) sin x - x, where X g R andx g [0, n/2).
(b)
(d)(c)
(d)(c)(a) [l.oo) (b) [0,oo)
Let
, then
Cd) 4
(d) Non-existent(c) 4(b) 2
(d) Non-existent(a) 7
21. lim lim f(x~) + lim (5/(x)) is equal to
x—>-2” x->-2‘
(where {•} denotes fraction part function)
(b) 8 (c) 12
2f(x)
l + /2(x)
, n g N and 
+ xz + l
(1 -cosx)(X-/(x)) 
cosx
(1 -cosx)(X-/(x))
’ (/(X))2
14. Identify the correct option :
(a) /(2) = e4 (b) /(2) = 2e2
15. Identify the correct option :
(a) g(2) = 2 (b) g(3) = 3 (c) g(3) = 9
16. The number of values of x, where fix') = g(x):
(a) 0 (b) 1 (c) 2
—, 00
.3
/(x) = lim -
n->® (
. 1 •g(x) = tan — sm
Paragraph for Question Nos. 19 to 21
X2 + 2(X-rl)2n
(X + I)2"* -! X2
17. g'(x) equals
(1-cosx)(/(x)-X) 
taj -----------------------------
cosx
(1 -cosx)(X-/(x))
7w
18. The exhaustive set of values of1X’ such that g'(x) > 0 for any x g [0, n/2) :
1
— , 00
2
20. lim ——~r is equal to : 
x->-3 sin (x + 3)g(x)
(a) 1
/(x)
tan2 x
19. The number of points where g(x) is non-differentiable V x g R is :
(a) 1 (b) 2 (c) 3
(x2 + 4x+ 3)
2
T rr COS XLet fM =---------------—
1 + C0SX+ COS X
68 Advanced Problems in Mathematics for JEE
is :
(b) 2V3 (c) V3
24. If ' [ — 4-1 ] equals to :
(c) 1 (d) 0
25. /(x) increases in the complete interval:
is :
(a) 0 (c) e
= 3
Paragraph for Question Nos. 22 to 24
Let f and g be two differentiable functions such that: 
/(x) =g'(l)sinx +(g"(2)-l)x
n) 
2J
(b) (-00, co)
(d) co, i) o ci, 00)
(d) -infinite
2n
T
Let /(x) be a polynomial satisfying lim - 
x-^00
/(2) = 5,/C3) = 10,/(-l) = 2J(-6) = 37
22. The number of solution(s) of the equation /(x) = g(x) is/are :
(a) 1 (b) 2 (c) 3
g(cosx))
(c) (-co, -l)o (-1, 0)
26. The value of the limit It (f(x))x 
X->00
(b) 1
(a) ^ + 1
2
U
(b)
(d) A
V3
Paragraph for Question Nos. 25 to 26
Suppose a function /(x) satisfies the following conditions
fl* + = ^Xfr+^\’ x,y eRand /'CO) = 1
l + /(x) /(y)
Also -1(b) 23. (b) 26. (b) (d) 28.22. 24. 27.
(b) (a)31. 32.
F
7.
Paragraph for Question Nos. 29 to 30
and g(x) = e2x. Let a and p be two values of x satisfying f (x) = g(x)
/(x)-x2-l
3(x + 6)
10. (c)
20. (b)
30. ’ (d)
I
= I then the value of c -1 equals to :
00 00
31. The minimum value of (cosec9) + where 0 * —; k g I is :
k=i k=i 2
(a) 1 (b) 2 (c) V2 (d) 4
32. The number of values of x for which g(x) is non-differentiable (x g R - {0}):
(a) 3 (b) 4 (c) 5 (d) 1
(d)
2
Paragraph for Question Nos. 31 to 32 
n
— V x, y g R - {0} where n g N and
J6
2
1_____
x2 + l-/(x)
(C) 1
xn + y 
xny " 
g(x) = max p2W>
1 (b)
. ’-15 51 in ----- , - equals:L 2 2j
(d) 0
' (a) 1 15. (c)
I (c) i 25. (b)
Consider f(x) = xlnx 
(a —|x| where [■], {•} are greatest integer and fractional part 
respectively then match the following List-I with List-II.
Continuity, Differentiability and Differentiation 71
(C) (R) 4
(D) (S) 2
4.
(A)
(B) (Q) 2
I
(C) 5
(D) 16
if
Column-1 Column-ll
a cannot has value
b has value
-1
1
? Answers
!
i
(A)
(B)
(C)
(D)
p has value
q has value
B—> S ;
B—> S ;
B-> Q ;
1/3
0
2.
3.
4.
5.
1 3
------\
(P)
Column-ll
1
2
x 
COS —
2
lim
(P) |
(Q)
(R)
(S)
Let /(x) = max. (cosx, x, 2x -1) where x > 0 then (R) 
number of points of non-differentiability of /(x) is
If /(x) = [2+ 3sinx], 0 P, Q, R; B-»Q; C-»P, Q; D->R, S,T
A-> P, Q, R, S, T ; B->P,T; C-»Q, S; D-»P, R, T 
A —> Q » B —> S’ C —> P j D —>R
A —> P j B —> S ; C —> Q j D —> R 
A-4-S; B->Q; C->P; D-> R
Cqlumn-J 
2r+l 
x2 + 2x-1 Yfr-1 = 
2x2 -3x-2y
Iogsecx/2C°SX 
lim —
x->0
72 Advanced Problems in Mathematics for JEE
Exercise-5 : Subjective Type Problems ___ /
1. Let /(x)
2. Ify = sin(8sin
6. If/(x) =
-. If F(x) is
8. Let /(x) =
b •
3
121 sin
7t I :
/(x) + (x2)ng(x) 
l + (x2)n
a— + cos
o
cosx ; x 0
then find the number of points where g(x) = /(| x|) is non-differentiable.
7. Let /(x) = x2 + ax + 3 and g(x) = x + b, where F(x) = lim - 
n->3
differentiable everywhere and f'M is continuous at x = 3 and |a + p + b + q| = k, then k = 
x) then (1 -x2)^-^- -x— = -icy, where k = 
dx2 dx
„ ,, o i_ d2y ka2 , , 2
3. If y = 4ax, then —= ——, where k =
dx2 y3
4. The number of values of x, x e [-2,3] where /(x) = [x2] sin (nx) is discontinuous is
(where [■] denotes greatest integer function)
5. If /(x) is continuous and differentiable in [-3, 9] and f'M e [-2,8] V x e (-3,9). Let N be the 
number of divisors of the greatest possible value of/(9) -/(-3), then find the sum of digits of 
N.
= n and
3/2
j /(x)dx = - + 1,
1/2
then find the value of
Continuity, Differentiability and Differentiation 73
h'(5) = 0, then
is equal to
X—>00
17. Let f (x)
18. If/(x) =
also /(I) = 0;/'(l) = 1
(where [•] denotes greatest integer function).
derivative of /(x) w.r.t. x, k e N. If
2
22. If x = cos0 and y = sin3 0, then
23. The value of x, x e (2,oo)where/(x) =7
for
C X \
16. Let f be a continuous function on [0, oo) such that lim /(x) + f f(t) dt exists. Find lim /(x).
X->oo J X->ao
k 0 J
dx
find lim
x->e
at 0 = - is :
2
V8x -16 + -Jx - V8x -16 is not differentiable is :
f 324. The number of non differentiability points of function f(x) = mini [x],{x}> x--
x e (0,2), where [■] and {•} denote greatest integer function and fractional part function 
respectively.
a + bx3'2 
x94
15. Ify =e2sin
x2 x3 X4 x5 _1
■ x + — + — + — + — and letg(x) = f 1 (x). Find g'"(0). 
2 3 4 5
cosx3 ; x 0
then find the number of points where g(x) = /(|x|) is non-differentiable.
19. Let f:R +---- > R be a differentiable function satisfying :
/(xy) = —+ — Vx,y gR+ 
y *
1
./(*).
20. For the curve sinx + siny = 1 lying in the first quadrant there exists a constant a for which
d2ylim xa —— = L (not zero), then 2a = 
x^o dx2
21. Let /(x) = xtan-1(x2) + x4. Let /fc(x) denotes kth
y2"1 (0) * 0, m g N, then m =
13. Let a(x) = /(x) - /(2x) and P(x) = /(x) -/(4x)
and a'(l) = 5 a'(2)=7
then find the value of 0'(1) -10 
x3 x2+ — + — + x +1 and g be inverse function of f and b(x) =
a2___ =
\ O J 
(x2 -l)y" 4-xy' 
y
14. Let /(x) = -4-e 2
x then
yd2y , 
dx2
74
□□□
3
2
4
3
64
4
0
4
16
3
1
3
3
2
2
2
9
3
17
5
2
2.
9.
16.
23.
3.
10.
17.
24.
5.
12.
19.
6.
13.
20.
7.
14.
21.
1.
8.
15.
22.
4.
18.
Advanced Problems in Mathematics for JEE
§ Answers J
8
7
2
Application of
Derivatives
Exercise-1 : Single Choice Problems
functionthe
(d) 4(c) 3
(d) 0(C) 2(b) 1
(b) 15 sec
(d) 30 sec
(b) —'—cm/min 
18n
(d) —cm/ min
6rt
(c) —— cm/min 
54n
of the1. The difference between
fix') = 3sin4 x - cos6 x is :
3 5(a) (b)
2 2
2. A function y = f (x) has a second order derivative f"(x) = 6(x -1). If its graph passes through
the point (2,1) and at that point the tangent to the graph is y = 3x - 5, then the function is :
(a) (x-1)2 (b) (x-1)3 (c) (x + 1)3 (d)(x + l)2-
3. If the subnormal at any point on the curve y = 3l~k ■ xk is of constant length then k equals to:
4. Ifx5 -5qx + 4r is divisible by (x-c)2 then which of the following must hold true V q,r,c eR?
(a) q = r (b) q + r = 0 (c) q5=r4 (d)q4=r5
5. A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts 
at a rate of 50 cm 3/min. When the thickness of ice is 5 cm, then the rate at which the thickness 
of ice decreases, is :
(a) -^—cm/min
36rt
6. If f (x) - , then number of local extremas for g(x), where g(x) = /(| x |):
(a) 3 (b) 4 (c) 5 (d) None of these
7. Two straight roads OA and OB intersect at an angle 60°. A car approaches O from A, where 
OA = 700 m at a uniform speed of 20 m/s, Simultaneously, a runner starts running from O 
towards B at a uniform speed of 5 m/s. The time after start when the car and the runner are 
closest is :
(a) 10 sec
(c) 20 secmaximum and minimum value
76 Advanced Problems in Mathematics for JEE
8. Let J(x) =
(c) (3,oo) (d) [3,oo)
9. /(x)
(d) 1 2(a) a g R
(a) 0 (d) None of these
(c) -
(a) (b) (c) (d) 1
i (d) x + 2y - 4 = 0 
value of the function
(b) (-oo,3]
, x has minimum at x = k, then :
(a) 2 (b) 0 (c) 3 (d) 1
12. If (a,b) be the point on the curve 9y 2 = x3 where normal to the curve make equal intercepts 
with the axis, then the value of (a + b) is :
d2y13. The curve y = f(x) satisfies —5- = 6x - 4 and /(x) has a local minimum value 5 when x = 1.
dx2
, > 20(c) T
(c) 2x-y + 2 = 0 
and the least
a2
a -3x ;
4x + 3 ;
(a) (-oo, 3)
3+|x-k|
_2 + sin(x-k)
(x-k)
(b) |a|1
23. Let /(x) = min - ——, —— for 0 1
x2
andg(x) = j(2t 
1
(a) g is increasing on (1, oo)
(b) g is decreasing on (1, oo)
(c) g is increasing on (1, 2) and decreasing on (2, oo)
(d) g is decreasing on (1, 2) and increasing on (2, oo)
19. Let /(x) = x3 + 6x2 + ax + 2, if (-3,-1) is the largest possible interval for which /(x) is 
decreasing function, then a = 
(a) 3 (b) 9
1A
------I. Then difference of the greatest and least value of/(x) on [0,1] is :
(a) n/2 (b) ji/4 (c) n (d) 7t/3
21. The number of integral values of a for which /(x) = x3 + (a + 2) x2 + 3ax + 5 is monotonic in 
V x g R.
(a) 2
2 -lnt)/(t)dt (x > 1), then :
2 meets they-axis atT; then
5
4
= [2-|x2 + 5x+6| x*-2
[ b2 + l x = -2
Has relative maximum at x = -2, then complete set of values b can take is :
(a) |b|> 1 (b) |b|l (d) b) (1,1
I 2 
(d) -,2 
lit
(d) None of ±ese
28. Angle between the tangents to the curve y = x2 - 5x + 6 at points (2, 0) and (3, 0) is :
(a) - (b) - (c) (d)
6 4 3 2
X
29. Difference between the greatest and least values of the function /(x) = j (cos21 + cost + 2)dt
o
1,2
31. Number ofintegers in the range ofc so that the equation x3 -3x + c = 0 has all its roots real and 
distinct is:
(a) 2 (b) 3 (c) 4 (d) 5
32. Let /(x) = | ex (x -1) (x - 2) dx. Then /(x) decreases in the interval:
(a) (2,oo) (b) (-2,-1)
(c) (1,2) (d) (-oo, 1) u (2, co)
33. If the cubic polynomial y = ax3 + bxz + ex + d (a,b,c,d e R~) has only one critical point in its 
entire domain and ac = 2, then the value of | b\ is :
(a) V2 (b) V3 (c) Vs
dythe point at which — is greatest in the first quadrant is:
dx
1
1 + x2’
(d)4“4-2 y x® 4=4-2 x y
1_
y2
> 2 V x g (0, co) is :
(c) 5
in the interval [0,2k] is Kit, then K is equal to :
(a) 1 (b) 3 (c) 5 (d) None of these
30. The range of the function /(0) = + - ^ ,6 e f 0, j is equal to :
Lagrange’s mean value theorem is applicable in [-1, 1] then ordered pair (m, c) is : 
« [1,-;
26. Tangents are drawn to y = cos x from origin then points of contact of these tangents will always 
lie on :
, . 1 1 n
(a) ~= —+ 1
xz
27. Least natural number a for which x + ax
Application of Derivatives 79
36. Let /(x) = sin , then which are correct ?
(d) [2,oo)
(c) a-—,b=-
40. Let fix') = then :
41. If m and n are
,2
2
OP,OQ have inclinations a, b where O is origin, then has the value, equals to :
(d) V2(b) -2(a) -1
(b) f has a local minimum at x = 0
(d) f is decreasing everywhere
1
8
1
8
1 + sinx, 
x2 — x +1,
(a) f has a local maximum at x = 0
(c) f is increasing everywhere
35. If/(x) = 2x,g(x) = 3sinx-xcosx, then for x e^0,•
(b) /(x) g(x)
(c) /(x) = g(x) has exactly one real root.
2g(x)
l + g(x)2
(i) /(x) is decreasing if g (x) is increasing and | g (x) | > 1
(ii) /(x) is an increasing function if g(x) is increasing and | g(x) | 1
(a) (i) and (iii) (b) (i) and (ii) (c) (i), (ii) and (iii) (d) (iii)
37. The graph of the function y = f(x) has a unique tangent at (eQ, 0) through which the graph
, .. ln(l + 7/(x))-sin(f(x)) .passes then hm ---------- --------------- ------- is equal to :
(a) 1 (b) 3 (c) 2 (d) 7
38. Let/(x) be a function such that/'(x) = log1/3 flog 3 (sin x + a)). The complete set of values of* a'
for which /(x) is strictly decreasing for all real values of x is : 
(a) [4, co) (b) [3,4] (c) (-co, 4)
39. If /(x) = a In | x| + bx2 + x has extremas at x = 1 and x = 3, then :
r \ 3 t 1 3,1(a) a = -,b=— (b) a = -,b=-
4 8 4 8
x > O’
X,. 3 , 1(d) a = —, b = — 
4 8
2m+1 dt, a * b,then :
3
4
80 Advanced Problems in Mathematics forJEE
(c) [0,1](a) (b)
du,forx > 0and/(0) = 0?
Ay
(b)(a)
>x >xO O
(d)(c)
>xO o
(b) (x-a)(y-b) = k 
Cd) (x-a)2 4-(y-b)2 =k
(1) f is increasing on [1,2]
(3) f is decreasing on [1,2] 
Which of the above are correct?
(a) 1 and 2 (b) 1 and 4
-1,-1
2.
Cd) i 1
.2
46. Which of the following graph represent the function /(x) = j e x 
o
Ay
47. Let /(x) = (x -a)(x - b)(x - c) be a real valued function where ato the curve y = —-— ; then 
x + 1
(c) 4
•it 3n . , if x g —, — is/are :
.2 2.
(c) 6
50. The function /(x) = sin3 x-msinxis defined on open interval and if assumes only 1 
maximum value and only 1 minimum value on this interval. Then, which one of the following 
must be correct ?
(a) 0 3 (d) m 3 V x > 0 and 
g'(0) =-l, then h(x) =g(x) + x Vx > Ois : 
(a) strictly increasing 
(c) non monotonic
Advanced Problems in Mathematics for JE82
= | x - a | will have atleast one solution then complete set of values of*
(c)(b)(a) [-1,1]
59. For any real number b, let /(b) denotes the maximum of sin x +
Cd) 1(c)
+ b VxeR. Then th 0 has two real solution for x g (0,1)
(c)
I 4 (d) M
(d) --1,- + 1
2 2
1-^1 + -
2 2
63. Let /: [0,2n] -> [-3,3] be a given function defined as /(x) = 3 cos A ■ The slope of the tangent tc 
the curve y = /-1 (x) at the point where the curve crosses the y-axis is :
2 11(b) -A (c) -A (d) -A
3 6 3
64. Number of stationary points in [0, n] for the function /(x) = sin x + tan x - 2x is :
(a) 0 (b) 1 (c) 2 (d) 3
G + 4 o n65. If a, b, c, d g R such that------- + - = 0, then the equation ax + bx + ex + d = 0 has
b + 3d 3
(a) atleast one root in (-1,0)
(c) no root in (-1,1)
2
---------- + b V x x g R .
3 + sinx
=
Application of Derivatives 83
I
then
Cb) a = -11, b = -6
Cb) fM =(a) /(x) = 2
a
Cb) a = ± (d) a = ± V2(a) a = ±1
2Cb) cot2 a sin a(a) cot2 a cos a
§ Answers |y '-------
(d) Cd) Cb)(d) Cb) (b) 6. (c) CO(a) (c) 7. 8. 9. 10.2. 3. 4.
(b) (a) (b) (b)(c) (c) (c) 16. (a)(c) (c) 17. 18. 20.12. 15. 19.13. 14.
Cb) (d)Cd)Cb) Cd) (d) (d) 26. CO Cc)21. Ca) 25. 27. 28. 29. 30.22. 23. 24.
Cb)Cb) (d) Cd) (a) Cc) (a) Cc) Ca)(c) 36. 37. 39. 40.31. 32. 33. 34. 35. 38.
(d) Cb)Ca) (b) Cb) Ca) 46. Ca) (a) Cd) Ca)41. 43. 45. 47. 48. 49. 50.42. 44.
Cd) Cb) Cb)Cb) Cb) Cc) (c)Ca) Ca) 56. Cc) 60.51. 52. 53. 54. 55. 57. 58. 59.
Cd)Cb) Ca) 63. Cb) 64. (c) 65. Cb) 66. 67. (c) 68. Ca) 69. (d) (a)61. 62. 70.
3— x
2
3— *2 J
(c) /(x) = (x-l)|x-l|
2
I
2V3 + 1 
V3
(d) a = 22, b = -6
sin(x-l) 
x —1 
1
Cb) f is decreasing if 0
(d) / is increasing if 4(2) > 0
67. If /(x) = x3 - 6x2 + ax + b is defined on [1, 3] satisfies Rolle’s theorem for c =
66. If /'(x) = 3
2
(c) tan2 a cos a
(a) a = -11, b = 6 (b) a =-11, b = -6 Cc) a = 11,beR
68. For which of the following functionCs) Lagrange’s mean value theorem is not applicable in 
[1,2]?
Cd) /(x)=|x-l| 
x2 y2 9
69. If the curves---- 1- -— = 1 and y = 16x intersect at right angles, then :
a2 4
Cc) a = ±-^=
V3
3 2 P70. If the line xcosa + ysina =P touches the curve 4x = 27ay , then — =:
a
Cd) tan2 a sin a
Advanced Problems in Mathematics forJEE84
Exercise-2 : One or More than One Answer is/are Correct
(d) x-y =(c) x-y =
(b) There exist some c e (0,8) where /'(c) = —
(c) There exist c!,c2
3. If/(x) =
x R is given by /(x) =
0
(b) / is a bounded function 
(d) /" is continuous V x g R
1
5
(a) / has a continuous derivative V x e R
(c) / has an global minimum at x = 0
5. If | /"(x) | 0
- x = 0, then 
2
cos-1 (cosx)
(a) x = 0 is a point of maxima
(b) /(x) is continuous V x e R
(c) global maximum value of /(x) V x g R is n
(d) global minimum value of /(x) V x g R is 0
x4
1. Common tangent(s) toy = x3 and x = y3 is/are :
1 12-2
(a) x-y =—= (b) x-y =—■= (c) x-y= —- (d)x-y= —-=
v3 v3 3/ 3 3 V 3
2. Let /:[0,8] -> R be differentiable function such that /(0) = 0, /(4) = 1, /(8) = 1, then which of 
the following hold(s) good ?
(a) There exist some q g (0,8) where/'(q ) = 1
1
2
6. Let /: [-3,4] -> R such that /"(x) > 0 for all x g [-3,4], then which of the following are always 
true ?
(a) /(x) has a relative minimum on (-3,4)
(b) /(x) has a minimum on [-3,4]
(c) /(x) has a maximum on [-3,4]
(d) if /(3) = /(4), then /(x) has a critical point on [-3,4]
85Application of Derivatives
2
(a) /(x,y) is maximum when x =
(c) maximum value of /(x, y) is
(d) maximum value of /(x, y) is
(d) y =(c) y =(a) y = xlnx
-X
2(x-l) 
x2
(c)Xg(-3,3)
2 is :
mk 
rn + n
(b) ffay') is maximum where x = y
1-x2
2x
7. Let /(x) be twice differentiable function such that /"(x) > 0 in [0,2]. Then :
(a) /(0) + /(2) = 2/(c), for atleast one c, c g (0,2)
(b) /(0) +/(2) 2/(1)
(d) 2/(0)+ /(2)> 3/1
8. Let g(x) be a cubic polynomial having local maximum at x = -1 and g'(x) has a local minimum 
at x = 1. If g(-l) = 10, g(3) = -22, then :
(a) perpendicular distance between its two horizontal tangents is 12
(b) perpendicular distance between its two horizontal tangents is 32
(c) g(x) = 0 has atleast one real root lying in interval (-1, 0)
(d) g(x) = 0, has 3 distinct real roots
9. The function /(x) = 2x3 - 3 (X + 2) x2 + 2Xx + 5 has a maximum and a minimum for :
(a) X e (-4, «>) (b) X g (-oo, 0) (c) X g (-3,3) (d) X g (1, co)
10. The function /(x) = 1 + xln^x + 71
(a) strictly increasing V x g (0,1) (b) strictly decreasing V x g (-1,0)
(c) strictly decreasing for x g (-1,0) (d) strictly decreasing for x g (0,1)
11. Let m and n be positive integers and x,y >0 and x + y =k, where k is constant. Let 
f(x,y) = xmyn, then:
mnnmkm+n 
(m + n)m+n 
km+nrnmnn 
(rn. + ri)m+n
12. The straight line which is both tangent and normal to the curve x = 3t2,y = 2t3 is :
(a) y +V3(x-l) = 0 (b) y-V3(x-l) = 0
(c) y + V2(x-2) = 0 (d) y-V2(x-2) = 0
13. A curve is such that the ratio of the subnormal at any point to the sum of its co-ordinates is equal 
to the ratio of the ordinate of this point to its abscissa. If the curve passes through (1,0), then 
possible equation of the curve (s) is :
Inx (b)y =----x
Advanced Problems in Mathematics forJEE86
14. A parabola of the form y = ax2 + bx + c(a>0) intersects the graph of J(x) =
(d) a e (-00,1)
at (x > 0) then :
4
o
o(b)(a)
number of possible distinct intersection (s) of these graph can be :
(a) 0 (b) 2 (c) 3 (d) 4
15. Gradient of the line passing through the point (2, 8) and touching the curve y = x3, can be :
(a) 3 (b) 6 (c) 9 (d) 12
16. The equation x + cosx = a has exactly one positive root, then :
(a) a e (0,1) (b) ae(2,3) (c) ae(l,oo)
17. Given that /(x) is a non-constant linear function. Then the curves:
(a) y = f(x~) and y = f~x (x) are orthogonal
(b) y = /(x) and y = (-x) are orthogonal
(c) y = f (—x) and y = J-1 (x) are orthogonal
(d) y = f(-x) and y = f~l (-x) are orthogonal
X
18. Let/(x) = |gt
o
(a) The number of point of inflections is atleast 1
(b) The number of point of inflections is 0
(c) The number of point of local maxima is 1
(d) The number of point of local minima is 1
19. Let /(x) = sin x + ax + b. Then /(x) = 0 has :
(a) only one real root which is positive if a > 1, b 1, b > 0
(c) only one real root which is negative if a xo 1 o
21. Consider /(x) = sin5 x + cos5 x -1, x e
7C(a) f is strictly decreasing in 0, -
(b) f is strictly increasing in -
(c) There exist a number ‘c’in 0,- such that /'(c) = 0
(d) The equation /(x) = 0 has only two roots in 0, -
x22. Let /(x) =
(d) -1
(a) /(x) is monotonically increasing in (4n -1) -, (4n +1) - V n e N
(b) /(x) has a local minima at x = (4n -1) - V n eN
[o, 1J whic’1 foUowin& is/are correct ?
(b) /(x) has a local minima at x = (4n -1) ~ V n g N
(c) The points of inflection of the curve y = f(x) lie on the curve x tan x +1 = 0
(d) Number of critical points of y = /(x) in (0,IOtc) are 19
24. LetF(x) = (/(x))2 + (/'(x))2, F(0) = 6, where/(x) is a thrice differentiable function such that 
| /(x) | 6, F'(c) = 0 and F"(c) £ 0
20+1 In x ; x > O' 
0 ; x = 0
If /(x) satisfies rolle’s theorem in interval [0,1], then a can be :
(a) -1 (b) to
2 3 4
xr cost23. Which of the following is/are true for the function /(x) = I-----dt (x > 0) ?
o f
irk.
2)
n
2
n 7i
.4’2.
. (- 7t
2
88 Advanced Problems in Mathematics forJEE
2
25. Let /(x) =
1 + cosx;
(c)
27. For the equation------ = X which of the following statement(s) is/are correct ?
(b) a > b (c) a 2Vx gR-{0}
(c) /(x) has a local minimum at x = 1
Z ------- ! Answers
(c, d) (a, c, d) (a, c) (b, c, d)2. 3. (a, c) (a, b, c, d)4. 5. 6.
(c, d) (b, d) (a, b, c, d)9. (a, c) (a, d) (c, d)7. 8. 10. 11. 12.
(a, d) (b, c, d) (a, d)15. 16. (b, c) (b, c)13. 14. (a, d)17. 18.
(a, b) (a, b, c, d)(a, b, c) (b, c)19. 21. (a, b, c) (a, b, d)20. 22. 23. 24.
(b, c, d)(b)(a, d) 26. 27. (a, d) (a, b, c, d)28. 29.25.
then /(x) has :
(a) local maximum at x = —
+ x -10x; -l/2 .7-/2
(a) — (b) —— (c) —— (d) ——
4 4 4 4
(d) a + b = y
x3
T, l_ • 6
(a) When X e (0, oo) equation has 2 real and distinct roots
(b) When X g (-oo, -e2) equation has 2 real and distinct roots
(c) When X g (0, oo) equation has 1 real root
(d) When X g (-e, 0) euqation has no real root
28. If y = mx + 5isa tangent to the curve x3y 3
. , ,18(a) a + b = —
29. If (/(x) -1) (x2 + x +1)2 -(/(x) +1) (x4 + x2 +1) = 0
V x g R - {0} and /(x) * ±1, then which of the following statement(s) is/are correct ?
(b) /(x) has a local maximum at x = -1 
n
(d) j (cos x) /(x) dx = 0
= ax3 + by 3 at P(l, 2), then
Application of Derivatives 89
Exercise-3 : Comprehension Type Problems
(d) 5(a) 1 (b) 2
Let /(x) =
Let the tangent to the curve y - g(x) at point P whose abscissa is - cuts x-axis in point Q.
(d) 2(c) 1
Paragraph for Question Nos. 1 to 2
Lety = /(x)such that xy = x + y + l,xeR-{l} andg(x) = x/(x)
6. If /"(x) > 0Vx g (-l,oo) and /'(0) = 1 then number of solutions of equation /(x) = x is :
(a) 2 (b) 3 (c) 4 (d) None of these
7. If /"(x) 0Vx g (0, 0 V x g (-1, co) also /'(-l) = 0 given lim /(x) = 0 and 
lim /(x) = co and function is twice differentiable.
X->oo
Paragraph for Question Nos. 3 to 5
; 0 8 
w Ts
15. If complete solution set of' x’ for which function h(x) =/(x) + 3x is strictly increasing is (-oo, k)
.3
(a) 1 (b) 2 (c) 3 (d) 4
x y 116. If acute angle of intersection of the curves — + + - = 0 and y = f (x) be 9 then tan 0 equals to:
16 f > 14 4
m S w 250(A)
(Q) 1(B)
(R) 2(C)
1. Column-I gives pair of curves and column-H gives the angle 9 between the curves at their 
intersection point.
May be positive or negative 
for some values of x
May result in zero for some 
of values of x
n
4
7t
2
8
2
it n
.? 2
-2x
4
V X 6 I 0, -
I 2
(cosx)sinx
x)sin
(sinx)sinx
x3-4 
(x-1)3
The number of possible distinct real roots of 
equation /(x) = c where c > 4 can be
The number of possible distinct real roots of 
equation g (x) = c, where c > 0 can be
The number of possible distinct real roots of 
equation h (x) = c, where c > 1 can be
Xjcos-’x
~(cosx)sinx
-(sinx)C0SX
x2=4y,y =
xz + 4
x V x e (cos 1, sin 1)
2
-Vx eR,hM--------- -
(x + 1)3
x^
Vx * 1, g(x) = —
^ + z!=i>x2 2=5
18 8
xy =l,x2-y2 =5
(In (lnx))ln(lnx) -Qnx)lnx
93Application of Derivatives
3(D)
4
4.
Column-ll
(A) 2
y =i +
(Q) 3(B)
(R) 4(C)
(S)(D) 5
(T) 0
5.
Column-llColumn-1
(P) 0(A) Maximum value of/(x) = log2
(Q) 1(B)
(R) 2(C)
(S) 3(D)
CT) |
The number of possible distinct real roots of (S) 
equation g (x) = c where -1 0 then number of points in 
(0,2) where /'(x) vanishes, is
lim
*->o’Lex -1.
([•] represent greatest integer function)
CO
The value of 4^ cot 
n=l
If a, P, y are
a -------+ •
x - a
4 
Vx + 2 + -J2-X
_____ Column-1___________ ’_____ \
roots of x3 -3x2 + 2x + 4 = 0 and (P)
Px _______ yx2_______
(x-a) (x-p) + (x-a)(x-P)(x-y)
then value of y at x - 2 is :
If x3 + ax +1 = 0 and x4 + ax +1 = 0 have a common 
roots then the value of |a| can be equal to
The number of local maximas of the function 
x2 + 4cosx+ Sis more than
If /(x) = 2|x|3 + 3x2-12|x|+l, where x e[-l, 2] then 
greatest value of /(x) is more than
94 Advanced Problems in Mathematics for JEE
Column-1 Column-ll
(A) /(x) gives a local maxima at (P)
/(x) gives a local minima at(B) (Q) a > 1; x =
(C) (R) 0 1; x =
3
Column-1 Column-ll
The value of ‘ a' is (P)(A) 0
The value of ‘ b’ is(B) (Q) 24
The value of‘c’ is Greater than 32(C) (R)
The value of ‘ d’ is (S)(D) -2
8.
Column-1
(A) (P)
(B) (Q) 2
(C) (R)
(S) 11
The ratio of altitude to the radius of the 
cylinder of maximum volume that can be 
inscribed in a given sphere is
The ratio of radius to the altitude of the cone of 
the greatest volume which can be inscribed in a 
given sphere is
The cone circumscribing the sphere of radius 
* f has the maximum volume if its semi vertical 
angle is 0, then 33 sin 0 =
The greatest value of x3y4 if 2x + 3y =7, 
x > 0, y £ 0 is
32
3
/(x) gives a point of inflection for
/(x) is strictly increasing for all 
xeR+
Column-ll
V2
6. Consider the function /(x) = — - ax + x2 and a > 0 is a real constant:
8
a + va
4
2-l
(D)
, 1a = 1; x = -
4
a -7a2 -1
7. The function /(x) = Vox3 + bx2 + cx + d has its non-zero local minimum and maximum values 
at x = -2 and x = 2 respectively. If ‘a’ is one of the root ofx2-x-6 = 0, then match the 
following:
95Application of Derivatives
j Answers |
2. A —> R, S ; B —> Q ; C —> R, S ; D —> Q
3. A—>Q, R; B-»R, S; C->Q, R, S ; D->P,R,T
C-*T; D->P,Q,R,TA->P; B—>P;
5. A —> Q; B —> S \ C-»R;
6. A —> Q ; B —> S j D-> R
7. A-> S ; B-»P; C^Q;
8. D->RA->Q; B->P; C-> S ;
I
i
--- --------
A. —> T j B —> R ; C —> Q j D —> Q
96 Advanced Problems in Mathematics for JEE
Exercise-5 : Subjective Type Problems
4. Let /(x) = ,3
6 = tan
- (X + l)ex + 2x is monotonically
= 0 is
13. Let the maximum value of expression y = for x > 1 is — where p and q are 
q
1. A conical vessel is to be prepared out of a circular sheet of metal of unit radius. In order that the 
vessel has maximum volume, the sectorial area that must be removed from the sheet is Aj and 
the area of the given sheet is A2. If — = m + y/n, where m, n e N, then m + n is equal to.
2. On [1, e], the least and greatest vlaues of /(x) = x2lnx are m and M respectively, then 
[>Jm + m] is : (where [ ] denotes greatest integer function)
2
3. If /(x) = — - — + xisa decreasing function for every x 0
; Find the value of a.
7. Let set of all possible values of X such that /(x) = e2x 
increasing for V x e R is (-co, k], Find the value of k.
8. Let a, b,c and d be non-negative real number such that a5 + b5 0 and r > 0. If the line through (p, q) and (r, s) is also tangent to both the curves at 
these points respectively, then find the value of (p + r).
10. /(x) = max|2siny -x| where y eRthen determine the minimum value of/(x).
X
11. Let /(x) = J ((a — 1) (t2 +t + I)2 - (a + l)(t4 +12 + l))dt. Then the total number of integral
o
values of ‘ a' for which /'(x) = 0 has no real roots is
12. The number of real roots of the equation x2013 + e2014x
.2
xe™ 
2 x + ax -x
k ais increasing is — . Then k + /is equal to
La l_
5. Find sum of all possible values of the real parameter ‘ b’ if the difference between the largest and 
smallest values of the function /(x) = x2 - 2bx + 1 in the interval [0,1] is 4.
x2 y26. Let ‘0’ be the angle in radians between the curves — + —=1 and x2 + y2=12. If
36 4
+ e
x4 -x'
x6 +2x3 -1
relatively prime positive integers. Find the value of (p + q).
5 + ds /(x) + 5 and f(x +1) «
Indefinite and Definite
72-1 
sinx 
sin (x - a)
(a) (sin a, cos a)
p
(a)
3. If J
■
V
1 
+ ■ ___
y/rt -J2n
(d) 2(72 + 1)
2x
+ C
x
+ c
I +c
.8
dx and I2
(a) ax Ini — 
lx.
(c) ax+lnl —
2. The value of:
(b) ax In! —
I e
, . V2 (a) —tan 
3
V2(c) — tan 
3
1
5. If = j 
0
(a) >1,I2 12
1 1 1 
.——. H—■=—-• H—-=.— -■ + 
\VnVn + l Vnvn+ 2 VnVn + 3
(b) 2(72-1) (c) 72 + 1
dx = Ax + B log sin (x - a) + C, then value of (A, B) is :
(c) 1 l
2
3 
/2
3
99Indefinite and Definite Integration
= y(x). Then the value
(d) {77,715}(c)(a) (b)
(d)(0(b) 1 - cos 2(a)
dx is equal to :
(c)
X
(c) —y=ln2
10. Let y(x) be a differentiable function such that y(x)
11.
i
Jz. TH 
3 ’ 3
77 715
2 ’ 2
1
18
sin-
x
tan
x
1-cosl
2
1 -cos2
2
6. Let /:(0,l) -> (0,1) be a differentiable function such that /'(x) * 0 for all x e (0,1) and 
ft x \ 
2ds
= x2 +
ds-jVl-C/Cs))
0___________
/(t)-/(x)
•MB- 
sin 2 
2
X 1
j e-tf (x -t) dt, then j /(x) dx = 
o o
r 1 hm —
n->oo n
(d) -ln2
2
(b) -—
12 36
(d)n
(b) -ln2
4
il 1 73/I — I = —. Suppose for all x, lim
(a)l (C) S
1
8. The value of J 
o
of y^i^ belongs to :
77 715]
4 ’ 4 J
7. Ify(0) =-|(l-cos6 9-sin6 0), then
(b) — ex +[ — |x + C 
3 — e \3 — e)
(d) —— ex + f-—-'ix + C 
2-e l3+ej
1
If y'(x) = y(x) + j y(x) dx and given y(0) = 1, then j y(x) dx is equal to : 
o
(a) —Z— ex + f ——x + C
3-e U-eJ
, . 3 x C1 + eA „(c) ----- e + ------ x + C
2-e 2
(d) - —
I X + l
f , f x + 1
(a) - ------
k x
^6
+c
(t>) Ai2
!4. f
dx
— ;x g(0,1):
(b) 6| —
\ x
(where C is an arbitrary constant.)
f^x + 72-x2 Y^1-x72^x
15. Evaluate j
a/1 -tan2 x
(a) V2 
r dx 
^x*2(x + l)7'2
1/6 
+C
I x
= j(sinx)n dx;n eN, then 5/4 -6I6 is equal to :
(a) sinx-(cosx)5+C
[1 + cos2 2x - 2 cos 2x] + C
x + Vx
101Indefinite and Definite Integration
dx equals to :
+ C+ C
+ C+ C
(b)(a) 33
(d) None of these
,5.13
dx = f (x) + C, then /(10) is equal to :21.
Cd) 2cos10(c) 2sinl0
22.
(b) (x-l)ex + C+ C
,x+x(d) xe
dx = ex gW + K, then g — =
(d) 2(c) -1
sinx+cosx dx =
X
+ C
+ C
dxis :
-1)
i
i
it
4
gx sinx+cosx
gx sinx+cosx
1
25. The value of the definite integral J 
o
24. P’
•x+vx + x2
Vx + Vl + x
(b) ^(21/2-l)
194
(d) exsinx+cosx[ 1---- — +C
I cosx)
20. J
(b) cXSinX+C0SX 1
X--------------
xcosx
(a) |(21/2-l)
(c) |(23/2-!)
3(x13 (x13
)ex+x
38
— dx =4
+ c
2tanx 2 ----------- + cosec
1 + tanx
cos3x4
a2 
a + fax
a2 
a + bx
a2 
a + bx
a2 
a + bx
(b) —I a + bx-2aln|a + bxl- 
b 3 I
(d) -M a + bx-2aln|a + ax|-
(d) _ (23/2
x2
(a + bx)2
(a) -^a + bx-aln|a + bx|-
1 (
(c) — a + bx + 2a In | a + bx|-
8x43 +13x
(x13 +x5 + 1)
x39
-3+C 
+ x5 +1)3 
x39
(c) ——s+c
5(x13 +x5 + 1)5
rf cos6x + 6cos4x + 15cos2x +10
J llOcos2 x + 5cosxcos3x + cosxcos5x
(a) 20 (b) 10
| (1 + x - x-1) ex+x 1 dx =
(a) (x + l)ex+x
(c) -xex+x
23. If Jex
(a) 0
x39---- - + c
+ x5 +1)
(b) 1 
x-xsinx + cosx 
x2 cos2
x--- —1 + C
cosx J
1—— 
xcosx
3
1
3
+ C 
5n
4
102 Advanced Problems in Methamatics for J EE
26. (21nx + l)dx
(d) (xx)x +C+ C
27. If J
equation = {x} in [0, 2k] is/are : (where {•} represents fractional part function)
(c) 2 (d) 3
28.
(a) x (b) xxQnx-x) + C
(c) xx (d) xx In x + C
.3 2
(a) + C (b)
(0 + C (d) + C
(b) (d) ex(c)+ C + C +c
+ C, then ‘ k’ is equal to :
(c) T (d) Z
dx = Plog|x| + Qlog|x7 +1| + C, then:
(c) 7P + 2Q = 0 (d) 7P-2Q=1
dx is equal to :
(a) sin2x + C (b) (c)+ C (d) -2sin2x + C+ C
x
2
dx is equal to :
(a) 0 (b) 1
f xx f (In x)2 + In x + — |dx is equal to :
J I xj
lnx-1 
(lnx)2 + l
sin2x
2
-sin2x
2
Inx 
(lnx)2 + l 
Jx-1 
Vx+ 2
- + C
(Inx)2 | + C 
xj
33.1 = j
31. I = j
30. I = j
32. j
Tzx4
x 
x2 +1
= =k 
5
-2x2 + 1
X
+ C; where j = 1; t^en the number of solutions of the
® 5
(c) x(xX)
(b) 2P + 7Q = 0 
x
j Xx2+1
X
l + (lnx)2
Qnx)_
2
x2-l is equal to : 
v2x4-2x2 + 1
4 -2x2+1
_x2
V2X4
29. Iff = f- 
Jx:
(b) x2lnx + C
dx = — 
(gW)2010
^2x4 -2x2 + 1 
x
-2x2 + 1
2x2
3
4
4
3
(a) + C 
xz+l
dx
^(x-1)3(x+2)
1-x7
x(l + x7)
(a) 2P-7Q = 0
sin8 x-cos8
1 -2sin2 xcos2 x
(a) x*+C 
cosec2x-2010 
’ cos2010x
. fM .
g(x)
Indefinite and Definite Integration 103
(b) ln(l + tan^3 x) + C
(d) —— ln(l + tan^3 x) + C
+ C
(d)+ C + C
+c +c
+c +c
37. (log(g(x)) -log(/(x)))dx is equal to :
2>
(d) log + C
(d) None of these
\
j
X 
73(x + l)>
vx 
3(x + l)
(where C is arbitrary constant.) 
rf /(x)g'(x)-/'(x)g(x) 
H /Wg(x)
x
V3(x + 1)
x
3(x + l)
2
I +c
2
+ c
34. I = j
. a 2(a) —tan 
V3
(c) 4= tan 
V3
38.
(a) exlnx + C1x + C2
, x Inx „ „. (c)------ i- C] X + C 2
X
36. J
(sin 2x)1//3 d (tan'
sin^3 x + cos
(a) —i—In (1 + tan1/3 x) + C 
2*3
(c) 21/3 Ind + tan^3 x) + C
(2012)sin’,(2012)X dx =
»
(a) log[^| + C 
l/wj
35. j
(b) ex In x + — + Cj x + C2 
x
(b) -^tan 
V3
2
(d) —tan 
V3
(2012)^ 
1-(2012)2x
(a) (log2012 e)2(2012)sin’1(2012)X
(c) dog2012e)2(2012)sin‘1(2012)X
i 2 1 L L lnx +------- - dx dx =* x2) J
(b) (log2012e)2(2012)x+sin'1(2012)X +C 
(2012)sin"1(2012)Z 
0°8 2012 e)2
J/3 X) 
:^3X
(where C denotes arbitrary constant.) 
(x + 2) dx .—-—:---------- —------is equal to :
(x2 + 3x + 3)Vx + l
/ \
(b) 1
g(x)
fM
1
2^3
Advanced Problems in Methamatics for JEE104
2
(a) 5
(a) 0
(x)dx will be :
.2
3 4 dx is equal to :
(c) 2e(a) 4e
dx is equal to (where [*] denotes greatest integer function.)
(d) None of these(c) 0
45. If/(x) =
(d) None of these(b) (c)(a)
dy is equal to :
(c) -2 (d) None of these
— Vx e 
00
i
39. Maximum value of the function /(x) = n2|t sin(x + 7tt)dr over all real number x : 
o
e2
= j (In x)n d(x2), then the value of 21 n + nl
1
4
46. j
o
(a) 0
(b) -Jn2 + 2 (c) Vn2 + 3 (d) Vn2 + 4
function, continuous on [0, 1] such that /(x) R be defined as /(x) = x + sinx. The value of j/-1 
o
(d) w
(b) - 
e
41. Let7n
x
1 + (In x)(ln x)...
e2 - 2e
2~
(b) —+ 21n2
2
(d) 2+21n —
2
3x - [x]
3x - [x]
, > 28(a) T
e2 +1
2
(c) 2 + ln| —
2
(b)|
(a) 2n2 (b) 2n2-2
1 , r —( 43. The value of the definite integral e
(c) 2n2 + 2 
2+ ln(x + 7x2 + 1J + 5xJ -8x
(d> a
e
(c) e2
2e
[1, co) then j /(x) dx equals is :
i
o
44. j
-io
e2-l
2
(y 2 - 4y + 5) sin(y - 2) 
(2y2-8y+ 11)
(b) 2
n_! equals to :
(a) 7tc2 +1
40. Let ‘ f’ is a
2 /(x) 0
is:
(a) 15
tt/2
50. If [•] denotes the greatest integer function, then the integral | 
o
X, then [X -1] is equal to :
(a) 0 (b) 1
(d) 3
[ (2-n + n) 
3n2
re In 3
2
Ttln 2
4
.2
dx
nln2 
3 
nln4 
3
(d) None of these 
esinx-[sinx]d(sin2x_[sin2y]) 
sin x - [sin x]
reciprocal of the limit lim f xe
X->oo J 
0
4
L x > 0. If J"
1
(b) 16
- j x"e 
o 
he^
54. The value of the definite integral j 
o
3 gSinx3 
X
Inx 
x2 + 4
n+he~l/h 
jx2e- 
o
2 _______
53. The value of the definite integral 71 + x3 + vx 
o v
3 
(d)
2
47. Let—F(x) = 
dx
(b) 1
(2-1 + n) 
l2 + n-l + n2
(a) 7t(l-7r2)e',tZ (b) 27r(l-n2)e-’tZ (c) n(l-7t)e"’‘ (d) nV”2
49. Letf:R+ -> R be a differentiable function with /(I) = 3 and satisfying :
xy x y
j f(f)dt =y| f(f)dt + xj f(f)dt Vx,y e R+, then /(e) =
1 1 1
(a) 3 (b) 4
[ (2-2 + n)
22 + n-2 + n2
esinx
2/l + x2)
dx ----------------------------------- is : 
tan x + cot x + cosec x + sec x
(d) 2O132013(c) 20132
(d) 5
4
n/2
57. The value of the definite integral | 
o
(d) —
4
, x n(a) —
16
W 2
, , 12S 
w V
(b) 5 + 1
4
, s n2
W 16
"2
\C 2013
S’-Iff Z
ok r=l
then k =
(a) 2013
60. /(x) = 2x-tan
(a) strictly increases V x e R
(b) strictly increases only in (0, oo)
(c) strictly decreases Vx eR
(d) strictly decreases in (0, 3(a) -
(0 i
71
(b) g(2)
7t
4k cos— equals :
n
Cd) -
n
e2
63. The value of the integral j
16
cosx2
J ~
x2
(a) it
cosec2x
JtgCddt
2
cosx2 + cos(lO-x)
1
2
x = 0
0oo J
0
2
(a) -g(2)
it
n _ j.
65. The value of lim V —— =o
0 r=1
(a) A = 2B
o
(where [•] denotes greatest integer function)
(a) 0 (b) 1
cosxdt 
yfl + t
2n 
j COS 
0
+ — 
n
+ — 
n
(n + 1) (n + 2)
4 4
68. For positive integers k = 1,2,3 
lentsuch that Z AOBk = — ,OA =1 and OBk 
2n
(a) -In3
3
(a)?
1 2014
(a) n2
(b)
71
1 2013
69. If A = j (r-x')dx and B = jn^ + x) dx, then :
o r=°
(b) 2A-B (c) A + B = 0
3 1 1
70. If/(x) = defined iri [0, 3], then j (/(x) + 2)dx =
(d) n3
(d) -L 
2n2
(d) 4
COSX z x
,g(x) = j (1 + sint)2 dt, then the value of /'I -I is equal to :
(c) 8
4......+ —
9n /
In 4
~3~
(n + 3) (d) (n + 4)
4 4
, n, let Sk denotes the area of AAOBk (where ‘O’ is origin) 
1 n= k. The value of the lim — is : 
n-*”n k=i 
n2
I2 
kn3+l
1 x
72. Let/(x) = -^ j (4t2 -2f’(t))dt, find 9/'(4) 
x o
(a) 16 (b) 4
22 
3+23
(b) ll’
(1-tan^l 
______ 2 
1 + tan2 —
I 2J 
•n-2
(b) 2L
32
3+33
x
120
In 6
3
1
2
Indefinite and Definite Integration 109
(c) 2 (d) 3
dx
(c)
dx equals to :
(d) None of these(c)
(c) 3(a) 1
(a) (c)(b)
(b)
(d) None of these(c)
X
, is :
(d) 3(b) 1(a) 0
1
2
(a) 0
3
71
4
n
2
tan
. x
n
76. If j 
o
i
79. | 
o
80. The value of
o
2x
1-x2
jc/2
[ —^—dx
J sinx o
n/2
[ — dx
J sinx o
| j \
[ 3x2 sin---- xcos— dx is :
J I x xj
(d) f
o
f x n
W 3
78. Lety ={x}[x]
77.
o
it/4 .
r sinx , --- dx
o X
i
75. Given a function ‘g’ continuous everywhere such that Jg(t) dt = 2 and g(l) = 5. 
o
W) i f2 J sinx o
24V2 
k3
V3/- Al r 1 d ( -tanJ I 2droo
3
— is :
88. If Jn
(a) - 
e
(d) 1 + - 
e
j is equal to (n e Z): 
2
(d)l
(d) -1
(b) 1
4
x284. Consider a parabola y = — and the point F (0,1).
4
Let Aj (x1,y1)M2(^2>y2)M3(x3,y3), 
xk >0and A0FAk = — (k = 1,2,3,
2n
82. lim 
n->oo
(a) ±30
(b) n
■1-. + C 
Vl + sinx
1 + 2* + 34 + 
n5
1 + 23 +33 + 
n5
Cb) - 
n
4
85. The minimum value of /(x) = j dt where x e [0,3] is :
o
Cb) e4-1
3 
COS X
X
’■ 1
, n). Then the value of lim — > FAk, is equal to : 
n->0° n k=i
. 8(c) -
7t
(C) 
e
(a) 2e2-l
86. IfT^dx = -, 
J x 2 o
(a) 2
j Vl + sinx ^cos^ -sin^ | dx =:
1 + sinx
2
 r sin(2nx)^ value of I 
J sin2x o
t ' nn(a) -
(d)
2
x
2
x
2
111Indefinite and Definite Integration
dx
o
(c)
equals to :
(c)
92.
+ n
(d)(b) (c)(a)
-dx, is :
(b) -ln(l + x4)— ln(l + x3) + c
(d) — ln(l + x
(c) -(b) -1(a) 0
(c) 0(b) sin (tan 1)(a) tan (sin 1)
_7_
12
96. Let /(x) = lim 
n->00
95. The value of Limit
it
2
n
97. The value of lim V| 
n->® k=l
cosx
1 + (tan-1
373-1 
3
is equal to :
X 1
90. Let f be a differentiable function on R and satisfies /(x) = x2 + j e-t/(x -t)dt; then j /(x) 
o
ic/2
91. The value of the definite integral j 
-0/2)
00
, then | /(x) dx = 
o
94./
COSX
j (cos
1_____
2x - sin 2x
t) dt
is equal to:
(d)
4
(d) —
12
. (tanl(d) sin ------
I 2
°o
372-1 272-1
—~— w —~—2 3
(x3 -1) 
(x4 +1) (x + 1)' 
(a) hn(l + x4) + |ln(l + x3) + c 
(c) iln(l + x4)-ln(l + x) + c 
4
4) + ln(l + x) + c
k 
n2 + n + 2k
ecot-1Wdx = /(x).ec°t-1(x)+c
9 
COS“ X
1 + 5X
1
4 
1
4
2
3
1
3
112 Advanced Problems in Methamatics for JEE
(0 (d) 1
(a) 0 (d) does not exist
. Find the value of
•: (you may or may not use reduction formula given)
(c)
dx = A sin 4x + B sin+ C, then A + B is equal to :
(c) 2
— + C where p, q, r g N then the value of (p + q + r) equals
(c) 6047 (d) 6021
e
1
n+l
(C)
1
2
1Z
2
3
4
1
= a, then j x2e-x dx is equal to 
o
3tt
32
101. J
dx
2-»n-l
.2 
dx
(Where C is constant of integration)
(a) 6039 (b) 6048
i
103. If J
0
102. J
(c) -(ea-1) 
e
a”
2
dx
(1 + x2)"
4 dx:
xq
1 + x'
*/4
100. Find the value of J (sin x) 
o
35(b) y
= lln 
P
(d)
4
, x 19(a) -
(d) -(ea + 1) 
e
W~4 (b)l
y
ji c -i i dt
98. The value of lim - ------------- is :
y->r tan(y -1)
(b) 1
99. Given that J
r1 dx
Jo(l + x2)4
. . 11 5it(a) — + —
48 64
. 3n
(a) 16
cos 9x + cos 6x 
2cos 5x-l
(Where C is constant of integration)
1 3(a) - (b) a
dx
x2014
(C) 2 
, (2n-3) r
2(n-l)J (i + x2)
x
2(n-1)(1 + x2)n-1
(a) -(ea-1) (b) — (ea + 1)
2e 2e
n+l 2
104. If/(x) is a continuous function for all real values of x and satisfies j /(x) dx = — V n el, then 
n
5
|/(| x|) dx is equal to :
-3
11 5n(b) — + —
48 32
c \ 1(c) — + —
,24 64
/•jx 1
(d) — + —
96 32
(d) ^-7_
16 8
(b) ^-1
32 4
113Indefinite and Definite integration
+ d, then= a In
(c) a=-,b =—,c = —
is equal to :(d) 2V2-1
(c)(a) 1
(d) 10a(c) 0
dx is :
(d)(c)(b) 2
,3111. Evaluate :
915
1
Vn V4n
4
3
2
107. Let fM = J
X
105. If J
1
110. The value of
0
b 
+?+
_2
(c) —ln2
Jlim EJ n->oo
(a) e2-l
3
(b) |(a) -In 2 
3
100 1001
109. If | fM dx = a, then j (/(r -1 + x) dx) = 
0
(a) 100a
e2-l
4
(b)|
c
1 + x3
(c) 2(V2-1) 
2
. The value of the integral J xfM dx is equal to : 
0
e2-l
2
dy
n/3
108. The value of the definite integral J ln(l + V3 tan x) dx equals 
0
2,1 1(b) a = —, b = — ,c = —
3 3 3
2 i. 1 1
(d) a = -,b=-,c = --
3 3 3
r=l 0
(b) a 
xk+22k 
kl
' ' 15'~ ' 9
(b) — (l + x3)3'2--(l + x3)3/2 + c
dx
x4 (1 + x3)2
(where d is arbitrary constant)
, . 1 L 1 1(a) a = -,b =-,x = -
3 3 3
. . 2,1 1(c) a = -, b = — ,c = —
3 3 3
1 1106. lim -=r-^= + -=^== + .. 
n~>“VnVn + l VnVn + 2
(a) 2 (b) 4
j x571 + x3 dx.
(a) A(i + x3)^2_l(1 + x)3)3/2 + c
" >0°k=0
(d) — In 2
2
15 9
(c) —(1 + x3)3'2- —(1 + x3 )3/2 + c
15 9
(d) — (l + x3)3'2--(l + x3)3/2 + c
n'
6
114 Advanced Problems in Methamatics for JEE
113. Evaluate
(b) -ln|x2 + 3|+tanx + c x + c
x + c
(b) 2| -(tanx)3/2 -(a) (tan x)3^2 --Aanx + C + C
3
sin x + -s/cosx + C(d)
dt equals to :
(c) e (d) Does not exist
(a) 1-A (c) >1-1 (d) 1 + A
1 
Vtanx
114. j dx equals to : 
x
3
(b) ~~A
2
(c) - (tan x)3/2 --Vtanx + C 
3
esin(tx)
X
x
115. lim f- 
x->oJ
0
(a) 1
it . »t/2
116. If A = j dx, then j
o x o
112. If /(x) = j S' — dt, which of the following is true ? 
o
(a) f(0)>/(l-l)
(b) /(0)/(2-l)
(c) /(0) /(3-l)
(d) /(0) /(4 • 1) 
fx3 + 3x2 + x+ 9 , 
J (x2 + l)(x2 + 3)
(a) ln|x2 + 3|+ 3 tan
(c) iln|x2 + 3|+ 3 tan
■J sec5 x
Vsin3
(b) 2
cos 2x , . .------- dx is equal to : 
x
1
2 '
(d) ln|x2 + 3|-tan
115Indefinite and Definite Integration
] Answers |
(d) (b)Cd) (d) Cd)(d) 6. (a) 8.Cb) Cb) Cb) 7. 9. 10.2. 3. 4.
Cb) (c) Cb)(a)Cb) (b) (a) 16. 18. 19. 20. (a)Cb) (a) 17.11. 15.12. 13. 14.
Cd) (d)(a)Cb) (c) (d) 28. (c)Cb) 26. 27. 29. 30.(a) (d) 25.21. 23. 24.22.
(a) (d)(c) (d)Cb) (d) Cc) 36. (a) 38. 39.Cd) Cc) 37. 40.31. 35.32. 33. 34.
Cd)Cd) Cd) Cc)Cb) Ca) Ca) 46. Ca) 47. 48. 49. 50.Cb)41. Ca) 45.42. 43. 44.
Cb) Cb)Cb)(b) Cc) 60. Ca)Cc) 56. 57. 58. 59.(c) Cb) Cc)53. 55.51. 52. 54.
Cd) Cd)Ca) 68. 69. Cb)(d) 66. Cd) 67.61. Cb) 64. Cc) 65. 70.(a) Ca) 63.62.
Cb) Cc) Cc) Cc)Cd) Cb) 76. Ca) 79. 80.77. 78.(d) Cb) Ca) 75.71. 74.72. 73.
(d) Cd) Cd) Cd)Cc) 86. Ca) 88. 89. 90.Cd) Cb) 87.Cb) Cd) 85.81. 84.82. 83.
Cb)Cb) Cc) Ca) Ca)Cd) 96. 97. 98. 99. 100.Cc)(d) Cc) Cb) 95.91. 94.92. 93.
Cd) Cb) 110. (d)Ca)Cc) 106. Ca) 108. 109.Cb) 107.(d) Ca) Ca) 105.101. 104.102. 103.
Cc)Cb) Ca) 116.Cc) (c)111. (d) 115.112. 113. 114.
116 Advanced Problems in Methamatics for JEE
f \
+ C, then :
(c) k2 =7 (d) k2 = 8
2ex + cos x In
(d) 4(c) 3
3. For a > 0, if I =
+ F, then:
(d) None of these(c) C = -20
dx = A sin
(b) B=a3/2 (d) B = al/2(c) A=±
then :
dx
2
(b)(a)
(d)(c)
(a) 1
________ 2r________ 
n-7(30n - 2r)(n0 + 2r)
(b) k. = 6 
7i+x
2011 2011
|F(x)dx = jF(x)dx
-2010 0
2011
J Fix') dx = 0
-2010
6. For a > 0, if I = j
0)
X-----2j
(c) /(0) is a constant function (d) y - f(0) is invertible
8. If /(x + y) = f(x)f(y) for all x,y and /CO) * 0, and F(x) = ——- then :
e?(b) /(0) = |J 
z o 02
(b) g(x) = sin xhas 6 solution for x e [-n, 2k]
(d) /(x) = g(x) has no solution
'd0 = (A sin3 0 + Bcos2 0+ C sin0 +Deos0 + E)esine
00
7. If/(0) = lim V
3 
dx >-, then possible value of a can be :
(a) /Cl) = 7 
o
x3/2'
—— + C, where C is any arbitrary constant, then :
x3/2)
—— + C, where C is any arbitrary constant, then :
/(x) 
1 + C/(x))2
2011 2010 2011
|F(x)dx- jF(x)dx= jF(x)dx
-2010 0 0
2010 2010
j(2F(-x)-F(x))dx = 2 |F(x)dx
-2010 0
(a) A=| (c) A-l
(a) A=|
(d) B=a1/2
-x3
C Q 14. Let J xsinx sec xdx = — (x-/(x)-g(x)) + k, then :
(a) /(x)g(-l,l)
(c) g'(x) = /(x),VxeR
5. If j (sin 30 + sin 0) cos 0 esin ®
(a) A=-4 (b) B=-12
x
7
(b) 2 
x
(b) B = a3/2
ndx = Asin 
-x3
Exercise-2 : One or More than One Answer is/are Correct
1 f fa 1 2
’ J (1 + V^)8
(a) ky =5
a
2. If
1
3
117Indefinite and Definite Integration
2
dx. Then which of the following altemative(s)9. Let J
is/are correct ?
(c) 2J-K = 3k(a) 2J + 3K = 8n
2 (b)dt(a)
2 (d) IMdt
o
. Then:
12. For a > 0, if I = | dx = Asin
,1/2(d) B = a
(c tan x) + D, then := a tan x + b tan
(c) c=V2(b) b=V2 (d) b =
(d) 4028(a) 0
where a e R then L can be :
(c) 0
1
2V2
15. Let L = lim 
n—>00
— + cot 
X
x 
gM = j 
0
X
14. The value of definite integral: 
2014 
-2014
n dx
2V2
sinx 
| sin x|
13. If J
7i+t
11. Let Zj = lim
X—>00
10. Which of the following function(s) is/are even ?
= equals : 
(x)
00
f
J 1 + n xa
(c) (2014)2
(b) 4J2 + K2=26n2
(b)
2
x3/21—— + C, where C is any arbitrary constant, then :
(c) A=i(a) A-j
dx
1-sin4 x
(a) a =i
2
(d) 2. = -
K 5
(a) Both \ and l2 are less than 22/7
(b) One of the two limits is rational and other irrational
(c) Z2>Z1
(d) l2 is greater than 3 times of
x
7
1 
and = lim f 
h->o+ J
(b) B = a3/2
1 + sin2015
-x3
dx
(x) + 71 + sin4030
(b) 2014
(a) it
fM = j In ft + 
0
X
(c) h(x)=j[7
0
2 
X-COS X
x + sin x
hdx
777
(2t +Dt .------------at
2' -1
(d)J
1 4-1 4-12 -77-
118 Advanced Problems in Methamatics for JEE :
Idx then correct statement(s) is/are :
(a) I + J = 2 (b) I-J = n
| Answers |
(b, c) (a, b, c, d) (a,b)2. 3. (a, c, d)4. (a, b, c) (a, b)5. 6.
(b.d) (a, b)(a, b, d) 8. 9. (a, b, c, d)10. (a, b, c, d) (a, b)11. 12.
(a, b, c)(b)(a, c) 16. (b,c)13. 14. 15.
r jTZvT
1-Vx
(d) J=^
2
/ \ T 2 + 71 (c) I = —- 
2
dx and J = J
o
16. Let I = j 
o
Indefinite and Definite Integration 119
Exercise-3 : Comprehension Type Problems
i
(d) None of these
Cd) 5
(c) 2 (d) 0it
(b) 1 (0 2
(d) Does not exist
i
1 I
6. Minimum value of f(x) is :
(a) 0 (b) 1
Let /(x) = j x
Paragraph for Question Nos. 3 to 4
Let /(x) be a twice differentiable function defined on (-oo, oo) such that /(x) = /(2 - x) and 
f'\ 1 ] = /'[ 1 | = o. ThenI2J UJ
Paragraph for Question Nos. 6 to 8
Consider the function /(x) and g(x), both defined from R R 
3 x 1
/(x) = — + l-xjg(t)dt andg(x) = x-j/(t)dt, then 
o o
(a) 1 
i
5. J/(I-t)e-C0SnC
■ o
COS nt J J.
(b) 
2
dt-J/(2-t)e
i
1. If/:R -(2n + 1)^---- > R then/(x) be a :
(a) even function (b) odd function
(c) neither even nor odd (d) even as well as odd both
2. The number of solution(s) of the equation /(x) = x3 in [0,2n] be :
(a) 0 (b) 3 (c) 2
3(C) 2
2
(a)
o
cosradt is equal to :
3. The minimum number of values where /"(x) vanishes on [0, 2] is :
(a) 2 (b) 3 (c) 4
i
4. j /' (1 + x) x2ex dx is equal to :
Paragraph for Question Nos. 1 to 2
2 cos2 x(2x + 6tan x - 2x tan2 x)dx and /(x) passes through the point (it, 0)
(d) it
.,(1
4.
120 Advanced Problems in Methamatics for JEE
(d) 3
CO
(b) (c) (d)e
2(c)
(b) — Inx-----
(a) 0 (b) 1 (c) -1 Cd) -e
i
j/(x)dx = 
o
7. The number of points of intersection of/(x) and g(x) is/are :
(a) 0 (b) 1 (c) 2
13. Value of lim 
n—>oo
3
I — 
2e
CLD/nCl) 
ln(n)
Paragraph for Question Nos. 9 to 11
Let J(x) be function defined on [0, 1] such that /(I) = 0 and for any a e (0,1], 
a 1
j /(x) dx - j /(x) dx = 2/(a) + 3a + b where b is constant, 
o a
12. f3M equals :
x3 f 5(a) — Inx--3 I (
(d) - 
e
Paragraph for Question Nos. 12 to 13
Let /0 (x) = Inx and for n > 0 and x > 0
Let /n+1 (x) = j /n(t)dt then : 
o
(a) - 
e
x3 ( 11(c) — Inx-—[3 I 6
8. The area bounded by g(x) with co-ordinate axes is (in square units) :
9 9 9(a) - (b) - (c) - (d) None of these
4 2 8
(b) 2-
2e
... 3 3
(d) + «2e 2
9. b =
3 3 3 3(a) -1-3 Cb) (c) ^ + 3
2e 2e 2 2e
10. The length of the subtangent of the curve y = f(x) at x = 1/2 is :
(a) Ve-1
x3 ( 5(d) — Inx--|3 I C
5
6
11
6
11
6
5
6
x3
3
121Indefinite and Definite Integration
Paragraph for Question Nos. 14 to 15
/(I) = 1
is equal to :
) + C- (d) sin 1(ex) + C(c) sin 1 (e
15.
+ C(a) cot + C
(c) tan + C+ C
2l
then :
16. The value of C is :
(d) 3(c) 2(a) 7
(c)
(a)7. (a) (c)(d) (a) 8. 9.denotes greatest integer function)
(c) [-2,-V3)
20. If f: R -> R, /(x) = ax + cos x is an invertible function, then complete set of values of a is :
(a)' (-2,-l]u[l,2) (b) [-1,1] (c) (-oo,-l]u[l,oo) (d) (^o,-2] u[2,oo)
X . X . X21. The range of function/(x) = [1 + sin x] + 2+sin— + 3 + sin— +...+ n + sin — Vxe[0,7r],
2 3 n
1 
----------is :
/(kl)
n(n +1) n2 + n + 2 n2 + n + 4
2 ’ 2 ’ 2
n 
X^" 4" ax 4* 1 -------------- , then the complete set of values of ‘ a’ such that /(x) is onto is: 
x2 + x + l
(a) (-00,00) (b) (-oo,0) (c) (0, co) (d) not possible
23. If /(x) and g(x) are two functions such that /(x) = [x] + [-x] and g(x) = {x} V x e R and 
h(x) = /(g(x)); then which of the following is incorrect ?
([•] denotes greatest integer function and {•} denotes fractional part function)
(a) /(x) and h(x) are identical functions (b) /(x) = g(x) has no solution
(c) /(x) + h(x) > 0 has no solution (d) /(x) - h(x) is a periodic function
"xir 15124. Number of elements in the range set of /(x) = —-----V x e(0,90); (where [■] denotes
15 x
x
3.
_15_] _
x
- ’ L 2.
n e N ([•] denotes greatest integer function) is :
n 2 + n - 2 n(n +1)
2 ’ 2
Advanced Problems in Mathematics for JEE6
O,(a) (b)
(c) (d)
26. Which of the following function is homogeneous ?
(c) h(x) = (d) |(x) =
If the range of /(x) = R (set of real numbers) then number of27. Let /(x) =
29. Range of the function /(x) = log2 is :
(a) (0, co) (b) (c) [1,2] (d) 4,1
(a) In
ri,l 
L4 J
4
-7x + 2 + f2-x
1,1 
.2
30. Number of integers statisfying die equation |x2 + 5x| + |x-x2|=|6x| is :
(a) 3 (b) 5 (c) 7 (d) 9
31. Which of the following is not an odd function ?
x4 + x2 +1 '
Jx2 + x + l)2,
(b) sgn(sgnfx))
(c) sin (tan x)
(a) /(x) = xsiny + y sinx
xy
x + y 2
2x + 3 ; x l
integral value(s), which a may take :
(a) 2 (b) 3 (c) 4 (d) 5
28. The maximum integral value of x in the domain of /(x) = log10 (logj/3 (log4 (x - 5)) is :
(a) 5 (b) 7 (c) 8 (d) 9
(d) /(x), where/(x) + /|-j = /(*)• f\ - | Vx eR-{0} and/(2) = 33 
lx 1 1 - 1
y ’ *
(b) g(x) = xex +yey
x-y cosx 
y sin x + y
1
x
7Function
; where [•] denotes greatest integer function(a) /(x) = cosx +
(b) g(x) =
33. Let f:N-----> Z and /(x) =
then g(x) be :34. Let g(x) be the inverse of /(x) =
y
>x (d)(c)>X (b)(a) ► xOo o
(d) [log 2 5, 3J
(b) fW is injective but not surjective
(d) f (x) is neither injective nor surjective
t: 2
35. Which of the following is the graph of the curve y/\y~\ = x is ? 
y y y
(c) log2^|^|J (d) log
; when x is odd
, then :
; when x is even
(a) f (x) is bijective
(c) f (x) is not injective but surjective 
2*+l -21-
2X + 2"x
(W, . 1. (2+x
(a) -log2 ------2 l2-x
36. Range of /(x) = log[x](9-x2); where [•] denotes G.I.E is :
(a) {1,2} (b) (-oo,2) (c) (-co, log 2 5]
37. If ex + = e, then for/(x) :
(a) Domain is (-00,1) (b) Range is (-00,1] (c) Domain is (-co, 0] (d) Range is (-co, 0]
38. If high voltage current is applied on the field given by the graph y + |y | -x -| x| = 0. On which of 
the following curve a person can move so that he remains safe ?
(a) y=x2 (b) y=sgn(-e2) (c) y = log1/3 x (d) y = m +|x|;m > 3
39. If|/(x) + 6-x2] =|/(x)| + |4-x2|+ 2, then/(x) is necessarily non-negaive for :
(a) xe[-2,2] (b) x g (-00,-2) u(2,(x) =|cosx|+ In(sinx)
x-1
2 
x
2
1
2
Advanced Problems in Mathematics for JEE8
(d) (-e, -1) u (1, e)(a) (b) (-e,-l) (c)
(d) Infinite(a) 0 (b) 1
x 044. The function, /(x) n e N is :
1 x = 0
m
fa) (d)
46. Let f(x) = of (x) is :
2
x X ■■■■ (a) (d)(b) (c)
2
1 + 1 +
n
47. Let f:R —> R, /(x) = 2x + |cosx|, then f is :
(a) One-one and into
(c) Many-one and into
n
S1
2
7
-1
2
x
TiT
X nx
71 + nx
-1,-*
e
(b) 3m
3m-l
f 1
ex
i
then fofofo 
n times
q2") 
(x2" sgnx) 2/1+1
(c) 3m-1
x2 x2
41. The domain of /(x) is (0, 1), therefore, the domain of y = /(ex) + /(In| x|) is :
1,1 
e
42. Let A = {1, 2, 3, 4} and f: A -> A satisfy /(l) = 2, /(2) = 3, /(3) = 4, /(4) = 1.
Suppose g: A -> A satisfies g(l) = 3 and fog = gof, then g =
(a) {(1, 3), (2, 1), (3, 2), (4, 4)} (b) {(1, 3), (2, 4), (3, 1), (4, 2)}
(c) {(1, 3), (2, 2), (3, 4), (4, 3)} (d) {(1, 3), (2, 4), (3, 2), (4, 1)}
43. The number of solutions of the equation [y + [y]] = 2cosx is :
(where y = i [sin x + [sin x + [sin x]]] and [•] = greatest integer function)
(c) 2
1
-e x
1
+ e x
(b) Even function
(d) Constant function
(a) Odd function
(c) Neither odd nor even function 
n—1
45. Let /(l) = 1, and /(n) = 2^/(r). Then ^/(r) is equal to : 
r=l r=l
3m -1
2
(b) One-one and onto
(d) Many-one and onto
48. Let/:R -> R, /(x) = x3 + x2 + 3x + sinx, then f is :
(a) One-one and into (b) One-one and onto
(c) Many-one and into (d) Many-one and onto
49. /(x) ={x} + {x +1} + {x + 2}+ + {x + 99}, then [/(V2)], (where {•} denotes fractional 
part function and [•] denotes the greatest integer function) is equal to :
(a) 5050 (b) 4950 (c) 41 (d) 14
Function 9
(a) [0, n] (b)
(c) (d)
(d) [2,272]
is :
(a) (b) (2,oo) (c)
(d) [-3,-2)o[2,3)
(c) y2=4ox (d) x2 = lay
(a)
does not exist(c) f (d)
(c) A =[2,273 
x
(c) 23 (d) 24
(2x2 + x-1), where [•] denotes the greatest integer
(c) 120 (d) 720
x-2 + 74-x, is invertible, then which of the
td)
(where [•] denotes greatest integer function)
(a) 21 (b) 22
55. The domain of function /(x) = logr j-i 
x+-L 2j
x
54. The number of positive integral values of x satisfying — =
50. If|cotx+ cosecx|=|cotx| + |cosecx|;x e[0,2n], then complete set of values ofxis :
3k1 r?7t „7t,— o —, 2n
2 4
_ 7t"i r 3tt „0,- u —, 2n2 J L 2
51. The function /(x) = 0 has eight distinct real solution and f also satisfy f (4 + x) = /(4 - x). The 
sum of all the eight solution of /(x) = 0 is :
(a) 12 (b) 32 (c) 16 (d) 15
52. Let /(x) be a polynomial of degree 5 with leading coefficient unity such that /(I) = 5, /(2) = 4, 
/(3) = 3, /(4) = 2, /(5) = 1. Then J(6) is equal to :
(a) 0 (b) 24
53. Let f: A -> B be a function such that /(x) = 
following is not possible ?
(a) A =[3,4] (b) A =[2,3]
= r2(a) x2+y2
function is :
3
— , 00
.2 )
56. The solution set of the equation [x]2+[x +1]-3 = 0, where [■] represents greatest integer 
function is :
(a) [-1,0)0[1,2) (b) [-2, -l)o [1,2) (c) [1,2)
57. Which among the following relations is a function ?
(b) 4+4=r2a2 b2
(where a, b, r are constants)
58. A funciton f;R -> R is defined as /(x) = 3x2 + 1. Then /-1 (x) is :
'x-1 „. 1 r .------ (b) -Vx-1
3 3
x-1
3
7t
2.
7ti
.4
Advanced Problems in Mathematics for JEE10
59. If/(x) = then /(/(x)) is given by :
(a) /(/(x)) = (b) f(/(x)) =
(c) /(/(x)) = (d) /(/(x)) =
(d) 4(c) 1
(d) [1, 7]
(d) [-2, 3]
(a) (b) (d)(c)
(b) (c) (d)
67. Range of the function, J(x) = , for x > 0 is :
(d) [6, co)
2 + x, x > 0
4-x, x k|_4
x > 0
, x 0
, x n
4
x2 + 5
2x-3
(l + x + x2)(l + x4) 
X3
x2 -5 
2x-3
2x2-5 
2x-3
4 + x , x > 0
x , x 0
4 + 2x , x , 3] -> (0, e7 ] defined by /(x) = ex3-3x2-9x+2 js :
(a) Many-one and onto (b) Many-one and into
(c) One to one and onto (d) One to one and into
3x2 + 3x-4
60. The function f:R->R defined as /(x) =--------------- is :
3 + 3x - 4x2
(a) One to one but not onto 0?) Onto but not one to one
(c) Both one to one and onto (d) Neither one to one nor onto
61. The number of solutions of the equation ex - log | x | = 0 is :
(a) 0 (b) 1 (c) 2 (d) 3
62. If complete solution set of e~x10.(a) (b) (c)3. 5.2.
(d) 16. (c)(b)(c) (c) (c)13. 14. 15.12.
/3 
4
2ey +1 
V3
2e* +1 
V3
7 
2ex -11 
V3 I
' 2ex -1
. V3
then J
17.t(d)
J Answers j
6. (b)i
Paragraph for Question Nos. 16 to 17
and fM = jte22 (1 + 3t2)1/2 
o
(b) I
-73(d)
2
Letg(x) = xceCx
17. The value of L is :
(a)l
Let /:/? -> ooJ be a surjective quadratic function with line of symmetry 2x -1 = 0 and
(a)
2(d) —= tan
V3
2(b) —cot 
V3
dx 
Vg(ex)-2
(a) sec-1(e_y) + C (b) sec-I(ex) + C 
(Where C is constant of integration) 
f eX a------- axJ/(ex)
14. ifg(x).^x)y(-^
f'Mdt. If L = lim ------ is non-zero finite number
x-^g'(x)
122 Advanced Problems in Methamatics forJEE
7Exercise-4 : Matching Type Problems -'V,.
Column-llColumn-I =
(P) 0
(A)
(Q) 1
(B)
(R) 2
dy =(C) x-y0
(S) 4
(D)
(T) 5
Column-ll
(A) (P) + C
(B) (Q) sin + C
8 (R)(C) /(*) =
X (S)(D) fW = -tan
x
(T) + C
lim 4 
n->co
00
I 
0
1 + ... —e
n
On / / .r | smx-sinyI lim y
J (x->ylo >• v
X+l
(x + 2)72y
(Vx - 2) -71 -x + cos-1 Vx + C
1-
b+
i ( 1 LIn x + — dx
\ x J
(1 + x2)
1 + -^- +C 
x2
n i i.= — In a, then a =
x6 
6(1 -x4)*2
x5 
5(1-x4)*2
3
3 - + —-en 
n2
1
2 
en 2 - —- + — en „2 n n
x4 + x' 
(l-x4)7/2
/(x)’----------t\...........
(x+2)Vx2 + 6x + 7
__________ Column-1
--------- 1 
(x2 +1) Vx2 + 2
i_______________________________________
2. Match the following j /(x) dx is equal to, if
Indefinite and Definite Integration 123
3.
Column-I Column-ll
(A) (P)
(B) (Q)
(C) (R) In4-ln3
dx =
(D) (S)
de =
4.
Column-ll
(A) (P) 6
(B) (Q) 1
is/are
(C) (R) 2
(D) (S) 3x +1
X
(Where C is integration constant), then A =
1
2
Number of solution of x - 2x sin — + 1 = 0
2
n
6
where [•] greatest integer function 
jt/2
o
n/2
41ir
4 
j|cosx|dx = 
o
1/2 Z
r [ r , , (1 + xJ [x] + ln| -----
-1/2
27 cos 0 
3(7sin0 + 7cos0)
______________ Column-1______________ \
If quardratic equation 3x2 + ax + l = 0 and 
2x2 + bx +1 = 0 have a common root then value 
of Sab -2a2 -3b2 =
Number of points of discontinuity y = — 
u
cosx , ------------------------dx = 
(1 + sin x)(2+ sin x)
, 1 ■ , where a =----- is/are
x-1
20 + 4=
72
-l/A
I +C-.4.
J^x5'2(l + x)7/2
1
2 +u-2
1 + x
1-x
124 Advanced Problems in Methamatics for JEE
5. :
Column-I Column-ll
(A) (P) -n
 S; C->P; D—> S1. B->P;
A->S; B->Q;2. C-»P; D->R
3. A —> R; B -> Q ; C->S;
D-> P4. A-> Q ; B-> R ; C->S;
C-> P ; D-> Q5. A-> S ; B-»R;
7
3
where {x} denotes the fractional part 
of x
2n
j[sinx + cosx] dx
o
j||sinx|-| cos x 11 dx 
o
1.5 
f[x2]dx 
0
4
j{Vx} dx 
o
125Indefinite and Definite Integration
Exercise-5 : Subjective Type Problems
3n
,44. If
cot
2
+ C6. The value of
) dx = N. Find the value of (N - 6).+ ln(x
+ C where /(x) = sin x + cosx find the9. If
,3 3
(rc-x)8 dx, then11. Let/
= — f 71-9x2 + (cos 
ki k
2 tan x 4-1
Va
00
2. If J 
o
i
5. If the value of the definite integral | cot
2
— dx
4xJ (1 + (x4)“ 
(1 + x4)2012
8. Let
1
1
7. Let j 
o
where Xand p are relatively prime positive integers. Find unit digit ofp. 
i„/-v2x2x2+1 
1 '
k2(Vq - Vb) 
4c
+ C, then k2 + k2 =
' l~3 1 2”
3. Let /(x) = xcosx; x e —,2tt and g(x) be its inverse. If jg(x)dx = an2 4-07t4-y, where a,0
L -* o
2x2+1
x + (arccos 3x)
71-9x2
(where C is an arbitrary constant.) 
x3 ± 1 . .. -
(a2 + x2)5 ka6
dx =x
71-(x2)W , 
where a, b,c,eN in their lowest from, then find the value of (a + b + c). 
r tan x , 2---------------------dx = x —-=tan 
J tan2x + tanx + 1 VA
Then the value of A is : 
3 ri /'v-4-\2Oio> 
dx - —
f
1
Ji-x:
f ————7— = A tan 1 (/(x)) + B In
J cos-5 x - sin x
value of (12A + 9V2B)-3.
10. Find the value of | a| for which the area of triangle included between the coordinate axes and 
any tangent to the curve xay = Xa is constant (where X is constant.)
" 7T15
= [ x6 (it - x)8 dx, then —------ =
Jo then
find the value of (0 4- y - a).
126 Advanced Problems in Methamatics forJEE
value of
14.
15. Let 1
is discontinuous where 0 g [0,2k].
19. The maximum value of
equal to :
, n e N, then evaluate
1
17. Find the number of points where /(0) = j
1 + —
Vn
/(15) + /(3) 
J(12)-/(10)
sin| n + - |x
dx where n g IV. If I2 + +13 +
sin2(n0)d0 
sin2 0
y 
16. If M be the maximum value of 72 I" ^x4 + (y -y2)2 dx for y g [0,1], then find —.
Jo 6
50
If j /(x) dx = I. Find [V7 - 3]. (where [•] denotes greatest integer function.) 
0
1
12. If maximum value of j (/(x))
0
(where p and q are relatively prime positive integers.). Find p + q.
13. Let a differentiable function /(x) satisfies /(x)/'(-x) = /(-x)-/'(x) and /(0) =1. Find the 
2 * r dx
J2l + /(x)’
1 5/(x) = - j (x-t)2g(t)dt, then find the value of f"'(l) + /"(l). 
0
*/2
21. If/(n)= — f
n J
0
100 91
If {x} denotes the fractional part of x, then I = j {Vx} dx, then the value of---- is :
0
n 4
0
+ ^20
1
20. Given a function g, continuous everywhere such that g(l) = 5 and jg(t)dt=2. If 
0
2
22. Let/(2-x) = /(2 + x) and /(4-x) = /(4+ x). Function /(x) satisfies j fWdx = 5.
0
= mn2, then find the
1
3 dx under the condition -1 oo yjn I % _ 
3n/2 
c Msin x • /(x) dx , subject to the condition | /(x) | » q
Vx 'I
• jg(x) h(x) dx 
Ju >
24. Let lim n
n—>&
where p and q are relative prime positive integers. Find the value of | p + q |. 
b
Jxsinx dx 
a
-e «
2n ) 
pq(p + q) 
10
.n"/
b a+b
J| sin x | dx = 8 and J | cosx | dx = 9 then the value of 
a 0
26. If /(x), g(x), h(x) and (x
J g(x) (Xx) dx - J /(x) («x) dx
11
x2 x3
1 + X + —+---- + 
2 3
AiAl
n' ■ (l1 • 22 • 33 •
(x A
J /(x) h(x) dx
a >
is divisible by (x -1)?' . Find maximum value of X.
2
27. If J (3.
0
Find the value of (a + b).
X
28. letf(x) = Jex~y/'(y)dy -(x2 - x + l)ex
0
Find the number of roots of the equation /(x) = 0.
n x \
29. For a positive integer n, let In = J I -|x| I cos nxdx
1
23. Let/n = J|x|
lx2 -3x+ l)cos(x3 -3x2 + 4x-2)dx = asin(b) , where a and b are positive integers.
Area Under Curves
U_____ .J
Exercise-1 : Single Choice Problems
(c) (d) 1
(c)
(a) 2 (c)
(b) 2 (c) (d) 1
1. The area enclosed by the curve
[x + 3y] = [x - 2] where x e [3,4) is :
(where [•] denotes greatest integer function.)
2 1(a) ± (b) ± • i
3 3
2. The area of region enclosed by the curves y = x
, , 1 2 4 ,,,16(a) - (b) - (c) - (d) —
3 3 3 3
3. Area enclosed by the figure described by the equation x4 +1 = 2x2 + y 2, is :
(b) (c) ? (d) i
3 3 3
4. The area defined by|y | 1; An represents the area of the region restricted to the following
2 2X 9 o Vtwo inequalities : — + y m
(a) 4 (b) 1 (c) 2 (d) 3
6. The ratio in which the area bounded by curves y 2 = 12x and x2 = 12y is divided by the line 
x = 3 is : .
(a) 7:15 (b) 15 : 49 (c) 1:3 (d) 17 : 49
7. The value of positive real parameter ‘a’ such that area of region bounded by parabolas
y = x -ax2, ay = x2 attains its maximum value is equal to : 
(a) 1 (b) 2 (c) 1
2 3
1
4
2 andy =71*1 is :
Area Under Curves 129
(0
2
(d) 1(c)
(c) 1
(d) 1
2
(d) none of these
(c)
(a)
(c)
(d) A 
ve
(d,T
(d)72— —
2
4 dx x4 dx
(a) -jL (b) ---- (c) -±=
y/8e -J4e y/2e
11. The area enclosed between the curves y = ax2 and x = ay 2 (a > 0) is 1 sq. unit, then the value 
of a is :
(a) 4= Cb) 7 (c) 1 (d) 1
73 2 3
12. Let /(x) = x3 -3x2 + 3x + l and g be the inverse of it ; then area bounded by the curve 
y = g(x) with x-axis between x = 1 to x = 2 is (in square units) :
11 3(a) A (b) A (c) A
2 4 4
13. Area bounded by x2y2 +y4 -x2 - 5y2 + 4 = 0is equal to :
(a) —+ V2 (b) —-72 (c) — + 273
3 3 3
14. Let f : R+ -> R+ is an invertible function such that f '(x) > 0 and /"(x) > 0 V x e [1, 5]. If 
/(l) = 1 and /(5) = 5 and area bounded by y = f(x), x-axis, x = 1 and x = 5 is 8 sq. units. Then 
the area bounded by y = f~x (x), x-axis, x = 1 and x = 5 is :
(a) 12 (b) 16 (c) 18 (d) 20
15. A circle centered at origin and having radius n units is divided by the curve y = sinx in two 
parts. Then area of the upper part equals to :
2 3 3
(a) T cb) T (c) T
16. The area of the loop formed by y 2 = x(l - x3 )dx is :
JoVx-x4 dx (b) 2Jo-/
j *7x-x4 dx (d) 4|q 7x-
x-x4 dx
8. For 0 0. Point B has 
coordinates (x, 0). If ‘O’ is the origin, then the maximum area of A AOB is :
(c)
130 Advanced Problems in Mathematics for JEE
, the area bounded by the curve y = f(x), x-axis, x = 0 and
(a)
(b)
(0 G
(d)
2^
| Answers |
(a)6. (03. 5. 7. 8. 10.2. 9.
16.14. 15. 17.12. 13. 18.11.
(b)
(d)
(b)
(b)
(0
(c)
(d)
(b)
(a)
(c)
(b)
(b)
(d)
(b)
(b)
(b)
' I 9 J
(c) (3k - 4) sq. units (d) (3k - 2) sq. units
K K 
3’2
2
X) , •. n
[[ sin—|dx + f
H 2) J
*2z \
2j rf • *i x dx + sm —
2y 
*1
x2 
j sin
*2 
jsin: 
*1
2n 
x-2K)2dx +
*2
ZJl x x
j(x-27t)2dx, where xx g i 0, j
x = 2k is given by
(Note : Xj is the point of intersection of the curves x2 and sin —; x2 is the point of intersection
X 9of the curves sin — and (x - 2k) )
IK, \ 
ff sin— |dx 
R 2J
—, — j and x2 6 (k, 2k)
2 3 J
. f 3k and Xj g —, 2kI 2 ,
-,2n
3
and x2 g
17. If /(x) = min x2, sin —, (x - 2k)2
2
o'
j x2JL. .
o
X1
Jx2dx + 
o 
j x2dx + 
o
It *2
x2dx + jI 
n 
2n 
dx +
X2 
2n
dx + | (x - 2n)2dx, where Xj 
*2 
2n
j (x - 2k) 2 dx, where Xj g 
*2
18. The area enclosed between the curves | x| + |y |> 2 and y 2 =4 1-— is :
(a) (6k-4)sq. units (b) (6k - 8) sq. units
Area Under Curves 131
Exercise-2 : One or More than One Answer is/are Correct
(a)
>x(b) eVa1 b
(c)
(d)
(c)
| Answers J
(d)1. (b, c, d) (a, b) (a, d)2. 3. 4.
yA2
r
7^77
(b) Snlln2
= aVx + bx passes through point (1,2) and the area bounded by curve, line x = 4
— , then V n g {1,2,3,...} :
, c 
dx >--- - f f M dxc-bJ
3n
2 ’ Sn = X
r=2n+l
1. Let /(x) be a polynomial function of degree 3 where a |a|) 
c c b
j /(x) dx = j /(x) dx + j /(x) dx 
aba 
c 
j /(x) dx hn2 
2
3. If a curve y 
and x-axis is 8, then :
(a) a = 3 (b) b = 3 (c) a = -1 (d) b = -1
4. Area enclosed by the curves y = x2 +1 and a normal drawn to it with gradient-1; is equal to:
2 1 3 4(a) (b) ± (c) -a (d) 1
3 3 4 3
132 Advanced Problems in Mathematics for JEE
fExercise-3 : Comprehension Type Problems
, where A represent domain set and B represent range set ofLet f: A -> B /(x) =
(b) -3 (c)
(a) 2V5 + In
(d) 3V5 + 21n(c) 3V5 + 41n
(c) -d-e )
equals :
(c) — e(a) e
(c) (d)(b)(a)
| l.|(a) 6. (b)5. (b)4. (b)2. Cd) | 3. [(b)
1. g(0) is equal to :
(a) -1
Paragraph for Question Nos. 4 to 6
For j = 0,1,2,... n let Sbe the area of region bounded by the x-axis and the curve yex = sin x 
for jn 0 
en(l + e”) 
2(e” -1)
4. The value of So is :
(a) — (1 + e71)
2
(d) i(e” -1)
2
3 
(d)
2
(b) +
2
(b) eK
1 + e71 
2(en -1)
Answers 5
1 + e” 
e” -1
1
2'
5. The ratio ^09 
$2010
e" (1 + e” ) 
(e7' -1)
1
2
Area Under Curves 133
Exercise-4 : Matching Type Problems
1.
Column-ll
(A) 48
(B) (Q) 27
(C) (R) 7
, then
(D) (S) 4
‘ a’ is equal to
(T) 3
j Answers |
’V3-V2 (V2 + 1)/
2 k
(P)
If the area bounded by the graph of y = xe- 0) and the abscissa axis is - then the value of
9
Column-1\
Area of region formed by points (x, y) satisfying 
[x]2 = [y]2 for 0 R j B —> Q j C —> P j D —> T
The area in the first quardant bounded by the 
curve y = sin x and the line 
2y-l 2., ..~— = - (6x - k) is 
V2-1 K
k =
134 Advanced Problems in Mathematics for JEE
Exercise-5 : Subjective Type Problems
.2
2^5
J Answers J
61 2. 7 3. 4. 5 85. 6. 22 7.
18.
□□□
. 2 X X y = sin — + cos— ,
then the numerical quantity k should be :
1. Let f be a differentiable function satisfying the condition / — * 0,f(y) * 0)
w Ay)
V x,y eR and /'(I) = 2. If the smaller area enclosed by y = /(x), x2 + y 2 = 2 is A, then find 
[A], where [•] represents the greatest integer function.
2. Let /(x) be a function which satisfy the equation /(xy) = /(x) + f(y) for all x > 0, y >0 such 
that /'(I) = 2. Let A be the area of the region bounded by the curves y = /(x), 
y =| x - 6x + 1 lx - 6| and x = 0, then find value of — A.
3. If the area bounded by circle x2 + y2 = 4, the parabola y =x
(where [ ] denotes the greates integer function) and x-axis is„;„2 *
' |_— 4
I 3 k)
4. Let the function f : [-4, 4] —> [-1,1] be defined implicitly by the equation x + 5y - y 5 = 0. If the 
area of triangle formed by tangent and normal to /(x) at x = 0 and the line y = 5 is A, find —.
5. Area of the region bounded by [x]2 = [y]2, if x g [1, 5], where [ ] denotes the greatest integer 
function, is :
6. Consider y = x2 and /(x) where /(x), is a differentiable function satisfying 
/(x +1) + f(z -1) = /(x + z) V x, z e R and f (0) = 0; /'(0) = 4. If area bounded by curve
2 ( 3 \
y=x andy =f(x)is A, find the value of I — A L
7. The least integer which is greater than or equal to the area of region in x -y plane satisfying 
x6 -x2 + y 2satisfied by family of curves y = Aex + Be3x +Ce5x 
arbitrary constants is :
(a) l!Z-9^Z. + 23^ + 15y =0 (b) + 9^-23^-15y = 0
dx3 dx2 dx dx3 dx2 dx
(c) ^ + 9^-23^ + 15y=0 (d) -9^ + 23^-15y = 0
dx3 dx2 dx dx3 dx2 dx
3. Ify = y(x) and it follows the relation +y cosfx2) = 5 theny'(O) is equal to :
(a) 4 (b) -16 (c) -4 (d) 16
4. (x2 + y 2) dy = xydx. If y (x0) = e, y(1) = 1, then the value of x0 is equal to :
Zzl
2
e2
136 Advanced Problems in Mathematics for JEE
7. A function y = /(x) satisfies the differential equation (x + 1) /'(x) - 2(x2 + x) /(x) =
(b)(a)
(d)(c)
8. The solution of the differential equation 2x2y — = tan(x2y2)-2xy2 given y(l)
(a) sin(x2y2)-l = 0
bywhose
-r-y =0
be solution of differential equation —y -4— + 4y =0; then a is :
(a) 0 (b) 1 (d) 2
(b) 2,1
(d) 1,3
^- = 0 
dx
5-6x
x + 1
6x + 5 
x + 1
(c being arbitary constant.)
(a) y = /(x) + cex (b) y = -fM + cex
(c) y = -f(x) + cex/(x) (d) y = c /(x) + ex
13. The order and degree respectively of the differential equation of all tangent lines to parabola 
x2 = 2y is :
(a) 1,2
(c) 1,1
V x > -1. If /(0) = 5, then /(x) is : 
3x + 5
x2 e
(x + 1)’
d2y-4—y -7x = 0 are a and p, then the
dx2
6x + 5
(x + 1)2
x2 • e
2,,2
10. If y = e(a+1)x
x2• e
x2 • e x2• e
value of (a + p) is :
(a) 3 (b) 4 (c) 2 (d) 5
dy ?12. General solution of differential equation of/(x)— = f (x) + fWy + f'Wy is : 
dx
(c) sin(x2y2) =ex-1
9. The differential
y =Cj cos(x + C2)-C3e
d4y d2y(a) —4----- + y = 0
dx4 dx2
d3y d2y dy
(c) —y + ^- + y- + y = o 
dx3 dx2 dx
solution is given 
..,C5 are constants is : 
d2y dy—+ — -y = 0 
dx2 dx
... « j d2y dy(d) —4 + -4 + -2L = 0
dx2 
dy
dx
dy
dx
(b) cos^ + x2y2
(d) sin(x2y2) = e2(x"1) 
equation whose general 
, -x+c4 + c 5 sin x, where C1,C2>__
(b)
dx3
(d) d3y
dx3
d2y
dx2
(c) -1 
fdyY/311. The order and degree of the differential equation —
I dxj
Differential Equations 137
(a) In (b) In
(c) 2 In (d) In
2(b) y sec
2
(b)
2
2
dx
 x
X~2~
, x 2 X(c) y cos x =----
X
X = -
2
(d) y cos2 x = —
2
+ 1 (c) + l (d)y2 =
dx dx
(a) y --Jx2 + y
(c) x-^x2+y
(x + y + I)(x + y-1)
(x + y)2
(x + y + I)(x + y -1)
(x + y)2
14. The general solution of the differential equation — + x(x + y) = x(x + y)3 -1 is : 
dx
= x2+C = x2 +C
= x2 + C
(a) y cos2
= cx2= cx2
dx
(x + y+ l)(x + y-l)
(x + y)4 
( (x + y+ l)(x + y-l)
(x + y)2
(where C is arbitrary constant.)
15. The general solution of — = 2y tanx + tan2 x is : 
dx
sin 2x --------+ C
4
cos 2x „ . , . __ .. --------+C
2 4
(where C is an arbitrary constant.) 
r?2 v dv
16. The solution of differential equation —— = — ,y(0) = 3 and y'(0) = 2:
dx2 dx
(a) is a periodic function (b) approaches to zero as x -> -oo
(c) has an asymptote parallel to x-axis (d) has an asymptote parallel to y-axis
17. The solution of the differential equation (x2 +1')^-^- = 2x\ — | under the conditions y (0) = 1
dx2 {dxj
andy'(O) = 3, is :
(a) y=x2 + 3x + l (b) y=x3+3x + l
(c) y=x4+3x + l (d) y = 3 tan-1 x + x2 + 1
18. The differential equation of the family of curves cy 2 = 2x + c (where c is an arbitrary constant.) 
is :
(a) ^=1
dx
19. The solution of the equation — + — tany = — tany siny is :
dx x x2
(a) 2y = siny (1 -2cx2) (b) 2x = coty (1 + 2cx2)
(c) 2x = siny(l-2cx2) (d) 2xsiny = 1 -2cx2
20. Solution of the differential equation xdy -ydx-^x2 + y 2dx = 0 is : 
2 = ex2 (b) y + 7x2 +y2 = ex
(d) y + 7x2 +y2
sin 2x --------+ C
4
sin 2x + £
2 +
138 Advanced Problems in Mathematics forJEE
the interval (0, oo) such that /(I) = 1 andon
(a) —+ ——
(b) In 18 (c) 21nl8 (d) In 9
(a) 7
(c) 3 (d) 2(a) 9
lim 
t->x
Cb) 5 
3x2 -y. 
2y-x2’
Cb) 4
27. Lety = /(x)and —— 
y dx
(a) -In 18
2 
dy o 9
23. The solution of the differential equation sin2y — + 2 tan x cos y = 2 secx cos y is : 
dx
(d) y[%
I dx
= V x > 0, then f(x) is :
Cd) 27
f (1) = 1 then the possible value of /(3) equals :
, . ( dy
(c) y \dx 
dy y ?26. If y = f (x) satisfy the differential equation — + — = x ; /CD = 1; then value of /(3) equals : 
dx x
Cc) 9
t) A4x
22. The population p(t) at time ‘t’ of a certain mouse species satisfies the differential equation
— p(t) = 0.5 p(t) - 450. If p(0) = 850, then the time at which the population becomes zero is: 
dt
\+2y—~x
1 dx
(where C is arbitrary constant)
(a) cosy secx = tanx + C (b) secy cosx = tanx+ C
(c) secy secx = tanx + C (d) tany secx = secx+ C
dy ?24. The solution of the differential equation — = (4x + y +1) is :
dx
,, A 3 x3 Cb) — + — 
4x 4
. . 1 3x3(c) — +-----
4x 4
21. Let /(x) be differentiable function 
y(x2A/y + 3)
(c) xy(y2y[x + 3) = c (d) Ay(y 2Vx + 5) = c
7t
2
dy
dx 
following statement(s) is/are correct ?
(a) yf^^o\ 6 7
141Differential Equations
Exercise-3 : Comprehension Type Problems
.4
2(b) ^- + 2y = 12x2 + 2
(d) —+y =12x + 2
2 (d) Data insufficient(c) e
o
(d) x = 2
(d) 4(c) 3
Cc) e
(d) x + ln(x + l)(c) 2xe
(d) [0,oo)(c)(a) R 2
— ,00 
e
1
— , 00 
e
(a) x = -1 (b) x = 0
4. The value of/(g(0)) + g(/(0)) is equal to :
(a) 1 (b) 2
5. The largest possible value of h(x) V x e R is :
(a) 1 (b) e1/3
g'(/W)/'WVxe
3. The graph of y = h(x) is symmetric with respect to line :
(c) x = l
Paragraph for Question Nos. 6 to 8
Given a function ‘g’ which has a derivative g'W for every real x and which satisfy 
g'(0) = 2 and g(x + y) = eyg(x) + exg(y) for all x and y.
6. The function g(x) is :
(a) x(2 + xe*) (b) x(ex+l)
7. The range of function g(x) is :
(b)
1- y =g{x) satisfies the differential equation :
(a) ^-y=12x2 + 2
(c) ^ + y=12x2 + 2
dx
2. The value of g(0) equals to :
(a) 0 (b) 1
+ x2; V x > 0.
Paragraph for Question Nos. 1 to 2
A differentiable function y = g(x) satisfies |(x-t + l)g(t)dt =x 
o
Cd) e2
Paragraph for Question Nos. 3 to 5
Suppose f and g are differentiable functions such that xg(/(x))/'(g(x))g'(x) = /(g(x)) 
R and f is positive, g is positive V x 6 R. Also j/(g(t))dt = (1 -e-2r)
V x e R, gC/CO)) = 1 and h(x) = V x e R.
dy
dx 
dy
dx
142 Advanced Problemsin Mathematics for JEE
(a) 0 (b) 1 (c) 2 (d) Does not exist
J Answers |
5. (c)4. (b) 6. (c)1. (c) 3. (c) 7. (b)2. (a) 8. (a)
8. The value of lim g(x) is : 
X—>-00
------ r- .• - - ' —•
143Differential Equations
Exercise-4 : Matching Type Problems
(P)(A)
(Q) + 1 = cy(B)
(x + l)(l-y) = cy(R)
= 0(C)
(D)
(T)
2.
Column-1
(P) ) = 1(A)
(Q)
(B)
(R) = c
(C)
(S)
(D)
(T)
(where c is arbitrary constant).
j Answers ?
i.
2.
d3y 
dx3
Solution of differential equation 
[3x2y + 2xy - ex (1 + x)]dx + (x3 + x2)dy = 0is :
I
4y 
dx
Solution of differential equation 
^(x2y3 +xy) = 1 is : 
dx
Solution of differential equation 
= xy(x2y2-l) is : 
dx
\ Column-11 Solution (Integral curves) 
y=Axx + A2x + A3dy 
y-x— = ydx
— = l-y2+ce~y/2 
x
(S)
/ , \2
I d2y -3-4 l^dx2 >
(x2y 2 -l)dy + 2xy3dx = 0
_______ Column-ll
y2(x2 +1 + cex2
\ Column-1 (Differential equation) \
2 + ^_ 
dx '
(2x-10y3)—+ y =0
dx
2 2 xy
Solution of differential equation
9 2
ydx - xdy - 3xy ex dx = 0 is :
x = Ayy 2 + A2y + A3 
xy 2 = 2y 5 + c
i = 2-y2 +ce 
x
A —> Q j B —> R j C —> P j D —> S
A->R; B—>T; C->S; D->Q
(x2 + x3 )y - xex = c
x 3 x2 ------- e
y 2
144 Advanced Problems in Mathematics for JEE
Exercise-5 : Subjective Type Problems
2. Lety = /(x) satisfies the differential equation xy(l + y)dx = dy. If /(0) = 1 and /(2) =
then find the value of k.
6. Let y = f(x) satisfies the differential equation xy (1 + y) dx = dy. If /(0) = 1 and /(2) =
then find the value of k.
• Answers |
■621 3. 4. 1 6.2. 5. 4 2
□□□
1. Find the value of | x+i /(t)-/(x + 4) x—>1 x-1
5. Let y = (a sin x + (b + c) cosx) ex+d, where a, b, c and d are parameters represent a family of 
curves, then differential equation for the given family of curves is given by y" - ay' + Py = 0, 
then a + p =
e2 
k-e2
Algebra
8. Quadratic Equations
9. Sequence and Series
10. Determinants
11. Complex Numbers
12. Matrices
13. Permutation and Combinations
14. Binomial Theorem
15. Probability
16. Logarithms
II
Mk.
i
r»i*%
Quadratic Equations
2.
(d) [0, oo)(c) [2, oo)(b) [1, oo)(a)
(d) (-1, 0)
Cd) 20(c) 15
Exercise-1 : Single Choice Problems
1. Sum of values of x and y satisfying the equation 3X - 4y = 77; 3x/2 -2y = 7 is :
(a) 2 (b) 3 (c) 4 (d) 5
3 3
If fto = {J (x - a,) + £ ai ~ 3x where at X2) are two values of X for which the expression
f(x, y) = x2 + Axy + y2 -5x -7 y + 6 can be resolved as a product of two linear factors, then 
the value of 3Xa + 2X2 is : 
(a) 5 (b) 10
148 Advanced Problems in Mathematics for J EE
(a) (b)
Cd)(c)
(c) 5 (d) 6
12. For x e R, the expression
is :
(b) 3 (d) 5
18. For real values of x, the value of expression
(c) 4
3
(c) 4 
llx2 -12x-6
x2 + 4x + 2
(a) (1, 2) Cb) [0, 1]
(0 (0,1) (d) (-1, 0)
13. If 2 lies between the roots of the equation t2 -mt + 2 = 0 , (m eR) then the value of
?3|x]
(9+x2
1
3 m
(b) 4
Y"2 4- ?Y 4- r
—------------- can take all real values if c e:
x2 + 4x + 3c
(where [•] denotes greatest integer function)
(a) 0 (b) 1 (c) 8 (d) 27
14. The number of integral roots of the equation x8 -24x7 -18x5 + 39x2 + 1155 = 0 is :
(a) 0 (b) 2 (c) 4 (d) 6
15. If the value of m4 + = 119, then the value of m
m4
(a) 11 (b) 18 (c) 24 . (d) 36
16. If the equation ax2 + 2bx + c = 0 and ax2 + 2cx + b = 0, a * 0, b * c, have a common root, 
then their other roots are the roots of the quadratic equation:
(a) a2x(x +1) + 4bc = 0 (b) a2x(x + 1) + 8bc = 0
(c) a2x(x + 2) + 8bc = 0 (d) a2x(l + 2x) + 8bc = 0
17. If cos a, cospand cos y are the roots of the equation 9x3 - 9x2 - x +1 = 0; a, p, y e [0, n] then 
the radius of the circle whose centre is (Za, S cos a) and passing through (2 sin-1 (tan it/4), 4) 
is: 
(a) 2
9. Let a, p be the roots of the quadratic equation ax2 + bx + c = 0, then the roots of the equation 
a(x + l)2 + b(x + l)(x-2) + c(x-2)2 =0are :
2a + l 2p + l 2a-l 2p-l
a-1 p-1 a + 1 P+1
a + ip + 1 2a + 3 2p + 3
a-2 p-2 a-1 p-1
10. If a, beR distinct numbers satisfying |a-l| + |b-l|=|a| + | b| =|a + l| + |b +1|, then the 
minimum value of | a - b | is :
(a) 3 (b) 0 (c) 1 . (d) 2 •
11. The smallest positive integer p for which expression x2 - 2px + 3p + 4 is negative for atleast 
one real x is : 
(a) 3
149Quadratic Equations
19.
(c) 1
(d) [-1, 1]
23. If the roots of the equation
(d) -pq(c) -
.4
(d) cotp(c) tanp
(b) does not lie between -17 and -3
(d) does not lie between 3 and 17
x + 3
2
(a) lies between -17 and -3
(c) lies between 3 and 17
S —— holds for all x satisfying: 
xz-x-2 *-4
(a) -2 4 (b) -1 4
(c) x -1 or 2+ kx + (k2 + 2k - 4) = 0, then the minimum 
value of a2 + p2 is equal to :
(a) 12 (b) (c) (d) f
9 9 9
32. Polynomial P (x) = x2 - ox + 5 and Q(x) = 2x3 + 5x - (a - 3) when divided by x - 2 have 
same remainders, then ‘a’ is equal to:
(a) 10 (b) -10 (c) 20 (d) -20
33. If a and b are non-zero distinct roots ofx2 + ax + b = 0, then the least value of x2 + ax + b is 
equal to :
2 9 9(a) (b) (c) (d) 1
3 4 4
34. Let a,p be the roots of the equation ax2 + bx + c = 0. A root of the equation 
a3x2 + abcx + c3 = 0 is :
(a) a + p (b) a2+p (c) a2-p (d) a2p
35. Let a, b, c be the lengths of the sides of a triangle (no two of them are equal) and k e R. If the 
roots of the equation x2 + 2(a + b + c)x + 6k(ab + be + ca) = 0 are real, then:
2 2 1(a) k - (c) k > 1 (d) k 4
!
43. If a, P, y, 8 e R satisfy
(d) -3 OVx e(-oo, oo)is:
(a) k > -- (b) k > 4 (c) -- 6 is (-oo, a] u [fc>, co), then 
a + b = :
(a) 2 (b) 3 (c) 6 (d) 4
45. If exactly one root of the quadratic equation x2 - (a + l)x + 2a = 0 lies in the interval (0, 3), 
then the set of value ‘ a’ is given by:
(a) (-oo, 0)u(6, oo) (b) (-oo, 0] u (6, oo)
(c) (-oo, 0]o[6, oo) (d) (0, 6)
46. The condition that the root of x3 + 3px2 + 3qx + r = 0 are in H.P. is :
(a) 2p3-3pqr + r2 = 0 (b) 3p3 -2pqr + p2 = 0
(c) 2q3 - 3pqr + r2 = 0 (d) r3 -3pqr + 2q3 = 0
47. If x is real and 4y 2 + 4xy + x + 6 = 0, then the complete set of values of x for which y is real,
is:
(a) x 3 (b) x 3 (c) x 5 -3 or x £ 2
The solution of the equation log ^*2 (3 - 2x) , - 3) u (8, oo)
(c) (8, oo) (d) None of these
55. Two different real numbers a and p are the roots of the quadratic equation ax2 + c = 0 with 
a,c * 0, then a3 + p3 is :
(a) a (b) —c (c) 0 (d) -1
56. The least integral value of ‘ k’ for which (k -1)x2 -(k + l)x + (k +1) is positive for all real 
value ofxis:
(a) 1 (b) 2 (c) 3 (d) 4
57. If (—2, 7) is the highest point on the graph of y = -2x2 - 4ox + k, then k equals:
(a) 31 (b) 11 (c) -1
58. If a + b + c = 0, a,b,c eQ then
(b + c - a) x2 + (c + a - b) x + (a + b - c) = 0 are :
(a) rational (b) irrational (c) imaginary
59. If two roots of x3 '"'2
(a) a + be = 0
(c) ab = c
2X + X
x + 4
to :
3 11(a) A (b) A (c) A
4 2 4
51. Solution set of the inequality, 2-log2(x2 + 3x) > Ois :
(a) [-4, 1] (b) [-4, -3)u(0, 1] (c) (-oo, -3)u(l, oo) (d) (-oo, -4)u[l, co)
52. For what least integral ‘ k’ is the quadratic trinomial (k - 2) x2 + 8x + (k + 4) is positive for all
real values of x ?
(a) k = 4 (b) k = 5 (c) k = 3 (d) k = 6
53. If roots of the equation (m -2)x2 -(8 - 2m)x -(8-3m) = 0 are opposite in sign, then
number of integral values (s) of m is/are : 
(a) 0 (b) 1
(a) xg -oo, — u(ll,co)
I 2J
( 3")
(d) XG -00, -- U(-l, 11)
I 2)
(d) -i 8
9
which x2
(a) 0
60. If a and pare the real roots of x2 + px + q = Oand a4, p4 are the roots of x2 - rx + s = 0. Then
the equation x2 - 4qx + 2q 2 - r = 0 has always (a * p, p * 0, p, q, r, s eR): 
(a) one positive and one negative root (b) two positive roots
(c) two negative roots (d) can’t say anything
61. If x2 + px + 1 is a factor of ax3 + bx + c, then:
(a) a2 + c2 = -ab (b)a2 + c2=ab (c)a2-c2=ab (d) a2
ABC AC62. In a EABC tan —, tan — , tan — are in H.P., then the value of cot —cot — is :
2 2 2 2 2
(a) 3 (b) 2 (c) 1 (d) V3
63. Let /(x) = 10-|x-10| Vx g[-9, 9], if M and m be the maximum and minimum value of /(x) 
respectively, then:
(a) M + m = 0 (b) 2M + m = -9 (c) 2M + m = 7 (d) M + m =7
64. Solution of the quadratic equation (3|x|-3)2 =|x| + 7 , which belongs to the domain of the 
function y = -7(x-4)x is :
(a) ±i ±2 (b)i 8 (c) -2, -1
9 9 9
65. Number of real solutions of the equation x2 + 3| x| + 2 = 0 is :
(a) 0 (b) 1 (c) 2 (d) 4
66. If the roots of equation x2 - bx + c = Obe two consecutive integers, then b2 -4c =
(a) 3 (b) -2 (c) 1 (d) 2
3x2 + 9x + 17 .
------- 2------------------- 1S :
3x2 + 9x + 7
, x 17 (Oy
3x-2 .-------- is :
7x+5
70. Let A denotes the set of values of x for which ~t~ 
x
71. If the quadratic polynomial P(x) = (p - 3)x2 - 2px + 3p - 6 ranges from [0, oo) for every x e R, 
then the value of p can be :
(a) 3 (b) 4 (c) 6
72. If graph of the quadratic y = ax2 + bx + c is given below :
y
then:
(a) a 0, c > 0 (b) a 0, c 0 (d) a 0 (b) c (a - b + c) > 0
(c) b (a - b + c) > 0 (d) (a + b + c) (a - b + c) > 0
74. Minimum value of y = x2 - 3x + 5, x g[-4, 1] is :
(b)^
4
75. If 3x2 - 17x +10 = 0 and x2 - 5x + m = 0 has a common root, then sum of all possible real 
values of ‘ m’ is :
(b) (-oo, 4] u[4, oo)
(d) [4, oo) 
equation x
 26 .. 29(b) -— (c) —
9 9
76. For real numbers x and y, if x2 + xy -y 2 + 2x -y +1 = 0, then :
8 8 8(a) y can not be between 0 and - (b) yean not be between--and-
8 16(c) y can not be between -— and 0 (d) y can not be between —— and 0
77. If 3x4 - 6x3 + kx2 - 8x -12 is divisible by x - 3, then it is also divisible by :
(a) 3x2-4 (b)3x2 + 4
(c) 3x2 + x (d) 3x2 - x
78. The complete set of values of a so that equation sin4 x + asin2x + 4 =0 has at least one real 
root is : 
(a) (oo, - 5] 
(c) (-oo, -4]
79. Let r,s,t be the roots of the
(rs)2 + (st)2 + (rt)2 = b2 -kac, then k = 
(a) 1 (b) 2
Quadratic Equations 155
s
(a) a = 3, k = 8 (c) a = -3, k = -17 (d) a = 3, k = 17
(a) 19 (d) non-existent(b) 18
(b) (-co, 1) (c)
Cd) 4(c) 3
Cd) 950
-, then the value of a and k are :
} 7
’’ 3.
Cc) 150
a + 5i
2
80. If the roots of the cubic x3 + ax2 + bx + c = Oare three consecutive positive integers, then the 
i r a2 
value of------- =
b + 1
3
is equal to :
(a) 1 (b) 2 (c) 3 (d) 4
81. Let ‘ k’ be a real number. The minimum number of distinct real roots possible of the equation 
(3x2 + kx + 3) (x2 + kx -1) = 0 is :
(a) 0 (b) 2 (c) 3 (d) 4
1 11 (r82. If r and s are variables satisfying the equation------= - + -. The value of -
r+s r s 1 *
(a) 1 Cb) -1
(c) 3 (d) not possible to determine
83. Let /(x) = x2 + ax + b. If the maximum and the minimum values of /(x) are 3 and 2 
respectively for 0 50 
(C,T
89. The complete set of values of ‘ a' for which the inequality (a -1) x2 - (a +1) x + (a -1) > 0 is 
true for all x > 2.
(a) 1
V
90. If a, p be the roots of 4x2 -17x + X = 0, X eR such that 1 th611:
(a) fM 0 V x g R
(d) p > -1/3 
quadratic 
a2
(b) p > -1/4 (c) p > 1/3
the roots of the
]x-2(3log32-2log23) = 0, then the value of
(d) infinite
———— > -1 is satisfied Vx e R, 
x2 + mx + 4
(d) infinite 
taken by the expression,
equal to :
(a) 3 (b) 5 (c) 7 (d) 11
93. The minimum value of /(x, y) = x2 - 4x + y2 + 6y when x and y are subjected to the 
restrictions 0 1/4
a, P___
xz _i 3 + 2^log23
(a) 1 (b) -1 (c) 2
96. The number of integral values of k for 
x2 -2(4k-l)x + 15k2 -2k-7 > 0holds for all x gR is :
(a) 2 (b) 3 (c) 4
97. The number of integral values which can
x3 -1fM =------------z----------for x g R, is :
(x — 1) (x2 — x + 1)
(a) 1 (b) 2
157Quadratic Equations
a
110. If a, b and c are the roots of the equation x3 + 2x2 +1 = 0, find b 
c
(d) n
(c) /(x) takes both positive and negative values
(d) Nothing can be said about /(x)
101. If the equation x2 + 4 + 3 cos (ax + b) = 2x has a solution then a possible value of (a + b) 
equals
(a) (b) (c)
4 3 2
102. Let a, P be the roots of x2 -4x + A = 0 and y, 8be the roots of x2 -36x + B = 0. If a,p,y, 8 form 
an increasing G.P. and Ac = B then the value of*t’ equals
(a) 4 (b) 5 Cc) 6 (d) 8
103. How many roots does the following equation possess 3'*' (| 2 -1 x| |) = 1 ?
(a) 2 (b) 3 (c) 4 (d) 6
104. If cot a equals the integral solution of inequality 4x2-16x + 15 0, where e is the base of natural 
logarithm. The graphs of the functions intersect:
(a) once in (0, 1) and never in (1, co) (b) once in (0, 1) and once in (e2, co)
(c) once in (0, 1) and once in (e, e2) (d) more than twice in (0, oo)
106. The sum of all the real roots of equation
x4 - 3x3 - 2x2-3x + l = 0 is :
(a) 1 (b) 2 (c) 3 (d) 4
107. If a, p (a 1. Then the complete set of real values of x for which f'(x) > Ois:
(a) (1, oo) (b) (-1, co) (c) (0, co) (d) (0,1)
b c 
c a : 
a b
(a) 8 (b) -8 (c) 0 (d) 2
111. Let a,p are the two real roots of equation x2 + px + q = 0, p, q e R, q * 0. If the quadratic 
equation g(x) = 0 has two roots a + — ,P + - such that sum of roots is equal to product of 
a P
roots, then the complete range of q is :
158 Advanced Problems in Mathematics for JEE
(a) ±3
(d) 7
(d) -12
.X
(a) (-oo,!) (c) (d) None of these
(c) 6 (d) -22
’3
—. oo
.7
121. Let a,pbe real roots of the quadratic equation
value of (a2 + p2) is equal'to :
(a) 9 (b) 10 (c) 11 (d) 12
122. The exhaustive set of values of a for which inequation (a -1) x2 - (a + 1) x + a -1 > 0 is true 
Vx>2
(d) (-oo, 1 u (3, a>)(C) [I’3
(b)[|,oo
123. If the equation x2 + ax +12 = 0, x2 + bx +15 = 0 and x2 + (a + b) x + 36 = 0 have a common 
positive root, then b - 2a is equal to.
(a) -6 (b) 22
(b) [1,3
112. If the equation In (x2 + 5x) - In (x + a + 3) = 0 has exactly one solution for x, then number of 
integers in the range of a is :
(a) 4 (b) 5 (c) 6
2 1 6113. Let /(x) = x + — - 6x — + 2, then minimum value of/(x) is :
x2 x
(a) -2 (b) -8 (c) -9
114. If x2 + bx+ b is a factor of x3 + 2x2 + 2x + c(c * 0), then b -c is :
(a) 2 (b) -1 (c) 0 (d) -2
115. If roots of x3 + 2x2 +1 = 0 are a, P and y, then the value of (aP)3 + (Py)3 + (ay)3, is :
(a) -11 (b) 3 (c) 0 (d) -2
116. How many roots does the following equation possess 3^(|2-|x||) = l?
(a) 2 (b) 3 (c) 4 (d) 6
117. The sum of all the real roots of equation x4 -3x3 -2x2 -3x +1 = Ois :
(a) 1 (b) 2 (c) 3 (d) 4
118. If a and p are the roots of the quadratic equation 4x2 + 2x-l = 0 then the value of
£(ar +pr)is:
r=l
(a) 2 (b) 3 (c) 6 ■ (d) 0-
119. The number of value(s) of x satisfying the equation (201 l)x +(2012)x + (2013)x -(2014) 
= 0 is/are :
(a) exactly 2 (b) exactly 1 (c) more than one (d) 0
120. If a,P(a ’4_
Cd) -5.
45
(c) I?15-1 
2
133. Let /(x) = x2 + bx + c, minimum value of J(x) is -5, then absolute value of the difference of 
the roots of /(x) is :
(a) 5(b) V20
(c) V15 (d) Can’t be determined
134. Sum of all the solutions of the equation |x-3| + |x + 5| = 7x, is:
r ) 6 „ v 8 , y 58(a) - Cb) - Cc) -
q,b,c e {1,2, 
a + b + c is : 
(a) 196 (b) 284 (c) 182 (d) 126
128. Two particles, A and B, are in motion in the xy-plane. Their co-ordinates at each instant of
time t(t > 0) are given by xA =t, yA =2t, xB = 1 - t and yB = t. The minimum distance 
between particles A and B is : 
(a) | (b) ±
5 V5
129. If a * 0 and the equation ax2 + bx + c = 0 has two roots a and p such that a 2, 
which of the following is always true ?
(a) a(a+|b| + c) > 0 (b) a(a + |b| + c) 0 (d) (9a - 3b + c)(4a + 2b + c) 2and 0 0 V x > p (c) ac > 0
140. If/(x) =
is:
(a) 5 (b) -5 (c) 10
(c) 6 (d) -22
+-
1
(d) (0, l)u(2,co) 
(2-a)(2-p)(2-y) 
(2+a)(2 + P)(2 + y)’
138. If x2 + bx + b is a factor of x3 + 2x2 + 2x + c (c * 0), then b - c is :
(a) 2 (b) -1 (c) 0 (d) -2
139. The graph of quadratic polynomical f (x) = ax2 + bx + c is shown below
(a) -|p-a|-l 
a
—^X + 4, then complete solution of 0 1
sinx
(d) -(p + q + r)
+2
(d) None of these
(b) no real root
(d) exactly four real nroots
’3
—, oo
145. Consider the equation x3 - ax2 + bx-c = 0, where a, b, c are rational number, a * 1. It is
given that x15x2 and xxx2 are the real roots of the equation. Then JC1X2I ~— I = 
\b + c)
(a) 1 (b) 2 (c) 3 (d) 4
146. The exhaustive set of values of a for which inequation Qa — l)xc2-(ca + l)x + a- l>0is true
Vx>2.
(b) -,oo L3 J
147. The number of real solutions of the equation
x2-3|x|+2 = 0
(a) 2 (b) 4
148. The equation esinx -e"sin* -4 = 0 has
(a) infinite number of real roots
(c) exactly one real root
149. If a, p are the roots of the quadratic equation x2 - 2 (1 — sin 20) x - Z: cos2 20 = 0, (0 e R) then
the minimum value of (a2 + p2) is equal to :
(a) -4 (b) 8 (c) 0 (d) 2
150. If the equation |sinx|2+|sinx|+ b = 0 has two distinct roots in [Oi, nJ; then the number of
integers in the range of b is equals to :
(a) 0 (b) 1 (c) 2 (d) 3
151. If a * 0 and the equation ax2 + bx + c = 0 has two roots ct and p such that a 2.
Which of the following is always true ?
(a) a (a + | b| + c) > 0
(c) 9a - 3b + c > 0
152. If a,p are the roots of the quadratic equation x
x2 + px-r = Othen (a -y)(a-5) is equal to :
(a) q + r (b) q-r (c)-(q+r)
153. Complete set of solution of log1/3 (2X+2 - 4*) > -2 is :
(a) (-00,2) (b) (-oo,2 + V13) (c) (2,oo)
(b) a(a + | b|+ c))(a) (1, oo)
(c) |a| 0 for atleast one x > 0, then if k e S, then S contains:
3. The equation|x2 -x-6| =x + 2has:
(a) two positive roots (b) two real roots
(c) three real roots (d) four real roots
4. If the roots of the equation x2 - ax - b = 0 (a, b e R) are both lying between -2and 2, then:
(a) |a|2-^
(d)|a|>^-2
5. Consider the equation in real number x and a real parameter A, | x -11 -1 x - 2| +1 x - 4| = X
Then for X > 1, the number of solutions, the equation can have is/are : 
(a) 1 (b) 2 (c) 3 (d) 4
a 1 1— - - - then :
b b a
(a) a > 0 (b) a 0
7. Let /(x) = ax2 + bx + c, a > 0 and f(2 - x) = /(2 + x)Vx e R and /(x) = 0 has 2 distinct real ‘ 
roots, then which of the following is true ?
(a) Atleast one root must be positive
(b) /(2)/(l)
(c) Minimum value of /(x) is negative
(d) Vertex of graph of y = /(x) lies in 3rd quadrat
8. In the above problem, if roots of equation /(x) = 0 are non-real complex, then which of the 
following is false ?
(a) /(x) = sin — must have 2 solutions
4
(b) 4a - 2b + c 1V x e R
(d) All a, b, c are positive
9. If exactly two integers lie between the roots of equation x2+ax-l = 0. Then integral 
value(s) of‘a’ is/are :
(a) -1 (b) -2
164 Advanced Problems in Mathematics for JEE
(c) c > 0 (d) D > 0
Cd) -10
(d) 2 0 (b) b > 0
(where D is discriminant)
11. The quadratic expression ax2 + bx + c>0VxeR,then:
(a) 13a - Sb + 2c > 0 (b) 13a - b + 2c > 0
(c) c > 0, D b, D Oand x2-3x-4 2 (d)--------- if x > 2
(x-2) x + 2
If all values of x which satisfies the inequality log(1/3) (x2 + 2px + p2 + 1) > 0 also satisfy the 
inequality kx2 + kx~k2 0, the minimum value of —- + equals 24.
2
(d) (2,oo)
(b) The number of solutions of the equation tan 20 + tan 30 = 0, in the interval [0, n] is equal 
to 6.
128x?
— +(c) For Xj, x2, x3 > 0, the minimum value of + 13^1 + —
x2 x2 4XjX2
(d) The locus of the mid-points of chords of the circle x2 + y2 -2x-6y -1 =0, which are 
passing through origin is x2 + y 2 - x - 3y =0.
20. If (a, 0) is a point on a diameter inside the circle x2 + y 2 = 4. Then x2 - 4x - a2 = 0 has :
(a) Exactly one real root in (-1, 0] (b) Exactly one real root in [2, 5]
(c) Distinct roots greater than -1 (d) Distinct roots less than 5
21. Let x2 - px + q = 0 where p e R, q e R, pq * Ohave the roots a, 0such that a + 20 = 0, then:
(a) 2p2 + q = 0 (b) 2q2 + p = 0 (c) q 0
22. If a, b, c are rational numbers (a > b > c > 0) and quadratic equation 
(a + b - 2c) x2 + (b + c - 2a)x + (c + a - 2b) = Ohas a root in the interval (-1, 0) then which of 
the following statement(s) is/are correct ?
(a) a + c 0 and the
equation ax2 + bx + c = Ohas no real roots. Then the possible value(s) of + + C is/are:
(a) 2 (b) -1 (c) 3 (d) V2
25. Let /(x) = x2 - 4x + c VxgR, where c is a real constant, then which of the following is/are 
true ?
(a) f (0) > /(I) > /(2) (b) f (2) > /C3) > /(4)
(c) f (1) /(3)
26. If 0 l (c) |0|6,then :
(a) xe(-oo, 1) (b)xe(-oo, 0) (c) xc(4,»)
Advanced Problems in Mathematics for J EE166
2
AY AY
>x
>x■» x(a) (b) (c) (d)
(a) a2+p2 =
(c) 0 (d) 2
(b) -r + ^- 
a2 p2
34. If the equation ax2 + bx + c = 0; a, b,c eR and a * 0 has no real roots then which of the 
following is/are always correct ?
(a).(a + b + c)(a-b + c) > 0 (b) (a + b + c)(a - 2b + 4c) > 0
(c) (a-b + c)(4a-2b + c) > 0 (d) a(b2-4ac) > 0
35. If a and p are the roots of the equation ax2 + bx + c = 0; a, b,c e R; a * 0 then which is (are) 
correct:
28. If both roots of the quadratic equation ax2 + x + b- a = 0are non real and b > -1, then which 
of the following is/are correct?
(a) a > 0 (b) a 2 + 4b (d) 3a l (c)|p| 6, then
(a) x e (-co, 1) (b) x e (-co, 0) (c) x e (4, oo) (d) (2, co)
42. The value of k for which both roots of the equation 4x2 - 2x + k = 0 are completely in (-1,1), 
may be equal to :
(a) -1 (b) 0 (c) 2
43. Let a, P, y, 8 are roots of x4 -12x3 + Xx2 - 54x + 14 = 0
Ifa + p = y + 8, then
(a) X = 45
(c) If a2 + p2 ^ = - 
y8 7
Advanced Problems in Mathematics for JEE168
Exercise-3 : Comprehension Type Problems
(d) nothing can be said(c) 0
(d) None of these(c) (10, 13)
(d) None of these(c) (10, 13)
(d) 3/2(c) 1
(d) (-oo, co)
Paragraph for Question Nos. 1 to 2
Let J(x) = ax2 + bx + c, a * 0, such that /(-I -x) = /(-l + x) Vx g R. Also given that 
/(x) = 0 has no real roots and 4a + b > 0.
Paragraph for Question Nos. 8 to 9
Consider the equation log 2 x-4log2 x-m2 -2m -13 = 0, m g R. Let the real roots of the 
equation be Xj, x2 such that xxbx + c Vx g R, (b, c g R) attains its least value at x = -1 and the graph of f (x) 
cutsy-axis aty =2.
Paragraph for Question Nos. 3 to 4
If a, p are the roots of equation (k +1) x2 - (20k + 14) x + 91k + 40 = 0; (a 0, then 
answer the following questions.
1. Let a = 4a - 2b + c, p = 9a + 3b + c, y = 9a - 3b + c, then which of the following is correct ?
(a)p “Z
4
9. The sum of maximum value of Xj and minimum value of x2 is : 
, 513 „ . 513 , . 1025M V — w —
15. The value of I —
I 2
■2, x3,x4 such that
mi
(d) - (sin 9 + cos 0 + sin 0 cos 0) 
2
16. Number of values of 0 in [0, 2tt] for which at least two roots are equal, is :
(a) 2 Cb) 3 (c) 4 (d) 5
x" -x2 + (x-x2)sinO + (x-x2)cos0 + (x-l)
Paragraph for Question Nos. 15 to 16
Consider the cubic equation in x, ~3 -2 ■ -2' ' '
sin 0 cos 0 = 0 whose roots are a, p, y.
10. x1x2 + x1x3+x2x4+x3x4 = 
(a) 0
11. x2 + x3 =
2+5-75
8
170 Advanced Problems in Mathematics for J EE
(d) 15
(d) -5
Paragraph for Question Nos. 19 to 21
19. If the above cubic has three real and distinct solutions for x then exhaustive set of value of Xbe
(a) (b) (d) (-4,4)
(c)
X
(b)(c) (c) (d)(c) (a) 6. Cd) (d)(a) 3. 4. 5. (a) 10.2. 8. 9.
(b)(c) (b) (d)(d) (a) (b)(a) 14. 15. 16. (c) (c)11. 13. 17. 18. 19. 20.12.
(d)(a) (c)21. 22. 23.
-1 1
4 ’ 4
Paragraph for Question Nos. 22 to 23
Consider a quadratic expression f (x) = tx2 - (2t -1) x + (5t -1)
, -1
’’ 4
1 1 
.5’4.
22. If /(x) can take both positive and negative values then t must lie in the interval 
fl 1 >u — ,°o (c) 
 0 then t lies in the interval
(a)
Paragraph for Question Nos. 17 to 18
Let P (x) be a quadratic polynomial with real coefficients such that for all real x the relation 
2(1 + P(x)) = P(x -1) + P (x + 1) holds.
If P(0) = 8 and P (2) = 32 then :
(a)3
.4
-1 1 
.4’4.
h Answers
17. The sum of all the coefficient of P (x) is :
(a) 20 (b) 19 (c) 17
18. If the range of P (x) is [m, oo), then the value of m is :
(a) -12 (b) 15 (c) -17
Let t be a real number satisfying 2t3 - 9t2 + 30 - X = 0 where t = x + — and X g R. 
x
171Quadratic Equations
(P)(A) The least positive integer x, for which
(Q) 1(B)
6(R)(C)
(S) 16(D)
10
R
(C)
(S) R -{0}(D)
{1}(T)
3.
No real roots(P)(A)
(Q) -3,3(B)
-8,1(R)(C)
(A)
(B)
The inequality is valid for all values of x and k is
There exists a value of x such that the inequality is valid 
for any value of k is
There exists a value of k such that the inequality is valid 
for all values of x is
There exists values of x and k for which inequality is 
valid is
The real root(s) of the equation x4 - 8x2 -9 = 0 
are
(P)
(Q)
4
3
Exercise-4 : Matching Type Problems
(R) {0}
:^3-2 = 0
________________________ _______________ ________ (T)____________
2. Given the inequality ax + k2 >0. The complete set of values of * a’ so that
The real root(s) of the equation x2 3
are
The real root(s) of the equation V3x + 1 +1 = Vx 
are
2x-l 
~1-------5------1S2x3 + 3x2 + x
positive, is equal to
If the quadratic equation
3x2 + 2(a2 + 1) x + (a2 - 3a + 2) = 0
possess roots of opposite sign then a can be equal to
The roots of the equation 
a/x+ 3-4>/x-1 + a/x + 8 - 6Vx -1 = 1 
can be equal to
If the roots of the equation 
x4 - 8x3 + bx2 - ex +16 = 0 are all real and positive 
then 2(c - b~) is equal to
172 Advanced Problems in Mathematics for J EE
(D) of the equation (S) 0,2
4.
(A) (P) 1
(B) (Q) 2
(C) (R) 3
(D) (S) 15
Answers I
U !.
A-> Q ; B->P;
2. B—> S ;
A —> Q; B —> R; C —> P; D —> S3.
lA—> Q; B—> Pj C—> S; D—> R4.
A-»Q;
. -
14k 15ttcosec---- + cosec-----
8 8
sin , then P + Q is equal to
The real root(s) 
9V -10(3v) + 9 = Oare
If a, b are the roots of equation x2 + ax + b = 0 
(a, b g Rf, then the number of ordered pairs (a, b) 
is equal to
T, _ k 2k 3k 13kIf P = cosec— + cosec— + cosec— + cosec-- +
8 8 8 8
, ■ K . 5kand Q = 8sin — sin —
18 18
C -> R ; D -> P
C-»R; D —> S
Let dj, a2, a3 be positive terms of a G.P and 
a4,l, 2,a10 are the consecutive terms of another 
12 ™
G.R If PI a, = 4 n where m and n are coprime, 
i=2
then (m + n) equals
For x, y eR, if x2 - 2xy + 2y 2 - 6y + 9 = 0, then 
the value of 5x - 4y is equal to
Quadratic Equations 173
Exercise-5 : Subjective Type Problems
ifare
5k
of satisfying‘x’
3 2
3
least 
(ex - 2) I sinf *
n 
x
x-8 
n-10
a, b, c are integers.
= /^cos^j, then find the value of/(2):
1. Let
. K
-x2 + 2x + 3 has the remainder 4x + 3 when divided by
has no solution.
11. The number of negative integral values of m for which the expression x2 + 2(m -l)x + m + 5 
is positive V x > 1 is :
12. If the expression ax4 + bx 
x2 + x-2, then a + 4b =
13. Find the smallest value of k for which both the roots of equation x2 -8kx +16 (k2 -k + 1) = 0 
are real, distinct and have values atleast 4.
14. If x2 - 3x + 2 is a factor of x4 - px2 + q = 0, then p + q =
15. The sum of all real values of k for which the expression x2 + 2xy + ky2 + 2x + k = 0 can be 
resolved into linear factors is :
16. Thecurvey = (a + l)x2 + 2 meets the curve y =ax+3, a* -linexacdyone point, then a2 =
K
4
7. The integral values of x for which x2 + 17x + 71 is perfect square of a rational number are a 
and b, then | a - b | =
8. Let P(x) = x6 -x5 -x3 -x2 -x and a, p, y, 6 are 
x4 -x3 -x2-l = 0, thenP(a) + P(P) + P(y) + P(8) =
9. The number of real values of ‘ a’ for which the largest value of the function / (x) = x2 + ax + 2 
in the interval [-2, 4] is 6 will be :
10. The number of all values of n, (where n is a whole number) for which the equation
the roots of the equation
/ (x) = ax2 + bx + c where
n, 3k 3k 5k . 5k . ksin - • sin — + sin — sin — + sin — sin -
7 7 7 7 7 7
2. Let a, b, c, d be distinct integers such that the equation (x - a) (x - b) (x - c) (x - d) - 9 = 0 
has an integer root ‘ r’, then the value of a + b + c + d - 4r is equal to :
3. Consider the equation (x2 + x + I)2 -(m -3)(x2 + x +1) + m = 0, where m is a real 
parameter. The number of positive integral values of m for which equation has two distinct 
real roots, is :
4. The number of positive integral values of m, mthen the largest value of m is pjq 
where p, q are coprime natural numbers, then p + q =
6. The least positive integral value
(x-loge 2) (sinx-cosx) 1. If f M takes the value p, a prime 
for two distinct integer values of x, then the number of integer values of x for which f (x) 
takes the value 2p is :
28. If x and y are real numbers connected by the equation 9x2 + 2xy + y 2 - 92x - 20y + 244 = 0, 
then the sum of maximum value of x and the minimum value of y is :
29. Consider two numbers a, b, sum of which is 3 and the sum of their cubes is 7. Then sum of all 
possible distinct values of a is :
30. Ify 2 (y2 -6) + x2 -8x + 24 = Oand the minimum value of x2 + y4 ism and maximum value 
is M; then find the value of M - 2m.
31. Consider the equation x3 -ax2 + bx-c = 0, where a,b,c are rational number, a * l.Itisgiven 
that x1}x2 and XjX2 are the real roots of the equation. If (b + c) = 2(a +1), then
f a + - l^b + c J 
a satisfy the equation x3 + 3x2 + 4x+5 = 0 and 0 satisfy the equation
- 3x2 + 4x - 5 = 0, a, 0 e R, then a + 0 =
Quadratic Equations 175
33. Let x, y and z are positive reals and x2 + xy + y 2 = 2;y2 + yz + z
If the value of xy + yz + zxcan be expressed as and q are relatively prime positive
35. The real value of x satisfying v 20:
the value of
satisfied foris V x g R,
3
, 8,9, then
3 + ox2 + bx + c = 0, where A, B,C are the 
angles of a triangle then find the value of a2 - 2b - 2c.
45. Find the number of positive integral values of k for which kx 2 + (k -3)x + 1 max(-x2 + (a + b) x - (1 + a + b)) V x e R, is :
^20x +13 = 13 can be expressed as - where a and b are 
b
relatively prime positive integers. Find the value of b ?
36. If the range of the values of a for which the roots of the equation x2 -2x-a2 +1 = 0 lie 
between the roots of the equation x2 -2(a + l)x + a(a -1) = Ois (p,q), then find the value of
q— •I pj
37. Find the number of positive integers satisfying the inequality x2 - lOx + 16 1 + cosx
a g (-oo, ki ] kJ [k2, oo), then | ^ | +1 k2| =
40. a and p are roots of the equation 2x2-35x + 2 = 0. Find the value of 7(2a -35)3(2p-35)
41. The sum of all integral values of ‘ a’ for which the equation 2x2 - (1 + 2a) x +1 + a = 0 has a 
integral root.
42. Let /(x) be a polynomial of degree 8 such that F(r) = -, r = 1,2,3 , 8,9, then =
43. Let a, Pare two real roots of equation x2 + px + q = O,p,q g R,q * 0. If the quadratic equation
g (x) = 0 has two roots a + —, p + - such that sum of its roots is equal to product of roots, then a p
then number of integral values q can attain is :
44. If cos A, cosB and cosC are the roots of cubic x
2 = 1 and z2 + zx + x2 =3.
176 Advanced Problems in Mathematics for JEE
—...-----
Answers
9 2. 0 3. 6.1 7 5. 5 3 37. 8. 6 9. 0 10.
0 12. 9 2 911. 13. 14. 15. 2 16. 4 56 717. 0 99 20.18. 19.
22. 3 621. 2 23. 1 24. 26.25. 4 8 27. 29. 3 30. 40 28. 7
31. 1 32. 0 33. 34. 36.5 9 35. 5 85 40.37. 5 38. 2 39. 3
5 341. 1 42. 43. 44. 1 45. 0
□□□
6
: * -^7 -
Sequence and Series
Exercise-1 : Single Choice Problems
(a) 1 (c)
then the
(c)
(c) 3
3
4
_8_ 
27
0 ) (1 -c) 
is :
(H |
2. If xyz = (1 -x)(l-y)(l-z) where 
x(l-z) + y(l-x) + z(l-y)is :
(a) I (b) 1
2 4
(d)l
4. Let a, b, c, d, e are nbri-zero and distinct positive real numbers. If a, b, c are in A.R ; b, c, d are in
G. R and c, d, e are in H.R, then a, c, e are in :
(a) A.R (b) G.R
(c) H.R (d) Nothing can be said
5. If (m +1) *, (n +1)th, and (r +1)111 terms of a non-constant A.R are in G.R and m, n, r are in
H. R, then the ratio of first term of the A.P to its common difference is :
(a) (b) -n (c) -2n (d) +n
6. Iftheequatiohx4 -4x3 + ax2 + bx + 1 = Ohas four positive roots, then the value of (a + b) is :
(a) -4 (b) 2
(c) 6 (d) can not be determined
(d)l
3. If sec(a-2p), seca, secfa + 20) are in arithmetical progression then cos2a = Xcos2p 
(P * mr, n e I) the value of X is : 
(a) 1 (b) 2
(d)i
minimum value of
178 Advanced Problems in Mathematics forJEE
7. If S and S
(d) 0(c) 1
upto n terms then the sum of the infinite terms is :
(c) e(a) 1
12 12
+ k and Qn = where k, n e N
(b) 1 (c) 3 Cd) 0
(a) !>8 (d) None of these(c) 1
15. Let x1} x2, x3,
x-l +x2 + x3 +
xi
(c) 1. 
n
(d) 18
(including 1 and n). If
10. Let Sk = 1 + 2 + 3 +
lim Qn = 
n->oo
(b) 2 
n
>1, S 2 ana o 3 
respectively, then
(d)±
75
f y 75 
00 T
(a) 4
1 • 28. If Sn = — + 
n 3!
(b) 0 and x + y + z = 1 then----------—---------- is necessarily.
(l-x)(l-y)(l-z)
179Sequence and Series
16. If q, a2, a3, 
Cc) H.PCb) G.P
n
Cd) 1(c)
Cd) 0Cc) -25
(d) 80(c) 160
Cd) 4
______ __ _^3_ 
AD’ A2)’ A3)’
in:
(a) A.R Cd) None of these
' n '
+ pn, then lim V*Sr =.
n-x» ■“
17. If a, p be roots of the equation 375x2 - 25x -2 = 0 and sn = a
(b)l
,an are in H.P and f(k) = ^ar ~ak then
(a)3
,-^-are 
An)
(a) ± Cb) 1 (c) 1
18. If a,, i = 1, 2, 3, 4 be four real members of the same sign, then the minimum value of 
y^-,i,je{l,2,3,4},i*As:
aj
(a) 6 Cb) 8 (c) 12 Cd) 24
19. Given that x, y, z are positive reals such that xyz = 32. The minimum value of 
x2 +{0}
64. If domain of y = f(x) is x e [-3,2], then domain of y = /(| [x] |) : 
(where [•] denotes greatest integer function)
(a) [-3,2] (b) [-2,3) (c) [-3,3]
65. Range of the function /(x) = cot-1{-x} + sin-1{x} + cos-1{x}, where {•} denotes fractional 
part function :
3k 
— > n
4 J
( 3166. Let/:R-j|^
J2008 (x) + /2009 (x) ~
. . 2x2 + 5
(a) "7-----7T
2x-3
■37c ' 
—L4 ,
3x + S->R} /(x)=■Let fi M=fM> fn M=f(fnA (x)) for n 2 2> n e N’±en
Function 11
69. If fW = sin log ■; x e R, then range of /(x) is given by :
71. If the range of the function /(x) = tan
(b)(a) (c)
73. Let f;R -> 7? is defined by /(x) =
4-e1
Cb) 0,(a) [0,1]
(d)(c)
(c) 12 (d) 16
76. Consider the function f : R -{1} -> R -{2} given by /(x) =------ . Then :
(b) b2=4c (c) b2 =12c (d)b2=8c
;-1 x, then the set of values of k for which of | /(x) | = k has exactly two
(a) f is one-one but not onto
(c) f is neither one-one nor onto
.... 2x .
' ' ’ ' x —1
(b) f is onto but not one-one
(d) f is both one-one and onto
n 3it
2’~2
a/4-x
1 —x
gW = 
e
(c) [l,oo) (d) [-oo,4]
9 71 |(3x + bx + c) is 0, — ; (domain is R), then : 
2j
(d) "’T
(* + ’ x ~ 1. If /(x) is invertible, then the
Inx + (b2-3b +10) ; x>l
(a) b2 =3c (
72. Let /(x) =sin“1 x-cos 
distinct solutions is:
*1]
2 1
75. Consider all functions f : {1, 2, 3, 4} -> {1, 2, 3, 4} which are one-one, onto and satisfy the 
following property:
if f(k) is odd then f(k +1) is even, k = 1, 2, 3.
The number of such functions is :
(a) 4 (b) 8
e2 + l
set of all values of ‘ b’ is :
(a) {1,2} (b) (c) {2,5} (d) None of these
74. Let /(x) is continuous function with range [-1, 1] and /(x) is defined V x eR. If 
efM _ el f(x)|
——.then range of g(x) is:
(a) [-1, 1] (b) [0, 1] (c) (-1, 1) (d) None of these
70. Set of values of * a’ for which the function f:R -» R, given by/(x) = x3 + (a + 2)x2 + 3ax + 10 
is one-one is given by :
(a) (-oo,l]u[4,oo) (b) [1,4]
e2 + l 
e2-l
e2 +1
Advanced Problems in Mathematics for JEE12
(d) [-2, 2](a) [-2,4]
78. Let f : R R and /(x) =
(d) [-1,1](a) [0,1] (b) [-1,0] (c)
(d) both (b) and (c)(b) 8
81. Let f:R —> R be a function defined by /(x) = , then
(c) 12 (d) 16
(b) Many-one, onto
(d) Many one, into
0 [0,5] be an invertible function defined by /(x) = ax2 + bx + c, where a, b, c e R, 
abc * 0, then one of the root of the equation cx2 + bx + a = 0is:
(a) a ' . ■ (b) b (c) c (d) a + b + c
84. Let f (x) = x2 + lx + y cosx, 1 being an integer and y is a real number. The number of ordered 
pairs (l,y) for which the equation /(x) = 0 and /(/(x)) = Ohave the same (non empty) set of 
real roots is :
(a) 2 (b) 3 (c) 4 (d) 6
85. Consider all function /:{1,2,3,4} ->{1,2,3,4} which are one-one, onto and satisfy the 
following property:
if /(k) is odd then /(k +1) is even, k = 1,2,3.
The number of such function is :
(a) 4 (b) 8
13Function
y
(a) 1
■>x0
y
y
(d) o 71
>x0 n
(d) [-2,2](a) [-2,4]
, then /(x) is :89. Let f:R -> R and /(x) =
(d) Many one, into(c) One-one, onto
(d) [-1,1](b) [-1,0] (c)(a) [0,1]
of function f defined asx
(d) 8(c) 7
(a) One-one, into
90. Let /(x) be defined as
■_r r 
2’2.
in the domain91. The number of integral values of 
/(x) = 71n| In|x|| + ^71 x|-1x|2-10 is :
(a) 5 (b) 6
1
(b)
86. Which of the following is closest to the graph of y = tan (sin x), x > 0 ? 
y
87. Consider the function /: R - {1} -> R - {2} given by /(x) = ^X-. Then
x-1
(a) / is one-one but not onto (b) f is onto but not one-one
(c) f is neither one-one nor onto (d) f is both one-one and onto
88. If range of function /(x) whose domain is set of all real numbers is [-2,4], then range of 
function g(x) = - /(2x +1) is equal to :
|x| 0 Bis an injective mapping satisfying/(i) * i.then 
number of such mappings are :
(a) 182 (b) 181 (c) 183 (d) none of these
98. Let /(x) = x2 -2x-3;x > landg(x) = 1 + Vx + 4;x > -4then the number of real solutions of
equation /(x) = g(x) is/are 
(a) 0 (b) 1
onto is :
(a) (-00,00) (b) (-oo,0) (c) (0,oo) (d) Emptyset
95. If A = {1,2,3,4} and / : A -> A, then total number of invertible function '/’ such that /(2) * 2 
/(4) * 4, /(I) = 1 is equal to :
(a) 1 (b) 2 (c) 3
96. The domain of definition of /(x) = log(X2_X+I)(2x2 -7x + 9) is :
(a) R (b) R-{0} (c) R-{O,1}
92. The complete set of values of x in the domain of function /(x) = ^logx+2{X} 
(where [•] denote greatest integer function and {•} denote fraction part function) is :
 2
Cb) f (x) = m has exactly two real solutions V m e (2, oo) {0}
(c) f (x) = m has no solutions V m4xy + 4y 2 + 2z 2 is equal to :
(a) 64 Cb) 256 (c) 96 (d) 216
20. In an A.P, five times the fifth term is equal to eight times the eighth term. Then the sum of the 
first twenty five terms is equal to :
25(a) 25 Cb) 4?
2
21. Let a, pbe two distinct values ofxlyingin[0, it] for which Vs sin x, lOsinx, 10(4sin2 x + l)are
3 consecutive terms of a G.P Then minimum value of | a - p| =
f x It ZLX It Z x 21t zjx 31t
(a) — (b) - (c) — (d) —
10 5 5 5
22. In an infinite G.P, the sum of first three terms is 70. If the extreme terms are multiplied by 4 and 
the middle term is multiplied by 5, the resulting terms form an A.P then the sum to infinite 
terms of G.P is :
(a) 120 (b) 40
23. The value of the sum X. nlV *s eclua^t0 :
k=in=i 2n
(a) 5 Cb) 4 Cc) 3 Cd) 2
24. Let p, q, r are positive real numbers, such that 27pqr > (p + q + r)3 and 3p + 4q + 5r = 12, then
p3+q4+r5 =
(a) 3 Cb) 6 Cc) 2
25. Find the sum of the infinite series — + — + — + — + — + 
9 18 30 45 63
180 Advanced Problems in Mathematics forJEE
(a) (-8,0) (b)
(d) (-8, l]-{0}(c)
3»
(c) 9 (d) 10
(a) (b) (c) (d)
upto n terms is equal:
(b) (d)
(a) 2k-3 (b) k + 1 (c) 2k+ 7
n(n + 3)
2
(n-l)2n 
n +1
-1- 2" + l
n + lj
29
= k, then the value of ^(1 • 5)n, is :
n=2
r . n2n q(c) ---- --1
30. Let Sk
32. If(1 - 5)30
-i 1) I1 1-1, -- o 1
r n2n(a) ---- -
n +1
S S26. If Sr denote the sum of first ‘r’ terms of a non constant A.R and — = — = c, where a, b, c are 
a2 b2
9
(d) 2k-- 
2
33. n arithmetic means are inserted between 7 and 49 and their sum is found to be 364, then n is:
(a) 11 (b) 12 (c) 13 (d) 14
34. The third term of a G.E is 2. Then the product of the first five terms, is :
(a) 23 (b) 24 (c) 25 (d) none of these
35. The sum of first n terms of an A.R is 5n2 + 4n, its common difference is :
(a) 9 (b) 10 (c) 3 (d) -4
distinct then Sc =
(a) c2 (b) c3 (c) c4 (d) abc
27. In an infinite G.R second term is x and its sum is 4, then complete set of values of ‘ x’ is :
L 8 8J
n(n + 2) 
2
31. The sum of the series + —— 21 +-!^-22 + -i^-23 +
CO
^(fc+1)'
n(n +1)
2
n2‘
n + 1
10
3-4
17
4-5
2
1-2
1
8
28. The number of terms of an A.R is odd. The sum of the odd terms (1st, 3rd etc.,) is 248 and the 
sum of the even terms is 217. The last term exceeds the first by 56, then :
(a) the number of terms is 17 (b) the first term is 3
(c) the number of terms is 13 (d) the first term is 1
29. Let A15 A2, A3, , An be squares such that for each n > 1 the length of a side of An equals 
the length of a diagonal of An+1. If the side of Ax be 20 units then the smallest value of ‘n’ for 
which area of An is less than 1.
(a) 7 (b) 8
n
, then ^kSk equal:
k=l
n(n -1)
2
5
2-3
Sequence and Series 181
36. If x + y =a and x2 + y 2 = b, then the value of (x3 + y 3), is :
(a) ab (d)
are
upto infinite terms =
(c)
39. The minimum value of
is :
20 taking two at a time is :
(c) a - b(b)
(a) (d)(b) (c)
2
’ 2n +1
(d) 34(b) 4
24
(c) 24(a) -i-
34
20
40. If £r3
i
1 +---------- +
S2S3S4
37. If Sj, S2> 
1,3,5,....
(b) a (c) a + b2
(d) a2-b2
20= a, y r2 = b then sum of products of 1, 2, 3, 4. 
1
r k a -b . a2 -b2
(a) —- (b) ——
2 2
41. The sum of the first 2n terms of an A.R is x and the sum of the next n terms is y, its common 
difference is :
 (b) (e) (d) 
3n2 3n2 3n 3n
42. The number of non-negative integers ‘ n’ satisfying n2 = p + q and n3 = p2 + q2 where p and q 
are integers.
(a) 2 (b) 3 (c) 4 (d) Infinite
43. Concentric circles of radii 1, 2, 3 100 cms are drawn. The interior of the smallest circle is 
coloured red and the angular regions are coloured alternately green and red, so that no two 
adjacent regions are of the same colour. The total area of the green regions in sq. cm is equals 
to:
(a) 1000n (b) 5050n (c) 4950n (d) 5151n
44. If log 2 4, log j?, 8 and log3 9k-1 are consecutive terms of a geometric sequence, then the 
number of integers that satisfy the system of inequalities x2 - x > 6 and | x| q
ABCD
2 + b
182 Advanced Problems in Mathematics for JEE
(a) Cb)
(c) Cd)
(a) 2008 O’) 3012 (d) 8032
247. The sum of the infinite series, 1 is :
(c) Cd)
(c)
2
(C)
466
(d) 22
 upto infinite terms is equal
(c)
46. If X?r =
1 x-------
+ TrTr+1Tr+2Tr+3 =
n(n + l)(2n + 1) 
6
n(n + l)(2n + 1)
12
Cd) 21
8
 (x -10) is :
(d) 2740
45. Let Tr be the rth term of an A.P whose first term is —| and common difference is 1, then
17 5 7
(a,T w 57 (c)F9
51. Number of terms common to the two sequences 17, 21, 25, 
is :
(a) 19
, . 1 25(a) - Cb) —
2 24
n r
48. The absolute term in P(x) = V x — , 
“A r A
to :
(a) - (b) - (c) -
53. The coefficient of x8 in the polynomial (x -1) (x - 2) (x - 3)
(a) 2640 03) 1320 (c) 1370
O’) 20
2 12 52. The sum of the series 1 + - + — + — +
3 32 33
5n
~~4
5n 1
--------4" —
4 2
5n 1--------4- —
4 4
-^ + 1
8
(c) 21
12 12
34 35 36 37
1-1 1-1(a) i Cb) —- (c) A (d) -A
2 2 • 4 4
49. Let a, b, c are positive real numbers such that p=a2b + ab2-a2c-ac2;q=b2c + bc2-a2b-ab2 
and r = ac +ac-cb -be and the quadratic equation px + qx + r = Ohas equal roots; then 
a, b, c are in :
(a) A.P O’) G.P (c) H.P (d) None of these
50. If Tk denotes the kth term of an H.P from the begining and — = 9, then — equals :
Te t4
Cd)y
 ., 417 and 16, 21, 26, 
n(n + l)(2n + 1)
6 
n(n + l)(2n + 1)
~6
n(n + l)(n4-2) >then Hm f 2008
3 n-»°° Tr
(c) 4016
 22 32 42 52 62 
5 + 52 53 + 54 5s +
25
54
x---— | as n approaches to infinity is :
r+ 2)
125
252
183Sequence and Series
(d) non-existent(c)
(a) V3 (c)(b) 1
is
(0 1
(d) 2(c) 4(a) 1
(d) 4
+... upto 60 terms :
3 3
(c) 4(a) 2
1
/2
is:
(a) 2 (b) 3
58. The sum of the series 22 + 2 (4)2 + 3(6)2 +
(a) 11300 (b) 12100
(c) 6 (d) 0
. upto 10 terms is equal to :
(c) 12300 (d) 11200
00
57. Ifa and pare the roots ofthe quadratic equation 4x2 + 2x-l = 0 then the value of (ar + pr)
Cd) *
V3
2 
— is :
2
63. Find the value of —- + 
I3
(d)
2
satisfying the equation
The sum of G.P is :
(d)l
equal to :
2 1(a) ± (b) A
3 3
60. The first term of an infinite G.P is the value of x 
log 4 (4X -15) + x - 2 = 0 and the common ratio is cos
® 5
(a) 4 (b) 23/4 (c) V2
62. If xy = 1 ; then minimum value of x2 + y 2 is :
(a) 1 (b) 2 (c) V2
6 12 20
l3 + 23 + l3 + 23 + 33 + I3 + 23 + 33 + 4'3 + n3
12
+ I3 +23 +3
+ b4 + c 
abc
(d) 2V2
4
61. Let a, b, c be positive numbers, then the minimum value of —
. .. (I3-l2) + (23 -22) + ...+(n3-n2) , .54. Let a = hm --------- -—----------------------------- , then a is equal to :
n->oo ft!
(a) 1 (b) i (c) 1
O *T
55. If 16x4 -32x3 + ax2 + bx + 1 = 0,a,beR has positive real roots only, then a -b is equal to :
(a) -32 (b) 32 (c) 49 (d) -49
ABC B56. If ABC is a triangle and tan — , tan —,tan — are in H.R, then the minimum value of cot — =
f 4 1
59. Ifa and b are positive real numbers such that a + b = 6, then the minimum value of - + —
Va b
184 Advanced Problems in Mathematics for JEE
(b) -Ft;(a) (c)
(d) 159(b) 153
66. Sum of first 10 terms of the series, S = is :
(b) (c) (d)(a)
4
(C)
68. Let rth term tr of a series is given by tr =
(b) 1 (c) 2
to infinite terms, is :
(c)
71. Ifx1,x2, x3,
(^-x2,) (Xj2 -xjj(a) (b)
(d)
55
111
(b) x2-18x-16 = 0
(d) x2-18x4-16 = 0
255
1024
88
1024
85
1024
2n
are in A.P, then ^(-l)r+1 x2 is equal to : 
r=l
(b) -±L
109
35
W,16
(d)
109
(a4
69. The sum of the series 1 + - + — + — + 
5 52 53
, ' 31 41 . . 45(a) — (b) — (c) —
12 16 16
70. The third term of a G.P is 2. Then the product of the first five terms, is :
(a) 23 (b) 24 (c) 25 (d) none of these10
67. ~—£l-3r2 + r
, ' 50(a)------
109
n
— .Then lim Vtr is equal to : 
n->oo —'
r=l
(d)l
7
22 ■ 52
64. Evaluate : V---------------------------------------
~(n +l)(n + 2)(n + 3)....(n + k)
11 11---------------- (b) — (c) ----- (d) - 
(k —l)(k —1)! k-k! (k-l)k!---------------- k!
65. Consider two positive numbers a and b. If arithmetic mean of a and b exceeds their geometric 
mean by 3/2 and geometric mean of a and b exceeds their harmonic mean by 6/5 then the value 
of a2 + b2 will be :
(a) 150
—-—(b) —-n-
(2n-l) (2n-l)
(c) -^(xf-x2,) (d) —l-fxf-x^)
n -1 2rt + 1
72. Let two numbers have arithmatic mean 9 and geometric mean 4. Then these numbers are roots
of the equation :
(a) x2 +18x + 16 = 0
(c) x24-18x-16 = 0
19 
+ 82112 +
(c) 156
13 
+ 52-82
264
1024
185Sequence and Series
(c)(a) 2
(c)
5
(C)
 up to infiniteT +3 31J + 2J +3J +4'
terms:
(c) 4(a) 2
78. The minimum value of the expression 2X + 22r+1 + —, x e R is :
(a) 7
is :
(c)
2
9
_1_
24
25
00
79. The value of
(d) (3.10)1'3
1
-5n3
20
3
(d)
W)2
(d) —
144
(d)l
(a) —
120
(a)|
(h) |
+ 4n
(b)
96
6
I3 7F +
co
76. The value of — 
n=3 n
2
77. Find the value of — + 
I3
(b) (7.2)1/7
(4r + 5)5~l
r (5r + 5)
2(b) i
73. If p and q are positive real numbers such that p2 + q2 = 1, then the maximum value of (p + q)is :
(b) - (c) A (d) V2
2 V2
74. A person has to count 4500 currency notes. Let an denote the number of notes he counts in the 
nth minute. If =a2 = = Ojo =150 and aio>an>ai2> are *n A-R with common 
difference -2, then the time taken by him to count all notes is :
(a) 34 minutes (b) 24 minutes (c) 125 minutes (d) 35 minutes
75. A non constant arithmatic progression has common difference d and first term is (1 - ad). If the 
sum of the first 20 terms is 20, then the value of a is equal to :
, . 2 19(a) — (b) —
19 2
12
I3 + 23 +33 +
5
2X
(c) 8
186
10.
20.
30.
40.
50.
60.
70.
2.
12.
22.
32.
42.
52.
62.
72.
Cc)
(c)
(d) 
Cd) 
Cb) 
(a) 
Cb) 
(d)
Cb)
Cb) 
(d) 
Cc)
(b)
Cb)
(c)
Cd)
Cb) 
Cc) 
(a) 
Cc)
(a)
(b) 
Cc)
(a)
5.
15.
25.
35.
45.
55.
65.
75.
(a)
(b)
(a)
Cb)
Cc)
Cb)
(d)
Cb)
7.
17.
27.
37.
47.
57.
67.
77.
(d)
(a)
(d)
Cb)
(c)
(d)
(d)
(c)
(a)
(c) 
Cb)
(d) 
(d)
Cb) 
Ca) 
(c)
(a)
Cc)
Cd)
Cd)
Cc)
Cd)
Cd)
Ca)
(c)
(d)
(d)
(a)
(b)
(0
CO
21.
31.
41.
51.
61.
71.
Cc) 
(d) 
Cb) 
(a) 
Cb) 
Cb) 
(d) 
Ca)
3.
13.
23.
33.
43.
53.
63.
73.
8.
18.
28.
38.
48.
58.
68.
78.
4.
14.
24.
34.
44.
54.
64.
74.
9.
19.
29.
39.
49.
59.
69.
79.
6.
16.
26.
36.
46.
56.
66.
76.
Advanced Problems in Mathematics for JEE
J Answers J
Cb)
Cc)
Cb)
Cd)
Ca)
(a)
Cd)
Cb)
Sequence and Series 187
Exercise-2 : One or More than One Answer is/are Correct
+ ax + b are always positive then possible value(s) of
(c) 3
(b) —0
, then :T = >
(a) = common difference
(0 Tpq =
(c) If a, Aj, A
are in A.Rb + c-a
a
(c) a5 + c5 > 2b
>A2n>
(d) T^q =
2> A3>
2n 
 b are in A.R then At- = n(a + b).
t=i
(d) If the first term of the geometric progression g1, g 2, g 3 , 
(a) 1 (b) 2
3. If a, b, c are in H.R, where a > c > 0, then :
, > l a + c(a) b> ——
2
(c) ac > b2
, 00 is unity, then the value of 
2 
the common ratio of the progression such that (4g 2 + 5g 3) is minimum equals —.
I
qCp + q)
(b) Tp+(J= — 
pq
1
2„2
r: 1 1 -o
a-b b-c
(d) bc(l - a), ac(l -b), ab(l -c) are in A.R
4. In an A.R, let Tr denote r111 term from beginning, T„ =---------- , Ta =----- ------
F nfnxnl 4 p(p + q)
I
p + q
5. Which of the following statement(s) is(are) correct?
(a) Sum of the reciprocal of all the n harmonic means inserted between a and b is equal to n 
times the harmonic mean between two given numbers a and b.
(b) Sum of the cubes of first n natural number is equal to square of the sum of the first n 
natural numbers.
6. If a, b, c are in 3 distinct numbers in H.R, a, b, c > 0, then : 
b + c-a c + a-b a + b-c . . b + c c + a a + b . . (a) -----------,----------- ,----------- are in A.R (b) ------- ,-------,------- are m A.R
be a b c
5 ■ (d)
b-c c
1. If the first and (2n -l)th terms of an A.R, G.R and H.R with positive terms are equal and their 
n111 terms are a, b and c respectively, then which of the following options must be correct:
(a) a + c = 2b (b) a > b > c
(c) ------ = b (d) ac = bz
a + c
2. Let a, b, c are distinct real numbers such that expression ax2 + bx + c, bx2 + ex + a and
a2 + b2 + c2ex2 + ax + b are always positive then possible value(s) of-----------------may be :
ab + be + ca
(d) 4
Advanced Problems in Mathematics for JEE188
If the sum of their
(c)
from an A.R
(b) A b > c
Sequence and Series 189
01 01
15. Given then which of the following is true
(c) m n + 1
r > 0, then which of the following is/are correct.
term
then+
(a) S5 = 5
(b) A b > c
1000.......
nzeroes 
1000. ~01
(n+1)zeroes
(a) m + 1 0. If— = - where p and
q are the product of the numbers in those sets A and B respectively.
(d) 86
(d) 74
(c) (21) (21) (21) (d) (23) (19) (15)
(b)(a) (c) (d) None of these
3. Sum of the product of the numbers in set A taken two at a time is :
(a) 51 , (b) 71 (c) 74
4. Sum of the product of the numbers in set B taken two at a time is :
(a) 52 (b) 54 (c) 64
1. The valueof (a2 +p2 + aP) is equal to :
(a) 1 (b) 2
2. The value of 5(A2 + B2) is equal to :
(a) 2 (b)- 4
Paragraph for Question Nos. 5 to 7
Let x, y, z are positive reals and x + y + z =60 and x > 3.
Paragraph for Question Nos. 3 to 4
There are two sets A and B each of which consists of three numbers in A.P whose sum is 15. D 
and d are their respective common differences such that D-d = 1, D > 0. If — = — where p and 
q 8
Paragraph for Question Nos. 8 to 10
, n are removed. ArithmeticTwo consecutive numbers from n natural numbers 1, 2, 3, 
mean of the remaining numbers is .
4
(d) 125 xlO3
(355)3 
32 • 63
5. Maximum value of (x - 3) (y + 1) (z + 5) is :
(a) (17) (21) (25) (b) (20) (21) (23)
6. Maximum value of (x - 3) (2y +1) (3z + 5) is :
(355)3 (355)3
33 • 62 33 • 63
7. Maximum value of xyz is
(a) 8xl03 (b) 27 x 103
Paragraph for Question Nos. 1 to 2
The first four terms of a sequence are given by 1\ = 0, T2 = 1, T3 = 1, T4 =2.
The general term is given byTn = Aan-1 + Bp"-1 where A, B, a, pare independent of n and A 
is positive.
(c) 64xl03
191Sequence and Series
(d) 49(c) 52
(d) 772
(d) 69
(d) 61024Cc) 1 + 31024
(d) 5(a) 2
(d)(c)
and p.
(d) 59(c) 57
(d) 4(c) 3(a) 2
lim /(n) and x2-2a--x + t= 0 has two positive roots a 
n->ao I 2 J
11. The value of a10 is equal to :
(a) 1 + 21024 (b) 41024
12. The value of n for which bn =
2Qqq1q2--'
—, a = 
2r+lc
2bn 
l + b2
3280 .
------- is : 
3281
(b) 3
n rLet /(n) = £—-7 
r=2 ^2
bnbn
l + 2b^
14. If value of /(7) + /(8) is — where p and q are relatively prime, then (p - q) is :
 0. Let the 
sequence {bn} is defined by formula b0 = and bn = " adcL y n > 1.
(c) 4
13. The sequence {bn} satisfies the recurrence formula :
W 
2
Advanced Problems in Mathematics for JEE192
(d) H.P(c) A.G.R
(C)
Answers 5
(c)(b) (d) (c) (a) (b)6. (c)(b) (a) 10.3. 4. 5. 7. 8. 9.2.
(b)(b) (d) 16. (a) (c)(c) (b) 14. 15. 17.12. 13.
82
1005
(a)
Q] 
a} +1 
Also aT + a2 + a3 +.
Paragraph for Question Nos. 16 to 17
Given the sequence of number aj, a2, a3, 
which satisfy ——— = °2 = ——— =. 
~ ' 1 u2 + 3 a3 + 5
+ aioos = 2010
16. Nature of the sequence is :
(a) A.R (b) G.R
17. 21st term of the sequence is equal to :
, , 86 . 83(a) - Cb) -
1005 1005
> a1005
a1005 
ai005 + 20 09
79
1005
Sequence and Series 193
Exercise-4 : Matching Type Problems
1.
Column-1 Column-ll
(A) 1
in G.P, then is equal to
(B) (Q) 4
(C) (R) 2
(D) (S) 0
2.
Column-1 Column-ll
6
2
(C) (R) 3
(D) (S) 20
(T) 40
3.
Column-ll
(A) 0
(Q) 11 1 1(B) Let S=(a2-a3)
an
I
(P)
=—— + 
al + Va2
(A)
(B)
a, > 0 Vi e {1, 2, 
S + 7 is equal to
The sequence a, b, 10, c, d are in A.P, then a+b+c+d=
Six G.M.’s are inserted between 2 and 5, if their product can be 
expressed as (10)". Then n =
If a, b, c be three positive number which form three successive 
terms of a G.P and c > 4b-3a, then the common ratio of the G.P 
can be equal to
Number of integral values of x satisfying inequality, 
-7x2 + 8x-9 > Ois
(P)
(Q)
2xyz
r
If three unequal numbers a, b, c are in A.P and b - a, c - b, a are 
a3 + b3 + c3 
3abc
Let x be the arithmetic mean and y, z be two geometric means
3 3y ° 4. 2 ° 
between any two positive numbers, then ------- is equal to
Column-1\ \
The number of real values of x such that three numbers 2X, 2X and 
3
2X form a non-constant arithmetic progression in that order, is
Let^, a2, a3,...... ,a10 are in A.P and/ix, h2, h3,....... , b10 are
in H.R such that cq = /q =1 and a10 = h10 =6, then a4h7 = 
( 7)
If log3 2, log3(2x -5) and log3 I 2 I are in A.P, then x =
——= +.... + - .........
a2+Va3 VGn-l +
where cq, a2, a3,....,an are n consecutive terms of an A.P and 
, n}. If cq = 225, an = 400, then the value of
Advanced Problems in Mathematics for JEE194
(R) 2(C)
A.R, then (S) 3(D)
(T) 4
4. Column-I contains S and Column-II gives last digit of S.
Column-llColumn-I
(A) (P) 0
(B) (Q) 1
(C) (R) 3
(D) (S) 5
(T) 8
5.
Column-ll
(A) 6(P)
(B) (Q) 3
(C) (R) 0
(D) (S) 2
In A ABC A, B, C are in A.R and sides a, b and c are 
in G.E then a2(b -c) + b2 (c-a) + c2 (a-b) =
In A ABC, (a + b + c)(b + c -a) = Xbc where X e I, 
then greatest value of X is
If a, b,c are three positive real numbers then the 
, rb+c a+c a+b.minimum value of-------Q; D—>S
2. A-» R, B—> R, C—> P, D-» R
3. A —> P, B —> R, C —> S, D —> Q
4. A-» Q; B—> P ;
5. A—> S; B —> R; C —> Pj D —> Q
6. A-»S; C-> Q ; D—>RB—> P ;
m
X
n=l
p(n)-2 
n
5m (m 4- 7)
2
3m (m 4- 7)
2
q(n)-3
2
Column-ll 
m(m 4-1) 
~ 2
m(m + 7) 
. 2
111 14-— such that P(n)/(n 4-2) =P(n)/(n) 4-q(n). Where P(n),
2 3 4 n
Q(n) are polynomials of least possible degree and P(n) has leading coefficient unity. Then 
match the following Column-I with Column-II.
m
L
n=l
yp(n) + q2(n)-ll 
n
^q2(n)-p(n)-7
i nn=l
l.A->R; B->P;
C->T; D->S
196 Advanced Problems in Mathematics forJEE
Exercise-5 : Subjective Type Problems
1. Let a, b, c, d are four distinct consecutive numbers in A.R The complete set of values of x for 
which 2(a -b) + x(b -c)2 + (c - a)3 = 2(a -d) + (b -d)2 + (c -d)3 is true is (-00, a] u[p, co), 
then | a| is equal to :
n
2. The sum of all digits of n for which y r 2r = 2+ 2n+1° is :
2r+1 r(r +1)
8r
4 is equal to : 
4r* +1
n
3. If lim V 
n->cc r=l
= —, then k = 
k
positive integers. Find (p + q).
14. A cricketer has to score 4500 runs. Let an denotes the number of runs he scores in the nth 
match. If a! -a2 =..... a10 =150 anda10, an, a12.......are in A.R with common difference (-2).
If N be the total number of matches played by him to score 4500 runs. Find the sum of the digits 
ofN.
5. Three distinct non-zero real numbers form an A.R and the squares of these numbers taken in 
same order form a G.R If possible common ratio of G.R are 3 ± Vn, n e N then n =
6. If /(llll.......1)-(222...... 2) = PPP........P then P =
\ 2n times n times n times
7. In an increasing sequence of four positive integers, the first 3 terms are in A.R, the last 3 terms 
are in G.R and the fourth term exceed the first term by 30, then the common difference of A.P 
lying in interval [1, 9] is :
1 x* 1 18. The limit of — y k(k + 2) (k + 4) as n -> 00 is equal to -, then X =
n k=i X
9. What is the last digit of 1 + 2 + 3 +......+ n if the last digit of l3 + 23 +.......... + n 3 is 1?
10. Three distinct positive numbers a, b, c are in G.R, while logc a, logb c, loga b are in A.R with 
non-zero common difference d, then 2d =
11. The numbers i, | logx y, ilogy z> ^°8Z x are *n H.R Ify = xr and z = xs, then 4(r + s) =
® ^2
12- = —; where p and q are relatively prime positive integers. Find the value of (p + q).
i^3k q
13. The sum of the terms of an infinitely decreasing Geometric Progression (GP) is equal to the 
greatest value of the function /(x) = x3 + 3x - 9 when x e [-4, 3] and the difference between 
the first and second term is /'(0). The common ratio r = — where p and q are relatively prime 
q
197Sequence and Series
neN then the remainder when /(I) + /(2) + /(3) + + /C60)16. Let/(n) =
jAnswers >
6.8 3 99 2 2 5.1. 8 3.2.
6 5 13. 512. 14. 78. 1 10. 34 9.
6 2 20.16. 19. 1 21. 815. 8 17. 3 18.5
□□□
100
15. Ifx = io£ 
n=3
, then [x] = (where [•] denotes greatest integer function) 
-4
34321. Let a and b be positive integers. The value of xyz is 55 and when a, x, y, z, b are inarithmatic
and harmonic progression respectively. Find the value of (a + b)
1
n2
4n + d4n2 -1
V 2n + 1 + -J 2n — 1
is divided by 9 is.
c .,111111117. Find the sum of senes 1 + - + - + - + — + — + — + — + 
2 3 4 6 8 9 12
reciprocals of the positive integers whose only prime factors are two’s and three’s :
18. Leta1,a2,a3j ,an be real numbers in arithmatic progression such that ax = 15 and a2 is
10 n
an integer. Given ^(ar)2 =1185. If Sn = ^ar and maximum value of n is N for which 
r=l r=l
Sn > S(n_X), then find N -10.
19. Let the roots of the equation 24x3 -14x2 + kx + 3 = 0 form a geometric sequence of real 
numbers. If absolute value of k lies between the roots of the equation x2 + a2x -112 = 0, then 
the largest integral value of a is :
C - i ,
20. How many ordered pair(s) satisfy log x + -y + - = log x + logy
\ 3 9 /
oo , where the terms are the
—. T-—
Determinants
Exercise-1: Single Choice Problems
1. if
(a) 1 (b) (c)
(b) 1 (c) 2 (d) 3
= 0 and vectors (1, a, a2)(l, b, b2) and (1, c, c2) are non-coplanar, then the
product abc equals :
(c) 1 (d) 0
3
8
(a) 2 (b) -1
4. If the system of linear equations
x + 2ay + az = 0 
x + 3by + bz = 0 
x + Acy + cz = 0 
has a non-zero solution, then a, b, c : 
(a) are in A.P
(c) are in H.R
410
cos a cosp 
cosy
1
cos a cosp
0 Ofor each i, then the determinant
logan+4
log an+io Is equal to :
log an+16
n +n
(b) log
I *=1 >
Qj + 2a2 + 3a 3 
c2 and Z)2 = b± + 2b 2 + 3b3 
ci + 2c2 + 3c3
(b) -10
I be 
l ac 
l ab
(b) Ax = 2A2
1 0 
a I 
b a
«1
10. IfDj = a2 
«3
(a) 10
1 
a 
a2
(ii) a, b, c e {1, 2, 3,...........,2001, 2002}
(iii) x +1 divides ax2 + 2bx + c
is equal to lOOOA., then find the value of X.
(a) 2002 Cb) 2001 (c) 2003 (d) 2004
6. If the system of equations 2x + ay + 6z =8, x + 2y + z = 5, 2x + ay + 3z = 4 has a unique 
solution then ‘a’ cannot be equal to : 
(a) 2 (b) 3
D.
Di
Advanced Problems in Mathematics for JEE200
= 10 is :
(d) 4(c) 2(a) 1
(d) x(b) e(a) a
15. The value of the determinant
2
3(b) (c)
n
2 then the minimum value of A is :17. Let ab = 1, A =
(d) 81(a) 3
18. The determinant
and
(d) 3
x-y — z 
2y 
2z
2x
2y 
z -x-y
x 
x2
2
(c) 27 
a + b + c + d 
2(a + b)(c + d) 
ab(c + d) + cd(a + b)
ab + cd
ab(c + d) + cd(a + b) = 0 for 
2a bed
(b) ab + cd = 0
(d) any a, b, c, d
(b) 9
2 
a + b + c + d 
ab + cd
«)£
9n
2ab 
1 -a2 + b
-2a
(a) a + b + c + d = 0
(c) ab(c + d) + cd(a + b) = 0 
m 
q
1
I m n
19. Let det A = p q r
111
if (Z-m)2 + (p-q)2 =9, (m-n)2 + (q-r)2 =16, (n-Q2 + (r -p)2 =25 , then the value of
(det. A)2 equals :
(a) 36 . (b) 100 (c) 144 (d) 169
20. The number of distinct real values of K such that the system of equations x + 2y + z -1, 
x + 3y + 4z = K, x + 5y + lOz = K2 has infinitely many solutions is :
(a) 0 (b) 4 (c) 2
-2b 
2a 
l-a2-b2
1 2
13. Sum of solutions of the equation 2 3
3 5
(c) d 
2x 
y -z -x
2z
(b) -1 
x + d x + e x + f
14. IfD=x + d + l x + e + 1 x + f + 1 then D does not depend on : 
x+a x+b x+c
(a) xyz (x + y + z)2 (b) (x + y-z)(x + y+ z)
(c) (x + y + z)3 (d) (x + y + z)2
16. A rectangle ABCD is inscribed in a circle. Let PQ be the diameter of the circle parallel to the side
AB. If Z.BPC = 30°, then the ratio of the area of rectangle to the area of circle is :
J?(a) — O: f5
k 2tc
l + a2-b2
2ab
2b
201Determinants
21. If is expressed as a polynomial in x, then the term independent of
(b) 2 (c) 12
is
(d) -2(c) 1
(c) H.R
(d) 2
26. If one of the roots of the equation = Ois x = 2, then sum of all other
(c) 2V5 (d) Vis(b) 0
(b)1. (a) (c) (b) (c) (a) 6. (c) (02. 3. 4. 8. 9.
(c) (b)11. (c) (d) (c) 16. (a) (c) (C)12. 14. 20.13. 15. 19.
(021. (b) (c) (a) (a)22. (c) 26.23. 24. 25.
five roots is : 
(a) -2
equal to :
(a) 0 (b) -1
23. If the system of linear equations
x + 2ay + az = 0
x + 3by + bz =0
x + 4cy + cz = 0
has a non-zero solution then a, b, c are in
(a) A.R (b) G.E
xis :
(a) 0
b 
c 
a
cosC 
-1 
cos A
cosB 
cos A 
-1
(d) None of these 
x 
a .
b
18. • (d)
7. (a)
17.1 (c)
6 
x2 -13
3
(x + 1)3
(•X + 2)3
(x + 3)3
7
2 
x2-13
2 a
24. If a, b and c are the roots of the equation x + 2x +1 = 0, find b 
c
(a) 8 (b) -8 (c) 0
25. The system of homogeneous equation Xx + (X + l)y + (X-l)z = 0, 
(A, + 1) x + Xy + (X + 2) z = 0, (X -1) x + (X + 2) y + Xz = 0 has non-trivial solution for 
(a) exactly three real values of X 
(c) exactly three real value of X
(b) exactly two real values of X
(d) infinitely many real value of X 
x2 -13
2
7
(d) 16 
-2
22. If A, B,C are the angles of triangle ABC, then the minimum value of cosC 
cosB
j Answers :
1 r
5.
I 
(a) 10.
(x + 1) (x + 1)2 
(x + 2) (x+2)2 
(x + 3) (x + 3)2
202 Advanced Problems in Mathematics for JEE
Exercise-2 : One or More than One Answer is/are Correct
1. Let /(a, b) = , then
2. If = 0 then 0 may be :
Cb)
then :3. Let A =
(c) 1 (d) 2
= ax3 +fix2 +yx + 5 then :5. Let DM =
(c) a + p + y + 8 = 0 (d) a 4- p 4- y = 0
= ax3 + px2 + yx + 8 then :6. Let D(x) =
(c) a + p + y + 8 = 0 (d) a + p + y = 0
(d) 4
a + d 
a + 2d 
a
2>/3 tan©
2V3 tanO 
1 + 2V3 tan 0
13
26
13
26
.. 571
' 6
a + 3d 
a
a + d
(a) a + p = 0
x2 + 4x-3
2x2 + 5x - 9 
qv2
x2 + 4x - 3 
2x2 + 5x-9 
Qv2
a 
a + d 
a + 2d
(b) (a + 2b) is a factor of /(a, b)
(d) a is a factor of /(a, b)
sin2 0 
1 + sin2 0 
sin2 0
(d)115 
o
f s 7n
(C) T
(b) A depends on d
(d) A = 0
4. The value (s) of X for which the system of equations
(1 - X)x 4- 3y - 4z = 0
x - (3 + X)y 4- 5z = 0
3x + y - Xz = 0 
possesses non-trivial solutions.
(a) -1 Cb) 0
2x4-4 
4x4-5
8xz-6x4-1 16x-6 104
(a) (2a + b) is a factor of f(a, b) 
(c) (a + b) is a factor of /(a, b)
1 + cos2 0 
cos2 0 
cos2 0
(a) A depends on a
(c) A is independent of a, d
(b) p + y = O
2x4-4
4x4-5
8xz-6x + l 16x-6 104
a a2 0
1 (2a+ b) (a + b)2
0 1 (2a + 3b)
(a) a 4- p - 0 (b)p-i-y = O
7. If the system of equations 
ax + y 4- 2z = 0 
x + 2y + z - b 
2x + y 4-az =0 
has no solution then (a 4- b) can be equals to : 
(a) -1 (b) 2 (c) 3
203Determinants
(d) 4(c) 3
| Answers |/ —
(a, b, d)(a, b) 6. (a, b, d)(a, b)Cb, c, d) (b, d) 5.3.1. 2.
(b, c, d) (b)7. 8.
8. If the system of equations
ax + y + 2z = 0
x + 1y + z = b
2x + y + az = 0
has no solution then (a + b) can be equal to 
(a) -1 (b) 2
Advanced Problems in Mathematics forJEE204
Z VExercise-3 : Comprehension Type Problems
The system ofequations has :
(d) X = 2, |i e R(c) X * 2, |i 3
(d) X = 2, ft g R(c) X * 2, |i = 3
(d) X = 2,peR(c) X 2, n = 3
§ Answers J
1. (b) 3. (d)2. (a)
Paragraph for Question Nos. 1 to 3
Consider the system of equations
2x + Xy + 6z = 8
x + 2y + |iz = 5
. x + y + 3z = 4
1. No solution if:
(a) X = 2, n = 3 (b) X * 2, |i = 3
2. Exactly one solution if :
(a) X * 2, n 3 (b) X = 2, p = 3
3. Infinitely many solutions if :
(a) X * 2, |i 3 (b) X = 2, ji ^3
205Determinants
Exercise-4 : Subjective Type Problems
«i
3. Find the co-efficient of x in the expansion of the determinant
4. If x, y, z e R and = 2 then find the value of
5. If the system of equations :
9. The minimum value of determinant A = V 9 e R is :
1 
cos 9
2
1 
-cos 9 
-1
cos 9
1 
-cos 9
1
1. If 3n is a factor of the determinant nCy
"C2
6ay 
b3 , A2 = 3by 
12q
a2 
b2 
c2
!2
3
4
-y3)
-z6)
-y3)
2a 3 
^3 
^3
3b3 
3c3
C1
(1 + x)
(1 + x)
(1 + x)
3aj +bi
3bi
3C|
3g2 4" ^2 3(^3 4" bg
3b2
3c 2
(1 + X)6 
(1 + X)9 
(1 + x)12
X 
x4 
x7
1
n+3Ci
n+3
C2
a3
&3
c3
(1 + x)4
(1 + x)6
(1 + X)8
and A3 =
al a2 a3
8. Let Ay = by b2
C1 c2 c3
then A3 - A2 = kAy, find k.
z3
z6
z9
y2 
y5 
y8
y5z6(z3 
y2z3(y6 
y 2z3(z3
2x + 3y-z =0
3x + 2y + kz = 0
4x + y + z =0
have a set of non-zero integral solutions then, find the smallest positive value of z.
6. Find a e R for which the system of equations 2ax -2y + 3z = 0; x + ay + 2z = Oand 2x + az = 0 
also have a non-trivial solution.
7. If three non-zero distinct real numbers form an arithmatic progression and the squares of these 
numbers taken in the same order constitute a geometric progression. Find the sum of all 
possible common ratios of the geometric progression.
2a 2 
b2 
4c2
2a3 + b3 
2b 3 + c3 = 
2c3 + a3
x4y5(y3 -x3) 
xy2(x6 -y6) 
xy2(y3 -x3)
x4z6(x3 -z3) 
xz3(z6 -x6) 
xz3(x3 -z3)
1
n+6C1 then the maximum value of rt is
n+6
c2
2ay + by 2a2 + b2
2. Find the value of X for which 2by + q 2b2 + c2 
2cy + ay 2c2 + a2
206 Advanced Problems in Mathematics for JEE
| Answers |
3 9 62. 3. 0 4. 4 5. 6.5 2 7.
8. 3 9. 3 10. 7 2 12. 2
□□□
12. If the system of linear equations
(cos0)x + (sin9)y + cos0 = 0
(sin0)x + (cos0)y + sin9 = 0
(cos9)x +(sin9)y -cos9 = 0
is consistent, then the number of possible values of 0,0 e [0, 2k] is :
’ --------------------—5^
10. For a unique value of p & A, the system of equations given by
x + y + z = 6
x + 2y + 3z = 14
2x + 5y + Az = p
has infinitely many solutions, then is equal to
11. Let lim n sin (2jie |n ) = kn, where n g N . Find k :
n-»-^ (b)
2 2
(c) >-
2
2. The number of points of intersection of the curves represented by
C z _ 5i
arg (z - 2 -7i) = cot (2) and arg ---------
I z + 2-i
208 Advanced Problems in Mathematics for JEE
7. If a, p are complex numbers then the maximum value of is equal to :
(d) less than 1
4
(b) 29 (d) 31(c) 30
9. If arg
12. If zn z2, z3 are vertices of a triangle such that |zj ~z2l=lz1 -z3|thenarg is:
(b) 0
their corresponding vectors is 60°, then
(d) 19
(c) (d) 17
(d) 3 + 272
(b) both zlt z2 are unimodular
(d) Z] -z2 is unimodular
(b) Maximum value of | z | is 672 + 3
(d) Maximum value of | z | is 1572 + 6
17
4
(a) 28
z -6-3i 
z - 3 - 61
Zl +Z2
zi “z2
a p + ap
laPl
(c) greater than 2
z4 + z3 + 2 = 0 , then the value of
number then n =
(a) 126 (b) 119 (c) 133
14. If all the roots of z3 + az2 + bz + c = 0 are of unit modulus, then :
(a) |a| 0 & 0 z3 are complex number, such that |zj|= 2, |z2| = 3, |z3| = 4, then maximum value of 
|zj-z2|2+|z2-z3|2+|z3-zx|2is :
(a) 58 (b) 29
21. IfZ =Z±L then findZ14:
3 + 4i
(a) 27 (b) (-2/ (c) (27)i (d) (-27)i
22. If | Z - 41 + | Z + 4| = 10, then the difference between the maximum and the minimum values 
of | Z | is : 
(a) 2
(d) -L
V2
4-z3 + z2-z + l = 0
210
Exercise-2 : One or More than One Answer is/are Correct
(c) Im 0
(d) If|zi -z2| = V2|z1 -z3| = V2|z2 — z3|, then Re = 0
(c) Im 0
(d) If | Zj -z2| = V2 | Zy — z3 | = V2 | z2 -z3 |, then Re = 0
- - - - - • IAdvanced Problems in Mathematics for JEE
z3 ~Z1
z3 ~z2
Z -Zj 
z-z2
Z -Zj 
z-z2
(b) \z1z2+z2z3 +z3z1\=\z1 +z2+z3\
(Z1 +z2)(z2 +z3)(z3 4-zJ^
Z1-Z2-Z3 )
> where | z | > 1
> where | z | > 1
(a) If arg — = - then arg 
kz2 J 2
(a) If arg — = — then arg
\z2j 2
(b) | z1z2 +z2z3 +z3zy\ =| zj +z2 +z3 |
(zt +z2) (z2 +z3) (z3 +Z1)>[_ 
ZyZ2-Z3 J
7
1. LetZi and Z 2 are two non-zero complex number such that | Zx + Z2| = |Z1| = |Z2|, then—-may
^2
be :
(a) 1 + co (b) 1 + co2
(c) co (d) co2
2. LetZj, z2 andz3 be three distinct complex numbers, satisfying |z1| = |z2| = |z3| = 1. Which of 
the following is/are true :
n
2
z3 ~Z1
z3 -z2
3. The triangle formed by the complex numbers z, iz, i2z is :
(a) equilateral (b) isosceles
(c) right angled (d) isosceles but not right angled
4. If AfjZy), B(z2), C(z3), D(z4) lies on | z | = 4 (taken in order), where Zj + z2 + z3 + z4 = 0 
then:
(a) Max. area of quadrilateral ABCD = 32
(b) Max. area of quadrilateral ABCD =16
(c) The triangle AABC is right angled
(d) The quadrilateral ABCD is rectangle
5. Let Zj, z2 and z3 be three distinct complex numbers satisfying \ z1 | = | z2 | = | z3 | = 1. Which of 
the following is/are true ?
n
2
Complex Numbers 211
(0
Cd) 196
' Answers S
(C,d) (b, c, d)1. (b, c) (a, b, c)2. 3. 4.
(d) (a, b, c)7. (c) Cb)8. 9. 10.
(a, c, d) 5. (b, c, d)
(d) Re =0
- 62 2 - 4iz -i = 0 represent
6.1
6. If 2]= a + ib and z2 =c + id are two complex numbers where a, b, c, d e R and | 2j | = | z2 | = 1 
and Im (2X22) = 0. If = a + ic and w2 = b + id, then :
(a) Im(w1iv2) = 0 (b) Im(w1w2) = 0
(c) ImfeJ = 0
7. The solutions of the equation 2 4 + 4 i 2 3
vertices of a convex polygon in the complex plane. The area of the polygon is :
(a) 21/2 (b) 23/2 (c) 2 5/2 (d) 25/4
8. Least positive argument of the 4th root of the complex number 2 - iV12 is :
(a) H (b) (c) (d)
6 12 12 . 12
9. Let co be the imaginary cube root of unity and (a + bco + cco2)2015 - r" f,z '2
where a, b, c are unequal real numbers. Then the value of a2 + b
(a) 0 (b) 1 (c) 2
10. Let n be a positive integer and a complex number with unit modulus is a solution of the 
equation2"+2 + 1 = 0 then the value of n can be :
(a) 62 (b) 155 (c) 221
= (a + bar + cco)
2 + c2 - ab - be - ca equals :
Cd) 3
212 Advanced Problems in Mathematics for JEE
J.Exercise-3 : Comprehension Type Problems
2.2
= 1, x, y eR, in (x, y) plane will represent:
(d) pair of line(c) ellipse
2
= 1, x, y g R in (x, y) plane, then point (1,1) will lie:2. Consider ellipses :
(c) 7 + V43 Cd) 7 + V41
(c) 5-V23 (d) 5 + V21
(d) 1
Paragraph for Question Nos. 1 to 2
Let /(z) is of the form az + 0, where a, 0 are constants and a, 0, z are complex numbers such 
that | a | * 101. /(z) satisfies following properties :
(i) If imaginary part of z is non zero, then /(z) + /(z) = /(z) + /(z)
(ii) If real part of z is zero, then /(z) + f(z) = 0
(iii) If z is real, then f(z)f(z) > (z + l)2 V z e R.
Paragraph for Question Nos. 6 to 7
Let z1 =3 and z2 = 7 represent two points A and B respectively on complex plane. Let the 
curve Cj be the locus of point P(z) satisfying | z - z^m- |z - z2|2 = 10 and the curve C2 be the 
locus of point P(z) satisfying |z -z1|2+ |z -z2|2 = 16.
(b) inside the ellipse S 
(d) none of these
(b) a circle with radius 7 and centre 04, 5)
(d) a square with side 7 and centre (-4, 5)
3. The complex number ‘m’ lies on :
(a) a square with side 7 and centre (4, 5)
(c) a circle with radius 7 and centre (-4, 5)
4. The greatest value of | m | is :
(a) 5>/21 (b) 5 + V23
5. The least value of | m | is :
(a) 7-V41 (b) 7-V43
Paragraph for Question Nos. 3 to 5
Let z2 and z2 be complex numbers, such what z2 - 4z2 = 16 + 20i. Also suppose that roots a 
and 0 of t2 + z^t + z2 + m = 0for some complex number m satisfy | a — 01 = 2^7, then :
+ 2(hn(0))2
4x" [ y
(/(l)-f(-l))2. C/(0))2 
(a) hyperbola (b) circle 
x2 
(Re (a))2
(a) outside the ellipse S 
(c) on the ellipse S
6. Least distance between curves and C2 is :
(a) 4 (b) 3 (c) 2
213Complex Numbers
2
is equal to :
(b)(a)
(d)(c)
(b)(a) (Z2-Z1)(Z3-Z1)
2(d) (Z2-ZJCZ3 -ZJ(C)
J Answers J
5. (a)4. (d) 6. (d)3. (b) 7. (d)1. (a) 2. (b)
AC
AB
7. The locus of point from which tangents drawn to C; and C2 are perpendicular, is :
(a) |z-5|=4 (b) |z-3|=2 (c) |z-5| = 3 (d) |z-5| = V5
Paragraph for Question Nos. 8 to 9
In the Argand plane Zx, Z2 and Z3 are respectively the vertices of an isosceles triangle ABC 
with AC = BC and Z.CAB = 0 . If I (Z4) is the incentre of triangle, then :
(Z2 -Zx) (Z3 -Zt) 
z4-z.
(Z2-ZI)(Z3 -ZJ 
(Z4-Zi)
(Z2 + Zx ) (Z 3 -4- Zx ) 
(Z4+Zi)
8. Cc) j
/" AR
8. The value of —
\IA
(Z2—Z1)(Z1 -Z3)
(Z4-ZJ
(Z2-Z!)(Z3 -ZJ
(Z4-Zx)2
9. The value of (Z4 - Zx)2 (1 + cos 0) sec 0 is :
(Z2-Z1)(Z3 -ZJ 
(Z4 -Z1)2
9.* (a) |
214 Advanced Problems in Mathematics for JEE
7Exercise-4 : Matching Type Problems
(A) (P) 2
= 2 arg
(Q) 3(B)
(C) 4
(D) (S) 6
(T)
Column-ll
(A) 8
(D) (S) 3
(T) 2
(B)
(C)
then the value of s - c
(where s is the semiperimeter) a = BC, b = CA, c - AB
If the altitudes of AABC are in harmonic progression then the 
side length ‘b’ can be
Let a, b and c be the position vectors of vertices A, B and C 
respectively. If (c-a)(b-c) = 
| ax b+ bx c+ ex a| equals to
Let the equations ajX + bjy+Cj =0 and a2x + b2y + c2 = 0 (R) 
represent the lines AB and AC respectively and 4
3
(P)
(Q)
(R)
7
5
zi ~z3 
z2~z3 
then biggest side of the triangle is
Q] b2 -a2b^ 
ai a2 + ^2
z3 ~Z1 
z2~zl
1. In a AABC, the side lengths BC, CA and AB are consecutive positive integers in increasing order.
Column-1\ \ Column-ll
If z}, z2 and z3 be the affixes of vertices A, B and C respectively 
in argand plane, such that arg
12
2. Let ABCDEF is a regular hexagon Atz^, Btzf), C(z3), D(z4), E(z5), f(z6) in argand plane 
where A, B, C, D, E and F are taken in anticlockwise manner. If z1 = -2, z3 = 1 - 73 i.
_____________________Column-1_____________________\
If z2 = a 4- ib, then 2a2 + b2 is equal to
The square of the inradius of hexagon is
The area of region formed by point P(z) lying inside the incircle
of hexagon and satisfying - S;B—>T;C—>S;D-^Q, R, S, T1.
2. A —> R j B —> S j C —> Q ; D —> P
3.
Let cube a non real cube root of unity 
then the number of distinct elements 
in die set {(1 + co+co2 + ... + con)m;
n, m e N} is :
Let co and co2 be non real cube root of 
unity. The least possible degree of a 
polynomial with real co-efficients 
having roots
2co, (2+ 3co), (2 + 3co)2, (2-co-co2) 
is
. z -a
segment of a circle whose radius is
Let zn z2, z3 are complex numbers 
denoting the vertices of an 
equilateral triangle ABC having 
circumradius equals to unity. If P 
denotes any arbitrary point on its 
circumcircle then the value of 
^((PA)2+ (PBJ2+(PC)2) equals to
A —> S ; B —> R \ C —> Q ; D —> P
Let a = 6 + 4i and 0 = 2 + 4i are two 
complex numbers on Argand plane. 
A complex number z satisfying amp 
it■g moves on a major
Advanced Problems in Mathematics for JEE216
Exercise-5 : Subjective Type Problems
of| 2zx + 3z2 + 4z3 |= 9, valuethenand
1/3
J Answers
66. 06 44. 5.2. 6 24 3.
8. 5
□□□
Let complex number ‘z ’ satisfy the inequality 2 then
.0; i * j
(where {■} denotes fractional part function)
2
5
cos2 a
0
0
1. Let A =BBt
+ B’3numbers and C = (A 
(a) 0
(a) A (b) A2 (c) A3 (d) A4
6. Let M = [a,7]3x3 where a17 e {-1,1}. Find the maximum possible value of det(M).
(a) 3 (b) 4 (c) 5 (d) 6
3 -3 4
5. If A= 2 -3 4 ; then A 
0-11
ro oiL° °J
' 0 0-11
0-10. The only correct statement about the matrix A is : 
l-l 0
(a) A is a zero matrix
(c) A-1 does not exist
218 Advanced Problems in Mathematics for JEE
2
(b)(a)
(d)Cc)
8. If the trace of matrix A =
(d) 2 or 3(a) -2 or 3
is equal to :9. If A = [fly ] 2x2 where = then A
0(c) 3 1
10. If
12. Let matrix A =
(d) 110
, m e N, a, b e R, for some matrix M, then which one of the following14. A
(b) M=(a2+b2)(a) M =
(c) M=(am+bm)
2
1
-3
-1
X 0 O’
0X0
0 0 X
tan0
0 
X + l 
0
0
0 
X+l
tan0 
1
0 
X+2 
0
0
0 
X+2
-tanO 
1
1 
-tan 9
1
9L-3
X+2
0
0
X2
0 :
0
0 
x2
(d) (A"1)2
1 0
0 1
0
0
0 X2
x y
1
1
2> 771 Fl 0
J 0 1
(d) ir° 3
3
b2"1 
-a2"1
(d) M = (a2 + b2)m-1
of such matrices A is :
[Note : adj. A denotes adjoint of square matrix A.]
(a) 220 —> —> —> —> 
c are unit like vectors and | a | = 4. If a+ X c = 2 b 
then the sum of all possible values of X is equal to
222 Advanced Problems in Mathematics for J EE
(C) (R) 4
If P = and
(S)(D) 5
4.
Column-ll
(P)(A) 1
(B) 2
(C) (R) 9
(D) (S) 8
that /(%) x = 0
0R be invertible matrices of second order (Q) 
such that A = PQ-1 ,B = QR~1,C =RP”1,then the 
value of det. (ABC + BCA + CAB) is equal to
The perpendicular distance of the point whose 
position vector is (1, 3, 5) from the line
r = i + 2j + 31c + X(i + 2j + 2k) is equal to
11 1
J2 V3 Vn
2 
t 
-2 3
Q = P-1, then the value of t is equal to
If y = tanu where u = v - — and v = Inx, then the 
v
dyvalue of — at x = e is equal to X then [X] is equal to 
dx
(where [•] denotes greatest integer function)
Let /(x) be a continuous function in [-1,1] such 
ln(px2 + qx + r) . _lsx P, Q, T; B -> S ; C -> P, R ; D -> R
A “> R; B —> P, Q, S ; C —> P, Rj D —> P, Q, R, S2.
3. A->Q;
4. A->S;
A->Q; B->S;5.
ay = 1 ■> 
0;
If 4X -2x+2 + 5+|| b-l| -3| = |siny I;
x, y, b e R
then the sum of the possible values of b is A 
then (X + 1) equals
___________
Let A = [ay] be a 3 x 3 matrix where 
2cost; if i = j 
if I = l 
otherwise
■|, -1 j then value of q - p is
B—>R; C->S; D-» P
B —> R j C —> P j D —> Q
then maximum value of det(A) is
Let /(x) = x3 + px2 + qx + 6; where 
p, q e R and f'M oo
[Note : Tr. (P) denotes the trace of matrix R]
3. Let A be a 2 x 3 matrix whereas B be a 3 x 2 matrix. If det. (AB) = 4, then the value of det. (BA), 
is :
4. Find the maximum value of the determinant of an arbitrary 3x3 matrix A, each of whose 
entries ay e {-1,1}.
5. The set of natural numbers is divided into array of rows and columns in the form of matrices as
7 81
10 11 and so on. Let the trace of A10 be X. Find unit digit of
1. A and B are two square matrices. Such that A2B = BA and if (AB)10 
k-1020.
2. Let An and Bn be square matrices of order 3, which are defined as :
21 + / 3i - i
An = [a(j-] and Bn = [by] where ay = —~ and by = — for all i and j, 1Lim Tr. (3Aj + 32A2 + 33 A3 + 
n->oo
= A k • B10. Find the value of
Permutation and
Combinations
-
Exercise-1 : Single Choice Problems
(b) 230
(d) 253
413'
21 z-»
L10
21r cio
1. The number of 3-digit numbers containing the digit 7 exactly once :
(a) 225 (b) 220 (c) 200 (d) 180
2. Let A = {xj, x2, x3, x4, x5, x6, x7, x8}, B = {yi> y-i, /3> The total number of function 
f : A -> B that are onto and there are exactly three elements x in A such that /(x) = jq is :
(a) 11088 (b) 10920 (c) 13608 (d) None of these
3. The number of arrangements of the word “IDIOTS” such that vowels are at the places which 
form three consecutive terms of an A.R is :
(a) 36 (b) 72 (c) 24 (d) 108
4. Consider all the 5 digit numbers where each of the digits is chosen from the set {1, 2, 3, 4}. Then 
the number of numbers, which contain all the four digits is :
(a) 240 (b) 244 (c) 586 (d) 781
5. How many ways are there to arrange the letters of the word “GARDEN” with the vowels in 
alphabetical order ?
(a) 120 (b) 480 (c) 360 (d) 240
6. If a * pbuta2 = 5a -3 and p2 = 5p- 3 then the equation havinga/pandp/aas its roots is :
(a) 3x2-19x+3 = 0 fb) 3x2+19x-3 = 0
(c) 3x2-19x-3 = 0 (d) x2-5x + 3 = 0
7; A student is to answer 10 out of 13 questions in an examination such that he must choose at 
least 4 from the first five questions. The number of choices available to him is :
(a) 140 (b) 196 (c) 280 (d) 346
8. Let set A = {1, 2, 3, , 22}. Set B is a subset of A and B has exactly 11 elements, find the sum 
of elements of all possible subsets B.
(a) 25221Cn
(c) 25321C9
Advanced Problems in Mathematics for JEE226
9. The value of
(0
is :
(d) (n + 2)2n-1
2009!+ 2006!
20081+ 2007!
(c) 2007 (d) 1
distinct prime numbers. Then the number of factors of
([•] denotes greatest integer function.)
(a) 2009 (b) 2008
10. If pp p2, p3 ......,pm+1 are
P1P2P3.......Pm+1 is:
(a) m(n + l) (b) (n + l)2m (c) n • 2m (d) 2nm
11. A basket ball team consists of 12 pairs of twin brothers. On the first day of training, all 24 
players stand in a circle in such a way that all pairs of twin brothers are neighbours. Number of 
ways this can be done is :
(a) (12)! 211 (b) (11)!212 (c) (12)! 212 (d) (11)! 2n
12. Let ‘m’denotes the number of four digit numbers such that the left most digit is odd, the second 
digit is even and all four digits are different and ‘n’ denotes the number of four digit numbers 
such that left most digit is even, second digit is odd and all four digits are different. If m = nk, 
then k equals :
4 3 S 4(a) (b) (c) (d) 2
5 4 4 3
13. The number of three digit numbers of the form xyz such that x2)100~fc 
k=o
(a) Number of ways in which 50 identical books can be distributed in 100 students, if each 
student can get atmost one book.
(b) Number of ways in which 100 different white balls and 50 identical red balls can be 
arranged in a circle, if no two red balls are together.
(c) Number of dissimilar terms in (xj + x2 + x3 + ... + x50)51.
2-61014....... 198
50!
3. Number of ways in which the letters of the word “NATION” can be filled in the given figure such 
that no row remains empty and each box contains not more than one letter, are :
(a) 11 [6 (b) 12 [6 (c) 13 [6 (d) 14 [6
4. Let a, b, c, d be non zero distinct digits. The number of 4 digit numbers abed such that ab + cd is 
even is divisible by :
(a) 3 (b) 4
230 Advanced Problems in Mathematics for JEE
Exercise-3 : Comprehension Type Problems
(d) 210
' Answers *
1. (d) 2. (c)
(a) 315426 (b) 135462 (c) 234651
2. Number of such six digit numbers having the desired property is :
(a) 120 (b) 144 (c) 162
1. A six digit number which does not satisfy the property mentioned above, is :
Cd) None of these
Paragraph for Question Nos. 1 to 2
• I-
Consider all the six digit numbers that can be formed using the digits 1, 2, 3, 4, 5 and 6, each 
digit being used exactly once. Each of such six digit numbers have the property that for each 
digit, not more than two digits smaller than that digit appear to the right of that digit.
Permutation and Combinations 231
Exercise-4 : Matching Type Problems
Column-I Column-ll
720
1800
5760
6000
7560
Column-llColumn-1
(A) (P) 28 (7!)
(B)
(C) (R) 210(7!)
(D) (S) 840 (7!)
(T)
1.
2.
The two A’s are not together
The two E’s are together but not two A’s 
Neither two A’s nor two E’s are together 
No two vowels are together
(A)
(B)
(C)
(D)
1. All letters of the word BREAKAGE are to be jumbled. The number of ways of arranging them so 
that:
The number of words taking all letters of the given word 
such that atleast one repeating letter is at odd position is 
The number of words formed taking all letters of the! (Q) 
given word in which no two vowels are together is
The number of words formed taking all letters of the 
given word such that in each word both M’s are together 
and both T’s are together but both A’s are not together is
The number of words formed taking all letters of the 
given word such that relative order of vowels and 
consonants does not change is
(ID!
(2!)3
4! 7!
(2!)3
■ Answers
A-> Q; B->R; C->P; D->T
A-> T ; B-> Q ; C -> R; D-> P
V
(P)
(Q)
(R)
(S)
(T)
2. Consider the letters of the word MATHEMATICS. Set of repeating letters = { M, A, T}, set of non 
repeating letters = { H, E, I, C, S }:
Advanced Problems in Mathematics for J EE232
Exercise-5 : Subjective Type Problems
10. There are n persons sitting around a circular table. They start singing a 2 minute song in pairs 
such that no two persons sitting together will sing together. This process is continued for 28 
minutes. Find n .
11. The number of ways to choose 7 distinct natural numbers from the first 100 natural numbers 
such that any two chosen numbers differ atleast by 7 can be expressed as nC7. Find the number 
of divisors of n.
12. Four couples (husband and wife) decide to form a committee of four members. The number of 
different committees that can be formed in which no couple finds a place is X, then the sum of 
digits of X is :
1. The number of ways in which eight digit number can be formed using the digits from 1 to 9 
without repetition if first four places of the numbers are in increasing order and last four places 
are in decreasing order is N, then find the value of —.
2. Number of ways in which the letters of the word DECISIONS be arranged so that letter N be
12
somewhere to the right of the letter “D” is —. Find X.
X
4. There are 10 stations enroute. A train has to be stopped at 3 of them. Let N be the ways in which 
the train can be stopped if atleast two of the stopping stations are consecutive. Find the value of 
■Jn.
5. There are 10 girls and 8 boys in a class room including Mr. Ravi, Ms. Rani and Ms. Radha. A list 
of speakers consisting of 8 girls and 6 boys has to be prepared. Mr. Ravi refuses to speak if Ms. 
Rani is a speaker. Ms. Rani refuses to speak if Ms. Radha is a speaker. The number of ways the 
list can be prepared is a 3 digit number n1n2H3, then | n3 + n2 -nx| =
6. Nine people sit around a round table. The number of ways of selecting four of them such that 
they are not from adjacent seats, is
7. Let the number of arrangements of all the digits of the numbers 12345 such that atleast 3 digits 
will not come in it’s original position is N. Then the unit digit of N is
8. The number of triangles with each side having integral length and the longest side is of 11 units 
is equal to k2, then the value of ‘k’ is equal to
9. 8 clay targets are arranged as shown. If N be the number of T T T
different ways they can be shot (one at a time) if no target O O O
can be shot until the target(s) below it have been shot. Find A A A
the ten’s digit of N. . Y T
233Permutation and Combinations
I Answers ! --------r-w x
6.5 99 8 3. 8 4. 8 5. 92.
8. 6 6 12. 7 13. 77 49. 10. 7
□□□
13. The number of ways in which 2n objects of one type, 2n of another type and 2n of a third type 
can be divided between 2 persons so that each may have 3n objects is an2 + pn + y. Find the 
value of(a + p + y).
14. Let N be the number of integral solution of the equation x+y+z+w= 15 where x > 0, y >5, 
z > 2 and w > 1. Find the unit digit of N.
Bionmial Theorem
7Exercise-1: Single Choice Problems
+1. Then which of the following
'Cra0 is
(c)
then ofthe seriessum
-1)
-1) -1)
(a) 1 (c)
+
(a) 3
1
3
9-20'
+ (-l)r ni
20C
— 2204
-3 
10
, r v-2010 
+ c2010x
20 r 
c20
2
Cd) 73
is not divisible by :
(d) 19
(d) v3
?|14|
— Cq + Cj X + C 2X2 +.., 
8 + + C2Oo9 equals to : 
2010-1)
cio +......... "
(C) 11
1. Let N = 21224 -1, a = 2153 + 277 +1 and 0 = 2408 
statement is correct ?
(a) a divides N but 0 does not (b) 0 divides N but a does not
(c) a and 0 both divide N (d) neither a nor 0 divides N
2n
2. If (l + x + x2)n = ^arxr, then ar -"Cj + nC2ar_2 ~nC3ar_3 +
r = 0
equal to : (r is not multiple of 3)
(a) 0 (b) nCr (c) ar (d) 1
3. The coefficient of the middle term in the binomial expansion in powers of xof (1 + ax)4 and of 
(1 - ax)6 is the same if a equals :
5 3(a) ~ (b)
3 5
4. If (1 + x)2010
^2 +C5 + C
(a) — (22010 -1) (b) 1(22010
2 3
(c) — (22009 -l) (d) 1(22009
2 3
5. Let an = (2 + V3)n. Find lim (an -[an]) ([■] denotes greatest integer function)
n—>oo
*4
. 20r
7 “ c8
(b) 7
6. The number N = 20C
235Binomial Theorem
(d) 14(a) 11 (b) 12
(d) 220
then sum of coefficients in the expansion
is divided by 7 is 3.
.128
th , r = 1, 2,... n thenroots of unity, ar = ea-i, a2,
+ nCnan is equal to:
(d) (a1+an_1)n-l(a)
(d) 6
14.
(d) 226 + —•(c) 213
2
(b) 169
1 + 50 term = —
ai + art-i
2
>an
ck
'C c13
1 117. The value of the expression log2 1 + - ^ 123, then xyznC0 -(x-l)(y-l)(z-1) nC1+(x-2)(y-2)(z-2)nC2-
(x-3)(y -3)(z -3) "C3 + + (-l)n(x-n)(y -n)(z-n) nC„ equals :
(a) xyz (b) x + y + z
(c) xy + yz + zx (d) 0
12. If 
11 + 50 term =---------------- ,
3! (k + 3)l
+ IOOxjqq)^ is :
• ••>*100 316 independent variables)
(b) (5050)51
(d) (5050)50
(128)128
(C) 13
. i Y28. The constant term in the expansion of x + — is :
\ x3 J
(c) 260
(b) ^-[(l + ai)n-l](c)
• 320 is divided by 7 is :
(b) 2 (c) 4
26C 2 + + 26C13 is equal to :
(b) 225 +l-26C13
(a) 26
o tc 3 4 5 
4! 5! 6!
(1 + 2x: + 3x2 +...
(where xn x2, x3, 
(a) (5050)49
(c) (5050)52
236 Advanced Problems in Mathematics for JEE
(d) n-3n-1
(b)(a) (0 (d)
(d) 91
/ 3 3
(C) (a) (c) (a) 6. (c) (a)2. 4. 5. (d) (d)7. (d)9. 10.
(b) (b) (d) (d) (d)(d) (a) 16.11. 14. 15. 17. (a) Cb)12. 13. (c)18. 20.19.
(d) (a)21. 22.
r8.
100
97
(d) 11
.+ (10C8)2-(10C9)2 + (10C10)2 equals:
-10C, (d) 10C
' Answers ■
(b)'3.
■+Q + [o)equals:
(S)
(d) 24(c) 23
(b) 71
+ 2r2 + 3r + 2
(r + 1)2
then the value of n is:
(a) 2 (b) 22
2f5+ —
x
17. The last digit of 9! + 39966 is:
(a) 1 (b) 3 (c) 7 (d) 9
18. Let x be the 7th term from the beginning and y be the 7,h term from the end in the expansion of 
n
. If y = 12x then the value of n is :
o4 
nCr=-
|3V3+ J_
I 41/3
(a) 9 (b) 8 (c) 10
19. The expression (10C0)2 - C10^ )2 + (10C2)2
(a) 10! (b) (10C5)2 (c) -1UC5 (d) u5
20. The ratio of the co-efficients to x15 to the term independent of xin the expansion of(x2 
is :
(a) 1:4 (b) 1:32 (c) 7:64 (d) 7:16
21. In the expansion of (1 + x)2(l + y)3(l + z)4(l + w)5, the sum of the coefficient of the terms of 
degree 12 is:
(a) 61
22. ~
(c) 81
24 +23 +22 -2
3
15. If ar is the coefficient of xr in the expansion of (l + x + x2)n (n eN). Then the value of 
(dj + 4a4 + 7a7 + 10a10 + ) is equal to :
(a) 3n-1 (b) 2n (c) —-2n
3
16. Let^ represents the combination of ‘ n’ things taken ‘ k’at a time, then the value of the sum
100^1
98 J
Binomial Theorem 237
f
300
(a) 5 (d) 13(c) 11
4. The expansion of Vx + is arranged in decreasing powders of x. If coefficient of first
(c) 3 (d) 2
666,222 .444 .888
C10 +..........
(C) 7
 + a3oox
nCr:
c7- is divisible by:
(d) 19 
then which of the following
100.- 
cio
1
2 a/x
three terms form an A.R then in expansion, the integral powers of x are :
(a) 0 (b) 2 (c) 4 (d) 8
n+4
= ^akxk . Ifaj, a2, a3 are in Al? then n is (given that nCr = 0, if n A,
The number of onto functions such that /(x) = x for atleast 3 
distinct x g A, is not a multiple of
cm-i>
If the sum of first 84 terms of the series 
4 + V3 + 8 + V15 + 12 + V35 * 
1 + V3 V3 + V5 V5 + V7 
equal to
where nCr = 0 if r > n, is maximum when y is
> i holds for x, then
2
(P)
n m
IfP = nCr;q - ^2 mCr ^15)r (m,n 6Nland if 
r=0 r=0
P = q and m, n are least then m + n =
Remainder when 11+31+51+ + 2011! is
divided by 56 is
|x| 
l+|x|
number of integral values of ‘ x’ is/are
3. Match the following
If n 1 Cr = (k2 - 3)nCr+1 and k g R+, then least value of 5[k] is 
(where [•] represents greatest integer function)
m
E2°c 
i=0
Advanced Problems in Mathematics for JEE240
(B) (Q) 2
is
(R)(C) 5
(S)(D) 9If(l + x)(l + x2)(l + x4)
J Answers ?
1. A-»Q;B->S;C->R;D->P, Q, R, S, T
D—> R2.
A-*Q; B->R; C->S; D->P3.
n
).= ^xr, then
r=0
If x,y gR, x2 + y 2 - 6x + 8y + 24 = 0, the greatest
i c 16 2value of — cos
5
If (V3 +1)6
24 . ----- sin 
5
+ (V3-1)6 =416, if xyz=[(V3 + l)6], 
x,y,zeN, (where [•] denotes the greatest integer 
function), then the number of ordered triplets (x,y, z) 
is
S r—
is equal to
(1 + x128
A-> S ; B->Q;
'x2+y2 x2 +y 2
Binomial Theorem 241
Exercise-4 : Subjective Type Problems
upto 2008 terms
= 1 V K > 2n. The find the value of
the
.+ 68. + 69.30
10. The remainder when
o
8. Find
3log3
Ozk-l
6
I is divided by 11, is :
•30 + '“'30
Find the sum of the digits of q.
s Z20 
k=l
9. Let q be a positive integer with q n + 3. 96C-,O + 30 r
c30 -
2007Co -8- 2007C2 -18-
too r31
c30
sum
llogyOl-2!-9
+ 7S
(x11 -11) the coefficient of x60
2007 Cj +13- 2007 rg3 +
1
11. Let a = 3223 + 1 and for all n > 3, let
fW=nC0.anA - nC1.an-2+ nC2.an~3 -....... + (-I)'*'1 "C^ a0.
If the value of /(2007) + /(2008) = 37 k where k g N then find k
12. In the polynomial (x-1) (x2 -2) (x3 -3) ...(x11 -11), the coefficient of x50 is :
13. Let the sum of all divisiors of the form 2P ■ 3q (with p, q positive integers) of the number 
1988 -1 be X. Find the unit digit of X.
for which the sixth term of
1. The sum of the series 3 •
is K, then K is :
2. In the polynomial function/(x) = (x-l)(x2 - 2)(x3 -3)
is :
3n 3n 3n 3n
3. If£ar(x-4)r = (x - 5)r and ak = 1 V K > 2n and dr (x - 8)r = ^Br (x - 9)r and
r=0 r = 0 r=0 r = 0
3n 3n a + D
^dr(x-12)r = ^Dr(x-13)r and dK = 1 V K > 2n. The find the value of — 
r=0 r = 0 ^2n
4. If 3101 - 2100 is divided by 11, the remainder is
5. Find the hundred’s digit in the co-efficient of x17 in the expansion of (1 + x5 + x7 )20.
6. Let x = (3a/6 + 7)89. If {x} denotes the fractional part of ‘x’ then find the remainder when 
x{x} + (x{x})2 + (x{x})3 is divided by 31.
3n n
7. Letn eN;Sn = £(3nCr) and Tn = £(3nC3r). Find |Sn - 3T„|.
r=0 r=0
of possible real values of x y
equal 567 :
242 Advanced Problems in Mathematics for JEE
n(n eN)
| Answers |
i 20 2. 3. 4. 2 25. 4 6. 0 7.
8. 9. 5 10. 3 9 (4)4 11. (1)12. (4)13. 14.
16. 615. 5
□□□
of possible 
logyO1'"21-9
sum
+7®
2-2x-k2+l. If
io
15. Letl + ^(3r -10Cr + r-10Cr) = 210 (a-45 +p) where a,0 gN and /(x) = x
r=l
a, p lies between the roots of /(x) = 0. Then find the smallest positive integral value of k.
c IS
16. Let Sn =^0^! + nCInC2 + + nCn_1nCn if—— = —; find the sum of all possible values of
for which the sixth term of14. Find the real values of xY
equals 567.
)[15J4
Probability
..0 Exercise-1: Single Choice Problems
(c) 1/9 ' (d) 2/9(b) 1/7
(a) 3 (b) 4 (c) 5 (d) 6
5. Three a’s, three b’s and three c’s are placed randomly in a 3 x 3 matrix. The probability that no 
row or column contain two identical letters can be expressed as —, where p and q are coprime q
then (p + q) equals to :
(a) 151 (b) 161 (c) 141 (d) 131
6. A set contains 3n members. Let Pn be the probability that S is partitioned into 3 disjoint subsets 
with n members in each subset such that the three largest members of S are in different subsets. 
Then lim Pn = 
n-^co
(a) 2/7
1. The boy comes from a family of two children; What is the probability that the other child is his 
sister ? :
112 1(a) i (b) 1 Cc) ± (d) i
2 3 3 4
2. If A be any event in sample space then the maximum value of 3-s/P(A) + 4y]P(A~) is :
(a) 4 (b) 2
(c) 5 (d) Can not be determined
------ 1 1 - 1 -3. Let A and B be two events, such that(n, 2it)
4. |loge|x||=|fc — 11 — 3 has four distinct roots then k satisfies : (where |x| 3
17Function
, then :9. The function f{x) = cos x + cos
7
(c) f-.R^R,fM =
(d) f:R->R,f(.x') =
13. If/(x) =
(b) Range of f contains two prime numbers 
(d) Graph of f does not lie below x-axis
(b) /(x) is many-one
(d) range of the function is R
73-3x
2
(d) /(x) is one-one for x e
8. Let fM be invertible function and let /-1 (x) be its inverse. Let equation /(/-1 (x)) - f~l (x) 
has two real roots a and p (with in domain of /(x)), then :
(a) /(x) = x also have same two real roots
(b) /-1 (x) = x also have same two real roots
(c) /(x) = y1 (x) also have same two real roots
(d) Area of triangle formed by (0, 0), (a, /(a)), and (P, /(P)) is 1 unit
x .n 0^2 'i 
— + 
2
-,1
.2
(-{-x}), where {x} is fractional part function. Then
(a) Range of /(x) is (b) Range of /(x) is ~ ~
(c) /(x) is one-one for x e -1, |
10. Let f:R -> R defined by /(x) = cos-1 
which of the following is/are correct ?
(a) / is many-one but not even function
(c) f is a periodic
11. Which option(s) is/are true ?
(a) f:R —> R, /(x) = -e"x is many-one into function
(b) f:R -> R, f(x~) = 2x + |sinx| is one-one onto
V-2 , Ay J_ 3Q
— ------------ is many-one onto
x2 -8x + 18
2x2 — x + 5— ------------ is many-one into
7x2 + 2x+10
e ... .X C12. If h(x) = In— + In— , where [•] denotes greatest integer function, then which of the 
ej |_ XJ
following are true ?
(a) range of h(x) is {-1,0}
(b) If h(x) = 0, then x must be irrational
(c) If h(x) = -1, then x can be rational as well as irrational
(d) h(x) is periodic function
x\ ’ xeQ,then:
-x3 ; x«Q
(a) /(x) is periodic
(c) /(x) is one-one
Advanced Problems in Mathematics for JEE18
(a) f(-4) = 40 (b)
16. For the equation------= X which of the following statement(s) is/are correct ?
(d) f3(x) > f5(x) for all x g 2kn, (4k +1)- I, k g I
2
(Where I denotes set of integers)
14. Let /(x) be a real valued continuous function such that
/(0) = | and /(x + y) = f(x)f(a-y) + fff(a-x^x,yeR, 
then for some real a :
(a) /(x) is a periodic function
(c) f(x) = |
(b) /(x) is a constant function
rr COSX(d) /(x)= ——
2x 0 1 f°r x e
(b) f2W = 1 for x = 2kn, k el
(c) f2W > f3 (x) for all x g
/(-13)-/(!!)_ 17 
/(13) + /(-ll) 21.
(d) Range of J(x) is [0, 50]
n
12
(c) f (5) is not defined 
e~x
1 + x
(a) when X g (0, oo) equation has 2 real and distinct roots
(b) when X g (-oo, - e2) equation has 2 real and distinct roots
(c) when X g (0, oo) equation has 1 real root
(d) when X g (-e, 0) equation has no real root
17. For x g R + , if x,[x],{x} are in harmonic progression then the value of x can not be equal to : 
(where [•] denotes greatest integer function, {■} denotes fractional part function)
(a) tan- (b) -^=cot- (c) -4= tan— (d) -^=cot —
V2 8 V2 8 V2 12 V2 12
18. The equation ||x-l| + a| = 4, a gR, has :
(a) 3 distinct real roots for unique value of a. (b) 4 distinct real roots for a g (-oo, - 4)
(c) 2 distinct real roots for | a| 4
19. Let /n(x) = (sin x)1/n + (cosx)1^, x g R, then :
2kjt,(4k +1)-| j, 6 f
19Function
21. The number of real values of x satisfying the equation ;
22. Let /(x) = sin
4
= 0(0) = -
(c) If/(x) = (2011-x
(d) The function f:R->R defined as /(x) =
++
(a, b, c, d)(a, b, c) 6.(a, b)(a, b, d) (a, b, c) (c, d) 5.2. 3.
(a, b, d) (a, c)(a, b, d)(b, c)(a, b, c) 12.(a, d) 8. 9. 10.7.
(a, b, c, d)(a, c, d)(b. c, d)(c, d) (a, b, c) (a, b, d) 16. 18.17.14. 15.13.
(a, b)a, b)(a, d) (a, c, d)(a, b) (a, b, c) 24.20. 23.21. 22.19.
i
3
8
71
2
7t
2
1 3- +-----
4x + 5~
6
' Answers j
i 4. I
(b) Z2015 (d) z2011
s=WJl+^-l 
|_4j L4 200J
(a) S is a composite number
(c) Number of factors of S is 10
2x + l
3
20. If the domain of fW = - cos 
it
(x2}log 3 — where, x > 0 is [a, b] and the range of f(x) is [c, d],
3x-l------- are
2
(b) 8 
+ cos6
fl 1991 u + - +----- then
.4 200.
(b) Exponent of S in 1100 is 12
(d) 25 Cr is max when r = 51
greater than or equal to {[•] denotes greatest integer function):
(a) 7 (b) 8 (c) 9 (d) 10
— . If ZnW denotes 0 th derivative of f evaluated at x. Then which6 [ *
14
of the following hold ?
(a) Z2014(0) = --? (b) Z2°15(0) = -|
8 8
23. Which of the following is(are) incorrect ?
(a) If Z(x) = sinxandg(x) = Inxthen range of gtfW) is [-1,1]
(b) If x2 + ax + 9 > x V x g R then -5 R with period 4 is such that 
max.(|x|,x2) ; 0, 1 o(2,oo)
0,|
II. „
3j - — ' ' '
12. Range of the values of ‘ k' for which /(x) = k has exactly two distinct solutions :
Paragraph for Question Nos. 13 to 14
Let /(x) be a continuous function (define for all x) which satisfies 
f3 (x) - 5/2(x) + 10/(x) -12 > 0, f 2(x) -4/(x) + 3 > 0and /2(x) - 5/(x) + 6P(A B) = -, P( A n B) = - and P(A) = -, where A stands
6 4 4
for complement of event A. Then events A and B are :
(a) equally likely and mutually exclusive (b) equally likely but not independent
(c) independent but not equally likely (d) mutually exclusive and independent
4. Let n ordinary fair dice are rolled once. The probability that at least one of the dice shows an
f 31>odd number is I — I than ‘ n ’ is equal to :
244
What is the
(d) None of these
(c)
relatively prime positive integers then the value of (p + q) is :
(a) 39 (b) 40 (c) 41 (d) 42
8. Mr. As T.V has only 4 channels ; all of them quite boring so he naturally desires to switch 
(change) channel after every one minute. The probability that he is back to his original channel 
for the first time after 4 minutes can be expressed as —; where m and n are relatively prime 
n
(Wls
(d’T
13. From a pack of 52 playing cards; half of the cards are randomly removed without looking at 
them. From the remaining cards, 3 cards are drawn randomly. The probability that all are king.
(b)
125
(d) I
numbers. Then (m + n) equals :
(a) 27 . (b) 31 (c) 23 (d) 33
9. Letters of the word TITANIC are arranged to form all the possible words, 
probability that a word formed starts either with a T or a vowel ?
. . 2 „ . 4 , „ 3(a) - (b) - (c) -
7 7 7
10. A mapping is selected at random from all mappings f : A -> A
where set A = {1, 2, 3,........, n}
3
If the probability that mapping is injective is —, then the value of rt is :
(a) 3 (b) 4 (c) 8 (d) 16
11. A 4 digit number is randomly picked from all the 4 digit numbers, then the probability that the
product of its digit is divisible by 3 is : 
, . 107 (a) ----
125
. . Ill(c) ----
125
12. To obtain a gold coin; 6 men, all of different weight, are trying to 
build a human pyramid as shown in the figure. Human pyramid is 
called “stable” if some one not in the bottom row is “supported by” 
each of the two closest people beneath him and no body can be 
supported by anybody of lower weight. Formation of ‘stable’ 
pyramid is the only condition to get a gold coin. What is the 
probability that they will get gold coin ?
45
Advanced Problems in Mathematics for JEE 
~~............
7. Three different numbers are selected at random from the set A ={1, 2, 3, , 10}. Then the 
probability that the product of two numbers equal to the third number is —, where p and q are 
q
245Probability
(b)(a)
(d)(c)
(c)(a)
(d)(c)(b)
’ Answers !7
8. (b)Cd) 7. (c)6. 9.(c)(a) (0 (c)(c) 4. 5.2. 3.
17.;(d)(c)16.(a) (a) (a)11. (a) (a) 15.12. 13. 14.
(d,n
, 100}. Probability that they
(d); 10. j (b)
l
18
91 c c3
16
91 c c3
17
91 r 
c3
------1------ (b) ------ -------
(25)(17)(13) (25)(15)(13)
------------- (d) ------1-----
(52)(17)(13) (13)(51)(17)
14. A bag contains 10 white and 3 black balls. Balls are drawn one by one without replacement till 
all the black balls are drawn. The probability that the procedure of drawing balls will come to 
an end at the seventh draw is :
15 105 r y 35 7-- (b) -- (c) -- (d) --286 286 286 286
15. LetS be the set of all function from the set {1, 2, 10} to itself. One function is selected from
S, the probability that the selected function is one-one onto is :
, > 9! , 1 , . 100 9!(a) —- (b) — (c) ----- (d) ——
109 10 10! IO10
16. Two friends visit a restaurant randomly during 5 pm to 6 pm. Among the two, whoever comes 
first waits for 15 min and then leaves. The probability that they meet is :
117 9(a) A (b) A (C) -C (d)
4 16 16
17. Three numbers are randomly selected from the set {10,11,12,
form a Geometric progression with integral common ratio greater than 1 is : 
, 15 -
3
1 -, 3(d) —243 243 216
10. The probability that a person crosses the first two obstacles but fails to cross the third obstacle.
f . 36 „. 116 . . 35 ... 143(a) — (b) --- (c) —— (d) -—243 216 243 243
249Probability
& Exercise-4 : Matching Type Problems S- -
Column-1
(A)
The number of elements in P is more than that in Q is(B) (Q)
(C) P n Q = Q;C->S;D->S
(T) I
1. A is a set containing n elements, A subset P (may be void also) is selected at random from set A 
and the set A is then reconstructed by replacing the elements of P. A subset Q (may be void also) 
of A is again chosen at random. The probability that
-A—-
Number of elements in P is equal to the number of elements in Q (P) 
is
Column-ll
2n
4"
,2n+l
(R)
H
(22n
2
CH4-1
4"
250 Advanced Problems in Mathematics for JEE
Exercise-5 : Subjective Type Problems
is, where [ ] denotes greatest integer function :
(p + q).
o o o m4. If a, b, c e N, the probability that a + b + c is divisible by 7 is — where m, n are relatively 
n
Find the unit digit of p.
3. Two hunters A and B set out to hunt ducks. Each of them hits as often as he misses when 
shooting at ducks. Hunter A shoots at 50 ducks and hunter B shoots at 51 ducks. The probability 
that B bags more ducks than A can be expressed as — in its lowest form. Find the value of 
q
1. Mr. A writes an article. The article originally is error free. Each day Mr. B introduces one new
2
error into the article. At the end of the day, Mr. A checks the article and has - chance of catching 
each individual error still in the article. After 3 days, the probability that the article is error free 
can be expressed as — where p and q are relatively prime positive integers. Let X = q - p, then 
q
find the sum of the digits of X.
2. India and Australia play a series of 7 one-day matches. Each team has equal probability of 
winning a match. No match ends in a draw. If the probability that India wins atleast three 
consecutive matches can be expressed as — where p and q are relatively prime positive integers.
q
prime natural numbers, then m + n is equal to :
5. A fair coin is tossed 10 times. If the probability that heads never occur on consecutive tosses be
— (where m, n are coprime and m, n g N), then the value of (n -7rri) equals to : 
n
6. A bag contains 2 red, 3 green and 4 black balls. 3 balls are drawn randomly and exactly 2 of 
them are found to be red. If p denotes the chance that one of the three balls drawn is green; find 
the value of 7p.
7. There are 3 different pairs (i.e., 6 units say a, a, b, b, c, c) of shoes in a lot. Now three person 
come and pick the shoes randomly (each gets 2 units). Let p be the probability that no one is 
able to wear shoes (i.e., no one gets a correct pair), then the value of , is :
4-p
8. A fair coin is tossed 12 times. If the probability that two heads do not occur consecutively is p,
, i f EV4096p -1]then the value of —-------------
2
9. X and Y are two weak students in mathematics and their chances of solving a problem correctly
are 1/8 and 1/12 respectively. They are given a question and they obtain the same answer. If 
the probability of common mistake is t^ien probability that the answer was correct is a / b
(a and b are coprimes). Then | a - b | =
251Probability
3 Answers 5
6.1 3 7. 28 5.3 4.3.1. 7 2. 7
77 12.78. 9 1 10.9.
□□□
the value of 5p - q.
12. Mr. B has two fair 6-sided dice, one whose faces are numbered 1 to 6 and the second whose 
faces are numbered 3 to 8. Twice, he randomly picks one of dice (each dice equally likely) and 
rolls it. Given the sum of the resulting two rolls is 9. The probability he rolled same dice twice is 
— where m and n are relatively prime positive integers. Find (m + n) .
n
10. Seven digit numbers are formed using digits 1, 2, 3, 4, 5, 6, 7, 8, 9 without repetition. The 
probability of selecting a number such that product of any 5 consecutive digits is divisible by 
either 5 or 7 is P. Then 12P is equal to
11. Assume that for every person the probability that he has exactly one child, excactly 2 children 
and exactly 3 children are —, and i respectively. The probability that a person will have 4
grand children can be expressed as — where p and q are relatively prime positive integers. Find 
9
Logarithms
y& Exercise-1: Single Choice Problems
ZZ-
(C)
(c) -5/2 (d) 5/2
(b) 2
5
(a) 1
5. Let P =
(d) 21 
(a2-3a + 3) > 0; (x > 0):1 x+—
(c) 3 (d) 4
and (120)p = 32, then the value of x be :1111 ---- +---- +-----+---- log2x log3x log4x log5x
(a) 1 (b) 2 (c) 3 (d) 4
2
6. If x, y, z be positive real numbers such that log^ (z) = 3, log5y (z) = 6 and log^ (z) = - then 
the value of z is :
1 13 4(a) i (b) -L (c) (d) Z
5 10 5 9
7. Sum of values of x and y satisfying logx(log3(logx y)) = Oand logy 27 = 1 is :
(a) TJ (b) 30 (c) 33 (d) 36
8. log001 1000+ log0j 0.0001 is equal to :
(a) -2 (b) 3
1. Solution set of the in equality loglo2 x - 3 (log10 x) (log10 (x - 2)) + 2 log]02 (x - 2) 0 is equal to : 
(a)
4 (log 2 17)
3 (log 3 23)
(b) 3| -
(3 + a
3 5^ ^^ kj(2, oo)
2 3j
9. If logI2 27 = a, then log6 16 =
(a) 2f—
\3 + a
10. If log2(log2(log3 x)) = log2(log3 (log2y)) = Othen the value of (x + y) is :
(a) 17 (b) 9 (c) 21 (d) 19
7
11. Suppose that a and b are positive real numbers such that log27 a + Iog9 b = - and
2
log 27 b + log 9 a = —. Then the value of a • b is :
, X .( 3-a
(c) 4 ~-----(3 + a
^,-f|u(2, oo) 
3 3)
|o(2, oo)
3 3)
(a) 81 (b) 243 (c) 27
12. If 2a = 5, 5b = 8, 8C = 11 and lld = 14, then the value of 2abcd
(a) 1 (b) 2 (c) 7 (d) 14
13. Which of the following conditions necessarily imply that the real number x is rational ?
(I) x2 is rational (II) x3 and x5 are rational (III) x2 and x3 are rational
(a) I and II only (b) I and III only (c) II and III only
14. The value of log8 17 - 1Og2^ 17
log9 23 log 3 23
|u(2, oo)
3 2J
16. If P is the number of natural numbers whose logarithm to the base 10 have the characteristic p 
and Q is the number of natural numbers logarithm of whose reciprocals to the base 10 have the 
characteristic -q then log10 P - log10 Q has the value equal to :
(a) p-q + 1 (b) p-q (c) p + q-1
17. If 22010 = an10" +an_110n-1 + + a2102+a1 -10 + a0, 
for all i =0, 1, 2, 3, , n, then n =
(a) 603 (b) 604 (c) 605 (d) 606
18. The number of zeros after decimal before the start of any significant digit in the number 
N = (O.15)20 are :
(a) 15 (b) 16 (c) 17
19. log2[log4(log10 164 + log10 258)] simplifies to :
(a) an irrational (b) an odd prime
(c) a composite (d) unity
20. The sum of all the solutions to the equation 2 log x - log (2x - 75) = 2:
(a) 30 (b) 350 (c) 75 (d) 200
7
2
254 Advanced Problems in Mathematics for JEE
(d) 3
(d) 8(b) 1 (c) 9
(d) ~(b) 3 (c)
(d) 4
48Then the value of (1 + x) is :
(d) 625
(a) 2
30. Let n g N, f(n) otherwise
(c) 6(a) 2011
31. If the equation = 0 has a solution for ‘x ’ when cof x satisfying the equation
^2X ^4x(0.125)1/x = 4(^2) is :
1
5
(d) x2
is equal to :
(b) y
22. Number of solution(s) of the equation xXy^
(a) 0 (b) 1
23. The difference of roots of the equation (log 27 x3 )2 = log 27 x6 is :
wi
24. If logj0 x + log10 y = 2, x-y = 15 then :
(a) (x, y) lies on the line y = 4x + 3
(c) (x,y)lies onx = 4y
(d) 
w
21 xloSxalo8ay4o&yz
(a) x
3
(b) 4 (c) 6
 f log 8 n if log 8 n is integer
| 0 otherwise
r 14(a) T ■
26. Sum of all values of x satisfying the equation
25(2x-x2+1) +9(2x-x2+1) _ 34(15(2r-*2)) js .
(a) 1 (b) 2 (c) 3
27. Ifax =by =c2 = dw, then loga (bed) =
r > fl 1 1(a) z — + — + — 
(x y w
• 428. If x = —=--------=------- -=———7=----- .
(V5 +1) (^5 +1) ( ^5 + 1) 0^5 +1)
(a) 5 (b) 25 (c) 125
29. If logx log18 (V2 + V8) =—, then the value of 32x =
3
5
255Logarithm
(d)
(d) 15(b) 11 (c) 13
the digits of M is :
CO 100
(d) none of theseCO x-y
(d) log 3CO
(d) 81
(d) l-31og27
(d) 30
is equal to :
Cd) 2(0 -1
■
l
32. If log Q 3 (x -1) y (b) x x2, then
(c) 2x1 = x% (d) x2 = xf
49. Let M denote antilog 32 0.6 and N denote the value of 490-1087 + 5-logs4. Then M.N is :
(a) 100 (b) 400 (c) 50 (d) 200
50. If log2(log2(log3 x)) = log3(log3(log2y)) = 0, then x-y is equal to :
(d) 9
(c) (30)1/8
(a) log1/2 2 (b) log 2 5 (c) log22
52. If log4 5 = a and log5 6 = b, then log3 2 is equal to :
(a) -J— (b) -A- (c) 2ab + l
2a +1 2b + 1
53. If x = loga bc;y = logb ac and z = logc ab then which of the following is equal to unity ?
(a) x + y + z
, . 1 1 1(c) ------+-------+------
1+x 1+y 1 + z
54. xlo^alogarlo^zisequalto:
(a) x (b) y (c) z (d) a
55. Number of value(s) of ‘ x’ satisfying the equation y108'/*3) = 9 is/are
(c) 2
45. The solution set of the equation : logx 21og2x 2 = log4x 2 is :
(a) {2-^,2^} (b) {1/2,2} (c) {1/4,22}
46. The least value of the expression 21og10 x - logx 0.01 is (x > 1)
(a) 2 (b) 4 (c) 6
47. If y]\og2x ~ 0-5 = log 2 Vx, then x equals to :
(a) odd integer
(c) composite number
48. If x2 and x2 are the roots of the equation e2xlnx
(a) Xj =2x2 (b) Xj = x2
1
1 + z
5
2
16
15
(a) 21/8
25 . 81
24 uu80
(b) (10)1/8
257Logarithm
(c)
equals to :
(c)
(d) 11
61.
(c)
(d) 6
(d)(c) 2k K
, the value of — + — + - is equal to :
(d) 3(c) 2
(d) 3
(a) 2 (0
13
12
15
6
59. The value of (—
IV27
(b) logs7
(c) 2
I */n 1
66. The value of logab — , if ldgab a = 4is equal to :
o
58. log8(128)-log9 cot^J =
. , 31(a) —
12
™ 19 ® F2 
logs16) 
21ogs9j
(c) log 3^ V3 > log 3^2 V5
68. Sum of all values of x satisfying the system of equations 5(logy x + logx y) = 26, ay = 64 is :
(a) 42 (b) 34 (c) 32 (d) 2
( > 5V2 ,,. V2 . . 4V2 .2V2(a) -- (b) — (c) --- (d) --27 27 27 27
60. The sum of all the roots of the Equation log 2(x -1) + log 2 (x + 2) - log 2(3x -1) = log 2 4
(a) 12 (b) 2 (c) 10
(log100 10)(log2(log4 2))(log4 log2(256)2) 
log4 8 + log8 4
,k+5"x
(b) 
6
67. Identify the correct option
(a) log 2 3 7 777196
a = logx/3 a is :
Cd) 27
= i + log2(x + 7)is :
(a) 0 (b) 1 (c) 2
73. If logfc xlogs k = logx 5, k * 1, k > 0, then sum of all values of x is :
, . c 24 / 26 ,37Ca) 5 (b) — Cc) — Cd) —o o o
74. The product of all values of x satisfying the equation | x -11log3 x "21og* 9 = (x -1)7, is :
„ . 162 , . 81Cb) (c) -=
73 -J3
75. The number of values of x satisfying the equation log2(9x-1
(a) 1 (b) 2 (c) 3
76. Which is the correct order for a given number a, a > 1
(a) log2 athen x can be rational.
(b) If a rational and b irrational then x can be rational.
(c) If a irrational and b rational then x can be rational.
(d) If a rational and b rational then x can be rational.
3. Consider the quadratic equation, (log10 8)x2 -(log10 5)x = 2(log210) 
following quantities are irrational ?
(a) Sum of the roots (b) Product of the roots
(c) Sum of the coefficients (d) Discriminant
4. Let A = Minimum (x2 - 2x + 7), x e R and B = Minimum (x2 - 2x + 7), x e [2, 00), then :
(a) log(B_X) (A+ B) is not defined (b) A + B=13
(c) log(2B-A) A 1
261Logarithm
Exercise-3 : Comprehension Type Problems
Let
(d) 47
(d) ±109(c) ±110
(d) ±110(c) ±2
Cd) 199(a) 31 (b) 28
7. The value of sin
(d) 1(c)
1. (C) 3. (d)2. (a) 4, Cc) 5. (b) 6.
73
2
5 Answers !
(a) I
\(25p2A1 +2)
6
(c) 3 + 4-
72
(b) 4=72
Paragraph for Question Nos. 1 to 3
log3N = 04
log5N = a2 + 02
log7 N = a3 +p3
where 04, a2 and a3 are integers and ft, 02, 03 e [0, 1).
1. Number of integral values of N if 04 = 4 and a2 = 2:
(a) 46 (b) 45 (c) 44
2. Largest integral value of N if 04 = 5, a2 = 3 and a3 = 2.
(a) 342 (b) 343 (c) 243 (d) 242
3. Difference of largest and smallest integral values of N if 04 = 5, a2 = 3 and a3 = 2.
(a) 97 (b) 100 (c) 98 - (d) 99
4A2 .
6. Aj + —- is equal to : 
7t
(a) I
Paragraph for Question Nos. 6 to 7
Given a right triangle ABC right angled at C and whose legs are given 1 + 41ogp2 (2p), 
1 + 2log2(log2p5 and hypotenuse is given to be l + log2(4p). The area of A ABC and circle
circumscribing it are A} and A2 respectively, then
7-|w|
Paragraph for Question Nos. 4 to 5
If logioIX3 + y 31 - log101x2 - xy + y 2| + log101 x3 -y 31 - log101 x2 + xy + y 2| = log10 221.
Where x, y are integers, then
4. If x = 111, then y can be :
(a) ±111 Cb) ±2
5. If y = 2, then value of x can be :
(a) ±111 (b) ±15
262 Advanced Problems in Mathematics for JEE
■FExercise-4 : Matching Type Problems Z-z'
Column-ll
(P) -1(A)
(Q) 1(B)
(R) 2(C)
(S)(D)
None of these(T)
5 Answers 5
3
2
__________________ __ Column-1_____________________ \
If a = 3(^8+2^7 - V8-2V7 ), b = ^(42) (30) + 36, then the 
value of loga b is equal to
If a = G/4+2-/3 -74-2V3), b = 711 + 6^2 -V11-6V2 then 
the value of loga b is equal to
If a = 5/3+ 2V2, b = V3-2V2, then the value of loga b is equal 
to
If a = \7 + a/?2 -1, b =\7 - 7? 2 -1, then the value of loga b is 
equal to
Logarithm 263
z v
, then
3log2x-12(xlog169) = log3
log(i +x }(l-2y+y2) + loga_y)(l + 2x + .
12. If logb n = 2 and log n(2b) = 2, then nb =
13. If logy x + logx y =2, and x2 + y = 12, then the value of xy is:
14. If x, y satisfy the equation, yx = xy and x = 2y, then x2 + y 2 =
15. Find the number of real values of x satisfying the equation.
log2(4x+1 + 4) • log2(4x +1)
x+5'
X+2,
110) = 9iog100*+io84 2 (where a > Oj a * 1), then
16. If Xj, x2(xj > x2) are the two solutions of the equation
/ \33 
f 1 1I— I , then the value of xT - 2x2 is :
2
I rJ
terms of a H.R for k e N, then:
7t 7t
3* 2
1. The ratio in which the line segment joining (2, - 3) and (5, 6) is divided by the x-axis is :
(a) 3:1 (b) 1:2
(c) V3 : 2 (d) V2 : 3
2. If L is the line whose equation is ax + by =c. Let M be the reflection of L through they-axis, and 
let N be the reflection of L through the x-axis. Which of the following must be true about M and 
N for all choices of a, b and c ?
(a) The x-intercepts of M and N are equal
(b) The y-intercepts of M and N are equal
(c) The slopes of M and N are equal
(d) The slopes of M and N are reciprocal
3. The complete set of real values of ‘a’ such that the point Pfa, sin a) lies inside the triangle 
formed by the lines x-2y + 2 = 0; x + y - Oand x-y -n = 0, is :
(a) (n (2n o 
\2 J I 2 J 
( it n'l 
U’ 2j
4. Let m be a positive integer and let the lines 13x + Uy = 700 and y = mx -1 intersect in a point 
whose coordinates are integer. Then m equals to : 
(a) 4 (b) 5 (c) 6
°--6r
X / \
—,p ;Q = — ,q = K J
where xk * 0, denotes the k * i
1
xp
268 Advanced Problems in Mathematics for JEE
(a) ar. (APQR) =
(d) g(l) + g(3)
2 2 2pgr
2
(b) APQR is a right angled triangle
(c) the points P, Q, R are collinear
(d) None of these
6. If the sum of the slopes of the lines given by x2 -2cxy -7y 2 = 0 is four times their product, 
then c has the value :
(a) 1 (b) -1 (c) 2 (d) -2
7. Apiece of cheese is located at (12,10) in a coordinate plane. A mouse is at (4,-2) and is running 
up the line y = -5x + 18. At the point (a, b), the mouse starts getting farther from the cheese 
rather than closer to it. The value of (a + b) is:
(a) 6 (b) 10
(c) 18 (d) 14
8. The vertex of right angle of a right angled triangle lies on the straight line 2x + y -10 = 0 and 
the two other vertices, at points (2, -3) and (4, 1) then the area of triangle in sq. units is:
i— 33(a) 710 (b) 3 . (c) y (d) 11
9. Given the family of lines, a(2x + y + 4) + b(x - 2y - 3) = 0. Among the lines of the family, the 
number of lines situated at a distance of 710 from the point M(2,-3)) is:
(a) 0 (b) 1
(c) 2 (d) oo
10. Point (0, p) lies on or inside the triangle formed by the lines y=0, x + y=8 and 
3x - 4y + 12 = 0. Then p can be :
(a) 2 (b) 4 (c) 8 (d) 12
11. If the lines x + y + l = 0j4x + 3y + 4 = 0 and x + ay + P = 0, where a2 + p2 = 2, are concurrent 
then:
(a) a = 1, p = —1 (b) a = 1, P = ±1
(c) a=-l,p = ±l (d) a = ±l,p = l
12. A straight line through the origin ‘O’ meets the parallel lines 4x + 2y = 9 and 2x + y = -6 at 
points P and Q respectively. Then the point ‘O’ divides the segment PQ in the ratio :
(a) 1:2 (b) 4:3 (c) 2:1 (d) 3:4
13. If the points (2a, a), (a, 2a) and (a, a) enclose a triangle of area 72 units, then co-ordinates of 
the centroid of the triangle may be :
(a) (4,4) (b) (-4,4) (c) (12,12) (d) (16,16)
14. Let g(x) = ax + b, where ais :
(a) ax + by=0 (b) ax-by=0 (c) bx+ay=O (d) x-y=O
16. If the equation 4y 3 -8a2yx2 -3ay 2x+ 8x3 = 0 represent three straight lines, two of them are 
perpendicular then sum of all possible values of a is equal to :
, . 3 ... —3 . . 1(a) - (b) — (c) -
8 4 4
17. The orthocentre of the triangle formed by the lines x-ly + 6 = 0, 2x-5y -6 = 0 and 
7x + y - 8 = 0 is :
Then y-intercept of the line is : 
3
(a) -- (b) 9
4
23. The combined equation of two adjacent sides of a rhombus formed in first quadrant is 
7x2 - 8>y + y 2 =0; then slope of its longer diagonal is :
(a) -i (b) -2 (c) 2 (d) 1
24. The number of integral points inside the triangle made by the line 3x + 4y -12 = 0 with the 
coordinate axes which are equidistant from at least two sides is/are :
(an integral point is a point both of whose coordinates are integers.)
(a) 1 (b) 2 (c) 3 (d) 4
, 3t3
22. The equations x = t +9 and y =-----+ 6 represents a straight line where t is a parameter.
4
(a) (8,2) (b) (0,0)
18. All the chords of the curve 2x2 + 3y 2
concurrent at:
(a) (0,1) (b) (1,0) (c) (-1,1) (d) (1,-1)
19. From a point P = (3, 4) perpendiculars PQ and PR are drawn to line 3x + 4y -7 = 0 and a 
variable line y -1 = m (x - 7) respectively, then maximum area of &PQR is :
(a) 10 (b) 12 (c) 6 (d) 9
20. The equation of two adjacent sides of rhombus are given by y = x and y = 7x. The diagonals of 
the rhombus intersect each other at the point (1, 2). Then the area of the rhombus is :
, , 10 „ , 20 . . 40 ... 50(a) — (b) — (c) — (d) —
3 3 3 3
21. The point P(3, 3) is reflected across the line y = -x. Then it is translated horizontally 3 units to 
the left and vertically 3 units up. Finally, it is reflected across the line y = x. What are the 
coordinates of the point after these transformations ?
(a) (0,-6) (b) (0,0)
(c) (-6,6) (d) (-6,0)
270 Advanced Problems in Mathematics forJEE
(c)
(b) 3 (c)(a) 4
through a fixed point, co-ordinate of fixed point is :
(b) (-1,-2) (c) (1,-2)(a) (-1,2)
P- (d) -
8 2
29. The vertices of triangle ABC are A(-l, -7), B(5,1) and C(l, 4). The equation of the bisector of 
the angle ABC of AABC is :
(a) y + 2x-ll = 0 (b) x-7y + 2 = 0
(c) y-2x+9 = 0 (d) y+7x-36 = 0
30. If one of the lines given by 6x2 - xy + 4cy 2 = 0 is 3x + 4y = 0, then c =
(a) -3 (b) -1 (c) 3 (d) 1
31. The equations ofLj and L2 are y = mx andy = nx, respectively. Suppose Lj make twice as large
of an angle with the horizontal (measured counterclockwise from the positive x-axis) as does L2 
and thatLj has 4 times the slope ofL2. If Lj is not horizontal, then the value of the product (mn) 
equals:
(a) (b)
2 2
(c) 2 (d) -2
32. Given A (0, 0) and B (x, y ) with x g (0,1) and y > 0. Let the slope of the line AB equals m1. Point
C lies on the line x = 1 such that the slope of BC equals m2 where 0 0, then the line whose equation is y = mx + b cannot 
contain the point:
(a) (0,2008) (b) (2008,0)
(c) (0,-2008)(d) (20,-100)
55. The number of possible straight lines, passing through (2, 3) and forming a triangle with 
coordinate axes, whose area is 12 sq. units, is:
(a) one (b) two
(c) three (d) four
56. If x1, x2, x3 and y1, y 2, y 3 are both in G.R with the same common ratio then the points (xt, yj), 
U2>J2)and (x3,y3)
(a) lie on a straight line (b) lie on a circle
(c) are vertices of a triangle (d) None of these
57. Locus of centroid of the triangle whose vertices are (a cost, asint), (bsint, -b cost) and (1,0); 
where t is a parameter is :
(a) (3x-l)2 + (3y)2 = a2-b2 (b) (3x-l)2 + (3y)2 = a2 + b2
(c) (3x +l)2 + (3y)2 = a2 + b2 (d) (3x +1)2 + (3y)2 = a2 -b2
58. The equation of the straight line passing through (4, 3) and making intercepts on co-ordinate
axes whose sum is -1 is : 
, v x y , , x y ,(a) — + — = -1 and — + — = -1
2 3 -2 1
r \ x y - 0 (b) 2x + y +1 > 0
(c) -2x + ll>0 (d) 2x + 3y-12>0
11. The slope of a median, drawn from the vertex A of the triangle ABC is -2. The co-ordinates of 
vertices B and C are respectively (-1,3) and (3,5). If the area of the triangle be 5 square units, 
then possible distance of vertex A from the origin is/are.
(a) 6 (b) 4 (c) 2V2 (d) 3^2
12. The points A(0,0), B(cos a, sin a) and C(cos0, sin 0) are the vertices of a right angled triangle if:
1
/2
/2
q-P
2
q-P
2
q-P]_
2 J
(c) cos ----- - =I 2 J
277Straight lines
•fExercise-3 : Comprehension Type Problems
Cd) 5y-l = 0
(d) 8(c) 6
J Answers J
1. Cd) 4. (d)3. (b)2. (a)
Paragraph for Question Nos. 1 to 2
The equations of the sides AB and CA of a A ABC are x + 2y = 0 and x -y = 3 respectively. 
Given a fixed point P(2,3).
Paragraph for Question Nos. 3 to 4
Consider a triangle ABC with vertex A (2, - 4). The internal bisectors of the angles B and C are 
x + y = 2 and x-3y =6 respectively. Let the two bisectors meet at I.
3. If (a, b) is incentre of the triangle ABC then (a + b) has the value equal to :
(a) 1 (b) 2 (c) 3 (d) 4
4. If (xj, y!) and (x2, y 2)are co-ordinates of the point B and C respectively, then the value of
(XjX2 +yiy2) is equal to : 
(a) 4 (b) 5
1. Let the equation of BC isx + py = q. Then the value of (p + q) if P be the centroid of the A ABC is:
(a) 14 (b) -14 (c) 22 (d) -22
2. If P be the orthocentre of A ABC then equation of side BC is :
(a) y + 5 = 0 (b) y - 5 = 0 (c) 5y +1 = 0
278 Advanced Problems in Mathematics for JEE
,7Exercise-4 ; Matching Type Problems
Column-1 Column-ll
(A) (P) (-4, -7)
(B) (-7, 11)(Q)
(C) (R) Cl, -2)
(D) (S) (-1, 2)
(T) (0, 0)
2.
Column-ll
n+1(A) (P) 1= 30, then n is equal to
(B) (Q) 4
(C) (R) 7
(D) (S) 10
3.
Column-1
(A) Exact value of cos 40° (1 - 2 sin 10°) = (P)
A point on the line x + y = 4 which lies at a unit distance 
from the line 4x + 3y = 10 is :
Orthocentre of triangle made by lines x + y =1, 
x-y + 3 = 0, 2x + y = 7 is
Two vertice of a triangle are (5, -1) and (-2, 3). If 
orthocentre is the origin then coordinates of the third 
vertex are
Column-ll
1
4
Column-1
Number of solutions of the equation 
sin 9x + sin 5x + 2 sin 2 x = 1 in interval (0, rc) is
If the roots of the equation x2 + ax + b = 0 
(a,b gR) are tan 65° and tan 70°, then (a + b) 
equals.
If a,b,c are in A.R, then lines ax+by + c = 0 are 
concurrent at:
, n
k=l
The number of integral values of g for which 
atmost one member of the family of lines given by 
(1 + 2X)x + (1-X)y + 2 +4A. = 0 (X is real 
parameter) is tangent to the circle 
x2 +y 2 + 4gx + 18x + 17y + 4g2 = Ocan be
Straight lines 279
(B) (Q)
(C) (R)
(D) (S)
] Answers |TTy............. ......
1» A —> R; B —> Q; C —> Sj D —> P
2. A -» Q; B —> R; C —> S; D —> P
3. A-> Q; B—>R; C-»S; D->P
3
2
1
2
1
2
Points (k,2-2k),(-k + l,2k) and (-4-k,6-2Jc) 
are collinear then sum of all possible real values of 
‘k’ is
Value of X for which lines are concurrent 
x + y + 1 = 0, 3x + 2Xy + 4 = 0, x +.y - 3X = 0 can 
be
00 
Value of y^sin^ 
k=3
Advanced Problems in Mathematics for JEE280
f XExercise-5 : Subjective Type Problems
2
| Answers
86 3. 1 0 6. 45. 32. 1 7.
6 58. 3 9. 10.□□□
for all permissible values of X] 
c.
1. If the area of the quadrilateral ABCD whose vertices are A(l, 1), B(7, -3), C(12, 2)andD(7, 21) 
is A. Find the sum of the digits of A.
2. The equation of a line through the mid-point of the sides AB and AD of rhombus ABCD, whose 
one diagonal is 3x - 4y + 5 = 0 and one vertex is A (3,1) is ax + by + c = 0. Find the absolute 
value of (a + b + c) where a, b, c are integers expressed in lowest form.
3. If the point (a, a4) lies on or inside the triangle formed by lines x2y + xy 2 - 2xy = 0, then the 
largest value of a is.
4. The minimum value of [(xj - x2)2 + (12--J1- x{
and x2 is equal to ajb -c where a, b, c e N, then find the value of a + b -
5. The number of lines that can be drawn passing through point (2, 3) so that its perpendicular 
distance from (-1, 6) is equal to 6 is :
6. The graph of x4 = x2y 2 is a union of n different lines, then the value of n is.
7. The orthocentre of triangle formed by lines x + y -1 = 0, 2x + y- l = Oandy = Ois (h, Jc),then
1
k2 ~
8. Find the integral value of a for which the point (-2, a) lies in the interior of the triangle formed 
by the lines y = x, y = -x and 2x + 3y = 6.
9. Let A = (-1, 0), B = (3, 0) and PQ be any line passing through (4,1). The range of the slope of 
PQ for which there are two points on PQ at which AB subtends a right angle is , X2), then
+ X2) is equal to.
10. Given that the three points where the curve y = bx2 - 2intersects the x-axis and y-axis form an 
equilateral triangle. Find the value of 2b.
-V4^)2]1/2
7[1S
Circle
2
(b) x2+y2-5x-3y + 6 = 0
(d) None of these
Exercise-1 : Single Choice Problems
1. The locus of mid-points of the chords of the circle x2 - 2x + y 2 - 2y + 1 = 0 which are of unit 
length is :
(a) (x -1)2 + (y-l)2 = — (b) (x-1)2 + (y-I)2 = 2
4
(c) (x-1)2 + (y-I)2 = i (d) (x-l)2 + (y-l)2=-?
4 3
2. The length of a common internal tangent to two circles is 5 and a common external tangent is 
15, then the product of the radii of the two circles is :
(a) 25 (b) 50 (c) 75 (d) 30
3. A circle with center (2, 2) touches the coordinate axes and a straight line AB where A and B lie
on positive direction of coordinate axes such that the circle lies between origin and the line AB. 
If O be the origin then the locus of circumcenter of AO AB will be: 
(a) xy = x + y + Jx2 + y2 (b) xy = x + y -Jx2 +y
(c) xy + x + y = yjx2 + y2 (d) xy + x + y + ^x2 + y2 = 0
4. Length of chord of contact of point (4,4) with respect to the circle x2 + y2 - 2x-2y -7 = 0 is:
(a) A (b) 3V2 (c) 3 (d) 6
V2
5. Let P, Q, R, S be the feet of the perpendiculars drawn from a point (1,1) upon the lines 
x + 4y =12; x-4y + 4 = 0 and their angle bisectors respectively; then equation of the circle 
which passes through Q, R, S is : 
(a) x2 + y 2 - 5x + 3y - 6 = 0 ■
(c) x2 + y 2 - 5x - 3y - 6 = 0
282 Advanced Problems in Mathematics for JEE
Cd)(c) 2
3
(b) x2 + y2-2x-2y-1 = 0
(d) None of these
$
6. From a point ‘P ’ on the line 2x + y + 4 = 0; which is nearest to the circle x2 + y 2 - 12y + 35 = 0, 
tangents are drawn to given circle. The area of quadrilateral PACB (where ‘C’ is the center of 
circle and PA & PB are the tangents.) is :
(a) 8 (b) a/110 (c) V19 (d) None of these
7. The line 2x-y + l = 0is tangent to the circle at the point (2, 5) and the centre of the circles lies 
on x - 2y =4. The radius of the circle is:
(a) 3V5 (b) 5V3
(c) 2V5 (d) 5J2
8. If A(cosa, sin a), B(sina, - cos a), C(l, 2) are the vertices of a triangle, then as a varies the 
locus of centroid of the AABC is a circle whose radius is :
(a) v i/I
9. Tangents drawn to circle (x -1)2 + (y -1)2 = 5 at point P meets the line 2x + y + 6 = 0 at Q on 
the x-axis. Length PQ is equal to :
(a) V12 (b) V10 (c) 4 (d) V15
10. ABCD is square in which A lies on positive y-axis and B lies on the positive x-axis. If D is the point 
(12,17), then co-ordinate of C is :
(a) (17,12) (b) (17,5) (c) (17,16) (d) (15,3)
11. Statement-1: The lines y =mx + l-m for all values of m is a normal to the circle 
x2 +y2 — 2x — 2y =0.
Statement-2: The line L passes through the centre of the circle.
(a) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for 
statement-1.
(b) Statement-1 is true, statement-2 is true and statement-2 is not the correct explanation for 
statement-1.
(c) Statement-1 is true, statement-2 is false.
(d) Statement-1 is false, statement-2 is true.
12. A(l, 0) and B(0,1) are two fixed points on the circle x2 + y 2 = l.C is a variable point on this 
circle. As C moves, the locus of the orthocentre of the triangle ABC is :
(a) x2+y2-2x-2y+1 = 0 (b) x2+y2-x-y=0
(c) x2 + y 2 = 4 (d) x2 + y 2 + 2x - 2y +1 = 0
13. Equation of a circle passing through (1,2) and (2,1) and for which line x + y = 2 is a diameter; 
is:
(a) x2 +y2 + 2x + 2y-11 = 0
(c) x2 + y 2 - 2x - 2y +1 = 0
4
3
Circle 283
2 2
(b) lOx/5
(d) 10+571
(b) 3x + 4y=45
(d) 3x + 4y=12
x2 +y2 -10x + 9 = 0, x2+y2-6x + 2y+ 1 = 0,
(b) 3x + 5y = 7
(d) 3x + 5y = 1
14. The area of an equilateral triangle inscribed in a circle of radius 4 cm, is :
(a) 12 cm2 (b) 9^3 cm2
(c) 8^3 cm2 (d) 12V3 cm
15. Let all the points on the curve x2 + y 2 - lOx = 0 are reflected about the line y = x + 3. The 
locus of the reflected points is in the form x2 + y 2 + gx + jy + c = 0. The value of (g + f + c) is 
equal to :
(a) 28 (b) -28 (c) 38 (d) -38
16. The shortest distance from the line 3x + 4y = 25 to the circle x2 + y 2 = 6x - 8y is equal to:
(a) 7/5 (b) 9/5 (c) 11/5 (d) 3^/5
17. In the xy-plane, the length of the shortest path from (0, 0) to (12, 16) that does not go inside 
the circle (x -6)2 + (y - 8)2 =25 is:
(a) 10>/3
(c) 10V3 + —
3
18. A circle is inscribed in an equilateral triangle with side lengths 6 unit. Another circle is drawn 
inside the triangle (but outside the first circle), tangent to the first circle and two of the sides of 
the triangle. The radius of the smaller circle is:
(a) 1/73 (b) 2/3
(c) 1/2 (d) 1
19. The equation of the tangent to the circle x2 + y 2 - 4x = 0 which is perpendicular to the normal 
drawn through the origin can be :
(a) x = 1 (b) x = 2 (c) x + y = 2 (d) x = 4
20. The equation of the line parallel to the line 3x + 4y =0 and touching the circle x2 + y 2 = 9in 
the first quadrant is : 
(a) 3x + 4y =15 
(c) 3x + 4y =9
21. The centres of the three circles 
x2+y2-9x-4y + 2 = 0
(a) lie on the straight line x - 2y = 5 (b) lie on circle x2 + y 2 = 25
(c) do not lie on straight line (d) lie on circle x2+y 2 + x + y-17 = 0
22. The equation of the diameter of the circle x2 + y 2 + 2x - 4y = 4 that is parallel to 3x + 5y = 4 
is:
(a) 3x + 5y = -7
(c) 3x + 5y = 9
284 Advanced Problems in Mathematics for JEE
(b)(a)
(d)(c)
or
(d) either (1, 0) or (-1, 0)or
(b)
(c) -J AB-AD (d)
(d) (2,3)
1
/2
1_
f2
AB-AD
AB + AD
( . ..(11 (a) either —, -
I 2 2
(b) either | ~| or 
IV2 V2J
AB-AD
Vab2-AD2
30. Radical centre of the circles drawn on the sides as a diameter of triangle formed by the lines 
3x - 4y + 6 = 0, x -y + 2 = 0 and 4x + 3y -17 = 0 is :
(a) (3,2) (b) (3,-2) (c) (2,-3)
, . . , { 1 1(c) either —=, —— 
IV2 V2
28. The circles x2 + y 2 + 6x + 6y = 0 and x2 + y 2 - 12x - 12y = 0 :
(a) cut orthogonally (b) touch each other internally
(c) intersect in two points (d) touch each other externally
29. In a right triangle ABC, right angled at A, on the leg AC as diameter, a semicircle is described.
The chord joining A with the point of intersection D of the hypotenuse and the semicircle, then 
the length AC equals to: 
. . AB-AD 
(a) ,----- -
7AB2 + AD2
23. There are two circles passing through points A(-l, 2) and B(2, 3) having radius -/5. Then the 
length of intercept on x-axis of the circle intersecting x-axis is :
(a) 2 (b) 3 (c) 4 (d) 5
24. A square OABC is formed by line pairs xy = 0 and xy +1 = x + y where' O'is the origin. A circle 
with centre C} inside the square is drawn to touch the line pair xy = 0 and another circle with 
centre C2 and radius twice that ofC1; is drawn to touch the circle Cj andthe other line pair. The 
radius of the circle with centre C2 is:
V2 . 2V2
■ r- r---------- (b) ------ 7=---------
V3(V2 +1) 3(72 + 1)
V2 (j) V2 + 1
3(V2 +1) 3x/2
5. The equation of the circle circumscribing the triangle formed by the points (3, 4), (1, 4) and 
(3, 2) is:
(a) 8x2 + 8y 2-16x-13y = 0 (b) x2+y 2-4x-8y + 19 = 0
(c) x2 + y 2 — 4x — 6y + 11 = 0 (d) x2+y2-6x-6y+ 17 = 0
26. The equation of the tangent to circle x2 + y 2 + 2gx + 2fy = Oat the origin is : ’
(a) fx + gy =0 (b) gx + jy = 0 (c) x = 0 (d) y = 0
27. The line y = x is tangent at (0, 0) to a circle of radius 1. The centre of the circle is :
1 _1
2’ 2
285Circle
(c)
2
(d) 3V2(C) 4V2
31. Statement-1: A circle can be inscribed in a quadrilateral whose sides are 3x-4y =0, 
3x - 4y = 5, 3x + 4y = 0 and 3x + 4y = 7.
Statement-2: A circle can be inscribed in a parallelogram if and only if it is a rhombus.
(a) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for 
statement-1.
(b) Statement-1 is true, statement-2 is true and statement-2 is not the correct explanation for 
statement-1.
(c) Statement-1 is true, statement-2 is false.
(d) Statement-1 is false, statement-2 is true.
32. Ifx = 3 is the chord of contact of the circle x 2 +y2 = 81, then the equation of the corresponding 
pair of tangents, is:
(a) x2-8y2 + 54x + 729 = 0 (b) x2 - 8y 2 - 54x + 729 = 0
(c) x2-8y2-54x-729 = 0 (d) x2-8y2 =729
33. The shortest distance from ±e line 3x + 4y = 25 to the circle x2 + y 2 = 6x - 8y is equal to :
7 9 11 7(a) 4 (b) £ (c) (d) i
3 5 5 5
34. The circle with equation x2 + y 2 =1 intersects the line y = 7x + 5 at two distinct points A and
B. Let C be the point at which the positive x-axis intersects the circle. The angle ACB is : 
. 4 . . ,4
(a) tan 1 - (b) cot 1 (-1) (c) tan 1 1 (d) cot-1 -
3 3
35. The abscissae of two points A and B are the roots of the equation x2 + 2ax-b2 = 0 and their 
ordinates are the roots of the equation x2 + 2px -q2 =0. The radius of the circle with AB as 
diameter is ::
(a) y]a2 + b2 + p2 + q2 (b) Ja2 + p
(c) yjb2 + q2 (d)
36. Let C be the circle of radius unity centred at the origin. If two positive numbers xx and x2 are 
such that the line passing through (xx,-l) and (x2,1) is tangent to C then:
(a) x1x2=l (b) xxx2=-l
(c) xx+x2=l (d) 4xxx2=l
37. A circle bisects the circumference of the circle x2 + y 2 + 2y - 3 = 0 and touches the line x = y 
at the point (1, 1). Its radius is :
3 9(a) 4 (b)
V2 y[2
38. The distance between the chords of contact of tangents to the circle x2 + y 2 + 2gx + 2jy + c = 0 
from the origin and the point (g, f) is:
a2 + b2 + p2 + l
2 2+ p +q
2
286 Advanced Problems in Mathematics for JEE
(b)
(c) (d)
(c)(a) (d) none0,
+ c
7
2-V2
4
Cb) I 0,-1=
I 2V2
(a) ^S2+/2
2
g2 + f2~c (d) Vg2+/2
2jg2 +f2 ' 27g2+ /
39. If the tangents AP and AQ are drawn from the point A(3,-1) to the circle 
x2 + y2 -3x + 2y -7 = OandC is the centre of circle, then the area of quadrilateral APCQ is:
(a) 9 (b) 4 (c) 2 (d) non-existent
40. Number of integral value(s) of k for which no tangent can be drawn from the point (k, Jc + 2) to 
the circle x2 + y 2 = 4 is :
(a) 0 (b) 1 (c) 2 (d) 3
41. If the length of the normal for each point on a curve is equal to the radius vector, then the curve:
(a) is a circle passing through origin
(b) is a circle having centre at origin and radius > 0
(c) is a circle having centre on x-axis and touching y-axis
(d) is a circle having centre on y-axis and touching x-axis
42. A circle of radius unity is centred at origin. Two particles start moving at the same time from the 
point (1,0) and move around the circle in opposite direction. One of the particle moves counter 
clockwise with constant speed v and the other moves clockwise with constant speed 3v. After 
leaving (1,0), the two particles meet first at a point £ and continue until they meet next at point 
Q. The coordinates of the point Q are:
(a) (1,0) (b) (0,1)
(c) (0,-1) (d) (-1,0)
43. A variable circle is drawn to touch the x-axis at the origin. The locus of the pole of the straight 
fine Ix + my + n = 0 w.r.t the variable circle has the equation:
(a) x(my-n)-Zy2 =0 (b) x(my + n)-ly2 = 0
(c) x(my - n) + ly 2 = 0 (d) none of these
44. The minimum length of the chord of the circle x 
through (1,0) is :
(a) 2 (b) 4 (c) 2V2 (d)
45. Three concentric circles of which the biggest is x2 + y 2 =1, have their radii in A.R If the line 
y = x +1 cuts all the circles in real and distinct points. The interval in which the common 
difference of the A.P will lie is:
2 + y 2 + 2x + 2y -7 = 0 which is passing
Circle 287
(a) (b)
(c) (d) None of these
(b) 1/4 (d) 16(c) 1
| Answers |
---- ----
6. 8. 9. 10.1. 7.2. 3. 4. 5.
16. 18. 19.17. 20.11. 15.12. 13. 14.
28.26. 29.21. 27. 30.22. 23. 24. 25.
36. 38. 39.37. 40.31. 35.32. 33. 34.
48.46. 47. 49.41. 50.43. 45.42. 44.
(a)
(a)
(c)
(d)
(b)
(b)
(a)
Cb)
(b)
(d)
(a)
(c)
(c)
(d)
(a)
Cb) 
(d) 
(c)
(c) 
Cb)
Cb)
(c)
(0
(a)
(c)
Cc)
(a)
Cb)
(a)
(c)
(a)
(c)
(c)
Cb)
Cb)
(a)
(d)
(d)
Cd)
(c)
Cd) 
(a) 
(d) 
CO 
(a)
Cb)
(a)
(d)
(b)
(c)
l + A-2-3
Vs Vs.
46. The locus of the point of intersection of the tangent to the circle x2 + y 2 = a2, which include an 
angle of 45° is the curve (x2 + y 2)2 = la2(x2 + y 2 - a2). The value of X is:
(a) 2 (b) 4
CO 8 (d) 16
47. A circle touches the line y = x at point (4,4) on it. The length of the chord on the line x + y = 0 is 
6V2. Then one of the possible equation of the circle is :
(a) x2 +y2 + x-y + 30 = 0 (b) x2+y 2 + 2x-18y + 32 = 0
(c) x2+y 2 + 2x +18y + 32 = 0 (d) x2+y2-2x-22y + 32 = 0
48. Point on the circle x2 + y 2 - 2x + 4y -4 = 0which is nearest to the liney = 2x +11 is :
1—^=,-2 + ^=
Vs Vs.
Vs Vs,
49. A foot of the normal from the point (4,3) to a circle is (2,1) and a diameter of the circle has the 
equation 2x - y - 2 = 0. Then the equation of the circle is:
(a) x2+y2-4y + 2 = 0 Cb) x2 + y2 -4y + 1 = 0
Cc) x2+y2 — 2x — 1 = 0 (d) x2+y2-2x + l = 0
50. If | a, - I | b, -1 (c, - I and | d, - | are four distinct points on a circle of radius 4 units then, abed is
\aj\bj\c) 25
(c) p > 27 (d) pq2
(c) p2 8q2
7. If a = max{(x + 2)2 + (y-3)2} and b = min{(x+ 2)2 + (y-3)2} where x,y satisfying 
x2 + y 2 + 8x - lOy - 40 = 0, then :
(a) a + b = 18 (b) a + b=178
289Circle
2
(d) (-6,-9)
I Answers}y - .T7 -
6-1 (b, d)(a, b)(a, b, c) 4.(a, c, d) 3.(a, b, c, d) 2.1.
(a, c)(b, d) 10.(a, c) 9.(b, d) 8.7. 1
—I--------5. (a, c, d)
8. The locus of pointsof intersection of the tangents to x2 + y2 = a2 at the extremeties of a chord 
of circle x2 + y 2 = a2 which touches the circle x2 + y 2 - 2ax = 0 is/are :
(a) y2 =a(a-2x) (b) x2=a(a-2y)
(c) x2+y2=(x-a)2 (d) x2+y2=(y-a)
9. A circle passes through the points (-1,1), (0,6) and (5,5). The point(s) on this circle, the
tangent(s) at which is/are parallel to the straight line joining the origin to its centre is/are 
(a) (1,-5) (b) (5,1) (c) (-5,-1) (d) (-1,5)
10. A square is inscribed in the circle x2 + y 2 - 2x + 4y - 93 = 0 with the sides parallel to the 
co-ordinate axes. The co-ordinate of the vertices are :
(a) (8,5) (b) (8,9) (c) (-6^5)
Advanced Problems in Mathematics forJEE290
Exercise-3 : Comprehension Type Problems
S2 = x2 + y 2 + 6x + y + 8 = 0,
(d) (-2,-1)(c) (-2,1)
(d) 2:3
(d) y =-x + 2
(a) (2,3) (c) (d) (3,2)
4. The chords in which the circle C cuts the members of the family 8 of circle passing through A 
and B are concurrent at:
Paragraph for Question Nos. 7 to 8
Let the diameter of a subset S of the plane be defined as the maximum of the distance 
between arbitrary pairs of points of S.
7. Let S = {(x,y): (y - x) 0, x2 + y 2 £ 2}, then the diameter of S is :
(a) 2 (b) 4 (c) V2 (d) 2V2
(d) (AP)2 + (BP)2
3^ 
’ 2
(b} 12,^
5. Equation of the member of the family of circles S that bisects the circumference of C is :
(a) x2+y2-5x-l = 0 (b) x2 + y2-5x+ 6y-1 = 0
(c) x2+y 2-5x-6y-1 = 0 (d) x2 +y 2 + 5x-6y -1 = 0
6. If O is the origin and P is the center of C, then absolute value of difference of the squares of the
lengths of the tangents from A and B to the circle C is equal to : 
(a) (AB)2 (b) (OP)2 (c) |(AP)2-(BP)2|
Paragraph for Question Nos. 1 to 3
Let each of the circles,
Si = x2 + y 2 + 4y —1 = 0,
S3 = x2+y2-4x-4y-37 = 0
touches the other two. LetP15 P2, P3 be the points of contact of Sx andS2, S2andS3, S3 andSj 
respectively and Ci, C2, C3 be the centres of Sj, S2, S3 respectively.
Paragraph for Question Nos. 4 to 6
Let A(3, 7) and B(6, 5) are two points. C : x2 + y 2 - 4x - 6y - 3 = 0 is a circle.
1. The co-ordinates ofPj are :
(a) (2,-1) (b) (2,1)
„ area (APiP2P3) . ,2. The ratio----------- —— is equal to :
area (ACjC2C3)
(a) 3:2 (b) 2 : 5 (c) 5 : 3
3. P2 and P3 are image of each other with respect to line :
(a) y = x + 1 (b) y = -x (c) y = x
291Circle
710 + 1275 y > 0, x2 + y2 0,(75-1) 
then the diameter of S is :
(a) -(75-1) (b) 3(75-1)
2
9. Circum centre of the triangle formed by Cx and two other lines which are at angle of 45° withCj 
and tangent to C 2 is :
(a) (1,1) (b) (0,0) (c) (-1,-1) (d) (2,2)
10. If S1; S2 and S3 are three circles congruent to C2 and touch both C2 and C2; then the area of 
triangle formed by joining centres of the circles S!,S2 and S3 is (in square units) 
(a) 2 (b) 4 (c) 8 (d) 16
(d) I 9. (b) 10. (c)
292 Advanced Problems in Mathematics for JEE
Exercise-4 : Matching Type Problems
1.
Column-ll
it/ 6(A) (P)
(Q)(B) n/ 4
(R) n/ 3(C)
(D) (S) 7t/ 2
(T) n
2.
Column-1
(A) (P)
(B) (4,-1)(Q)
(C) (-25, 50)(R)
(D) (S)
(-1, 2)(T)
-19 1
8 ’ 6
Two parallel tangents drawn to given circle are cut by a 
third tangent. The angle subtended by the portion of 
third tangent between the given tangents at the centre is
A chord is drawn joining the point of contact of tangents 
drawn from a point P to the circle. If the chord subtends 
an angle tt/2 at the centre then the angle included 
between the tangents at P is
If the straight line x - 2y +1 = 0 intersects the circle 
x2 + y 2 = 25 at the points P and Q, then the coordinate 
of the point of intersection of tangents drawn at P and Q 
to the circle is
The equation of three sides of a triangle are 
4x + 3y + 9 = 0, 2x + 3 = 0 and 3y - 4 = 0. The circum 
centre of the triangle is :
Column-ll
I 5 J
A ray of light coming from the point (1, 2) is reflected at 
a point A on the x-axis then passes through the point 
(5, 3). The coordinates of the point A are :
The equation of three sides of triangle ABC are x + y =3, 
x-y=5 and 3x + y=4. Considering the sides as 
diameter, three circles S1} S2, S3 are drawn whose 
radical centre is at:
__________________Column-1__________________ \
The triangle PQR is inscribed in the circle x2 + y 2 = 169.
If Q(5, 12) and R(-12, 5) then ZQPR is
The angle between the lines joining the origin to the 
points of intersection of the line 4x + 3y = 24 with circle 
(x-3)2 + (y-4)2 = 25
293Circle
■ Answers J
A—> Q; B—> S; C—> S; D—> S
2. A->P;B—>Q;C—>R;D—>S
t
1.
294 Advanced Problems in Mathematics for J EE
2
positive integers. Find the value of (p + q).
3. Lets = {(x, y) |x, y eR, x2 + y 2 -lOx +16 = 0}. The largest value of—can be putin the form 
x
Exercise-5 : Subjective Type Problems s
1. Tangents are drawn to circle x2 +y 2 = 1 at its intersection points (distinct) with the circle 
x2 + y 2 + (A. - 3)x + (2k + 2)y + 2 = 0. The locus of intersection of tangents is a straight line, 
then the slope of that straight line is.
2. The radical centre of the three circles is at the origin. The equations of the two of the circles are 
x2 + y 2 =1 and x2 + y 2 + 4x + 4y -1 = 0. If the third circle passes through the points (1,1) 
and (-2,1); and its radius can be expressed in the form of—, where p and q are relatively prime
q
8. If q and r2 be the maximum and minimum radius of the circle which pass through the point 
(4, 3) and touch the circle x2 + y 2 = 49, then — is
r2
9. Let C be the circle x2 + y2 - 4x - 4y -1 = 0. The number of points common to C and the sides 
of the rectangle determined by the lines x = 2,x = 5, y=-l and y = 5 is P then find P.
10. Two congruent circles with centres at (2, 3) and (5, 6) intersects at right angle; find the radius 
of the circle.
11. The sum of abscissa and ordinate of a point on the circle x2 + y 2 - 4x + 2y - 20 = 0 which is
f 3 Anearest to 2, - is :I 2)
12. AB is any chord of the circle x2 +y2 -6x-8y -11 = Owhich subtends an angle - at (1,2). If
2
locus ofmidpointof AB is a circle x +y -2ax-2by -c = 0; then find the value of(a + b + c).
— where m, n are relatively prime natural numbers, then m2 + n 
n
4. In the above problem, the complete range of the expression x2 + y 2 - 26x + 12y + 210 is [a, b], 
then b - 2a =
5. If the line y = 2 - x is tangent to the circle S at the point P(l, 1) and circle S is orthogonal to the 
circle x2 + y 2 + 2x + 2y - 2 = 0, then find the length of tangent drawn from the point (2, 2) to 
circle S.
6. Two circles having radii q and r2 passing through vertex A of a triangle ABC. One of the circle 
touches the side BC at B and other circle touches the side BC at C. If a = 5 and A = 30°; find 
7^2-
7. A circle S of radius ‘a’ is the director circle of another circle Sj. S1 is the director circle of S2 and 
so on. If the sum of radius of S, S1, S2, S3 .... circles is ‘2’ and a = (k - Vk), then the value of k is
295Circle
J Answers |
6.66 2 52 2. 5 3. 25 27.
68. 6 3 10. 3 12. 8 13. 89.
□□□
13. If circles x2 + y 2 = c with radius V3 and x2 + y2 + ax + by + c = 0 with radius V6 intersect at 
two points A and B. If length of AB =41- Find L
——»
Parabola
••
ix-a.-x. --------- -
fExercise-1: Single Choice Problems
(c) 4 (d) 5
(d) 13
m9j ;
1. Let PQ be the latus rectum of the parabola y 2 = 4x with vertex A. Minimum length of the 
projection of PQ on a tangent drawn in portion of parabola PAQ is :
(a) 2 (b) 4
(c) 2^3 (d) 2V2
2. A normal is drawn to the parabola y 2 = 9x at the point P(4, 6). A circle is described on SP as
diameter; where S is the focus. The length of the intercept made by the circle on the normal at 
point P is : 
r x 17 „ , 15(a)— (b) —
4 4
3. A trapezium is inscribed in the parabola y 2 = 4x, such that its diagonal pass through the point
25(1, 0) and each has length —. If the area of the trapezium be P, then 4P is equal to : 
4
(a) 70 (b) 71 (c) 80 (d) 75
4. The length of normal chord of parabola y 2 = 4x, which subtends an angle of 90° at the vertex is
(a) 6V3 (b) 7V2 (c) 8V2 (d) 9V2
5. If b and c are the lengths of the segments of any focal chord of a parabola y 2 = 4ox. Then the 
length of semi-latus rectum is :
f X be (a) ------
b + c 
r x b + c tc) —
6. Jhe length of the shortest path that begins at the point (-1,1), touches the x-axis and then ends 
at a point on the parabola (x -y )2 = 2(x + y - 4), is :
(a) 3V2 (b) 5 (c) 4-710
(b) — 
b + c
(d) VbZ
297Parabola
is :
(d) 2
(d) 8
OQ and have lengths q and r2 respectively, then the value of
ri
7. If the normals at three points P, Q, R of the parabola y 2 = 4ax meet in a point O' and S be its 
focus, then | SP | ■ | SQ | • | SR | is equal to :
(a) a3 (b) a2(SO')
(c) a(SO')2 (d) None of these
8. Let P and Q are points on the parabola y 2 = 4ax with vertex O, such that OP is perpendicular to 
r4/3 4/3 
'1 r2 
2/3 , _2/3 
I + r2
(a) 16a2 (b) a2 (c) 4 a (d) None of these
9. Length of the shortest chord of the parabola y 2 = 4x + 8, which belongs to the family of lines
(1 + X)y + (X-l)x + 2(1 - X) = 0, is :
(a) 6 (b) 5 (c) 8
10. If locus of mid-point of any normal chord of the parabola :
2 . . by2y = 4x is x - a = —- + —; 
y2 c
where a, b, c e N, then (a + b + c) equals to :
(a) 5 (b) 8 (c) 10 (d) None of these
11. Let tangents at P and Q to curve y 2 -4x-2y + 5 = 0 intersect at T. If S(2, 1) is a point such that 
(SP)(SQ) = 16, then the length ST is equal to :
(a) 3 (b) 4 (c) 5 (d) None of these
12. Abscissa of two points P and Q on parabola y 2 = 8xare roots of equation x2-17x +11 = 0. Let 
Tangents at P and Q meet at point T, then distance of T from the focus of parabola is :
(a) 7 (b) 6 (c) 5 (d) 4
13. If Ax + By = 1 is a normal to the curve ay = x2, then :
(a) 4A2(l-aB) = aB3 (b) 4A2(2 + aB) = aB3
(c) 4A2(l + aB) + aB3 =0 (d) 2A2(2-aB) = aB3
14. The equation of a curve which passes through the point (3, 1), such that the segment of any 
tangent between the point of tangency and the x-axis is bisected at its point of intersection with 
y-axis, is :
(a) x = 3y2 (b) x2=9y (c) x=y2+2 (d) 2x = 3y2 + 3
15. The parabola y = 4 - x2 has vertex P. It intersects x-axis at A and B. If the parabola is translated
from its initial position to a new position by moving its vertex along the line y = x + 4, so that it 
intersects x-axis at B and C, then abscissa of C will be : 
(a) 3 (b) 4 (c) 6
298 Advanced Problems in Mahematics for JEE
(d) 8V3(b) (c)
circletouches the
/ 5 Answers |
(d) (b) (d) (b)2. (a) (b)3. 4. 6. (a) (c) (a) (c)5. 10.8.7. 9.
Cb) (a) (d)11. (a) Cd)12. 13. 16. (c)14. 15. (c) (a)17. 18.
(b) 13xA + 8xA-21 = 0
(d) 13x2 + 21xA -31 = 0
16. A focal chord for parabola y 2 = 8(x + 2) is inclined at an angle of 60° with positive x-axis and
intersects the parabola at P and Q. Let perpendicular bisector of the chord PQ intersects the 
x-axis at R ; then the distance of R from focus is : 
. . 8 . 16^3 , . 16(a) - (b) —— (c) —
3 3 3
17. The Director circle of the parabola (y -2)2 = 16(x + 7)
(x -1)2 + (y +1)2 = r2, then r is equal to :
(a) 10 (b) 11 (c) 12 (d) None of these
18. The chord of contact of a point A(xa, y A ) of y 2 = 4x passes through (3,1) and point A lies on 
x2 + y 2 = 52. Then :
(a) 5xA + 24xA + 11 = 0
(c) 5x2 + 24xA + 61 = 0
299Parabola
Exercise-2 : One or More than One Answer is/are Correct
| Answers |
1. (a, b, c, d)
1. PQ is a double ordinate of the parabola y 2 = 4ax. If the normal at P intersect the line passing 
through Q and parallel to x-axis at G; then locus of G is a parabola with :
(a) vertex at (4a, 0) (b) focus at (5a, 0)
(c) directrix as the line x - 3a = 0 (d) length of latus rectum equal to 4a
Advanced Problems in Mahematics for JEE300
Exercise-3 : Comprehension Type Problems
(c) 64 (d) 32
Answers
2. (d) 3. (c)1. (a)
Paragraph for Question Nos. 1 to 3
Consider the following lines :
Li : x-y -1=0
L2:x + y- 5 = 0
L3 :y-4=0
Let Iq is axis to a parabola, L2 is tangent at the vertex to this parabola and L3 is another 
tangent to this parabola at some point P.
Let ‘C ’ be the circle circumscribing the triangle formed by tangent and normal at point P and 
axis of parabola. The tangent and normals at the extremities of latus rectum of this parabola 
forms a quadrilateral ABCD.
1. The equation of the circle ‘C ’ is :
(a) x2+y 2-2x-31 = 0 (b) x2 +y2-2y-31 = 0
(c) x2+y2-2x-2y-31 = 0 (d) x2 + y2 + 2x+ 2y = 31
2. The given parabola is equal to which of the following parabola ?
(a) y2=16V2x fb) x2=-4V2y
(c) y2 = —fix (d) y2 = &fix
3. The area of the quadrilateral ABCD is :
(a) 16 (b) 8
Parabola 301
Exercise-4 : Matching Type Problems
Column-I Column-ll
2(A)
(Q)(B) 3
(C) (R) 8
(D) (S) '41
then k is equal to
(T) 2
2.
Column-ll
(C) (R)
(D) (S)
j Answers!
i. A—> S; B —» P; C-> Q; D-> R
2. A—>P;B—>Q;C—>S;D—>T
(A)
(B)
Distance of the centroid from the circumcentre of APQR 
is equal to
Distance of the vertex from the centroid of APQR is equal 
to
(P)
(Q)
Column-1
Area of APQR is equal to
Radius of circumcircle of APQR is equal to
2
5
2
3
2
2
3
11
JS
(P)
The equation of the common tangent to parabola
2 2 ky = 4xandx = 4y isx + y + — = 0, then k is equal to 
V3
An equation of common tangent to parabola y 2 = 8x 
and the hyperbola 3x2 -y 2 = 3 is 4x - 2y + —= = 0,
V2
x2 y2
The equation of tangent to the ellipse — + — = 1 which, 
25 16
cuts off equal intercepts on axes is x - y = a where | a | 
equal to
The normal y=mx-2am-am2 to the parabola 
y 2 = 4ux subtends a right angle at the vertex if | m | equal 
to
(T)__ 1
302 Advanced Problems in Mahematics for JEE
| Answers j
2. 7 18 3.
□□□
of(X + p):
3. The chord AC of the parabola y 2 = 4ax subtends an angle of 90° at points B and D on the 
parabola. If points A, B, C and D are represented by (at?, 2ati), i = 1, 2, 3, 4 respectively, then 
find the value of C2 +t4 
fi +t3
.Exercise-5 : Subjective Type Problems f y ",
1. Points A and B lie on the parabola y = 2x2 + 4x - 2, such that origin is the mid-point of the line 
segment AB. If ‘V be the length of the line segment AB, then find the unit digit of I2.
2. For the parabola y = -x2, let a 0;P(a, -a2)andQ(b, -b2).LetM be the mid-point 
of PQ and R be the point of intersection of the vertical line through M, with the parabola. If the 
ratio of the area of the region bounded by the parabola and the line segment PQ to the area of
Xthe triangle PQR be —; where X and p are relatively prime positive integers, then find the value 
P
Ellipse
Exercise-1: Single Choice Problems Z-v
= 25 is :
(d) Sj + S 2F2 — ®
another ellipse having semi-axes :
(a) a + b and b (b) a - b and a (c) a and b (d) None of these
y 26. If and F2 are the feet of the perpendiculars from foci Sj and S2 of the ellipse —- + = 1 on
25 lo
x2 y21. If CF be the perpendicular from the centre C of the ellipse — + — = 1, on the tangent at any
12 8
point P and G is the point where the normal at P meets the major axis, then the value of (CF ■ PG) 
equals to :
(a) 5 (b) 6 (c) 8 (d) None of these
x2 y22. The minimum length of intercept on any tangent to the ellipse — + -— = 1 cut by the circle
4 9
the tangent at any point P of the ellipse, then :
(a) S1^+S2F2>2 (b) Sx^+S2F2>3 (c) S^+S2F2>6
5. The area bounded by the circle x2 + y 2
x2+y2
(a) 8 (b) 9 (c) 2 (d) 11
3. The point on the ellipse x2 + 2y 2 =6, whose distance from the line x + y = 7 is minimum is :
(a) (2, 3) (b) (2,1) (c) (1, 0) (d) None of these
4. If lines 2x+3y =10 and 2x - 3y = 10 are tangents at the extremities of a latus rectum of an 
ellipse; whose centre is origin, then the length of the latus rectum is :
, . 110 98 , . 100 ,120(a) -- (b) — (c) -- (d) --27 27 27 27
, x2 y2= a2 and the ellipse —- + = 1 is equal to the areaMathematics for JEE22
A defined byand
(d) [-5,-2](b) [-2,5] (c) [-5,2]
Answers
(c) (d) 3. (C) (d)2. (C)(a) (b)4. 6. (b) (b)5. (a) 9. 10.7. 8.
(b) (a)12. 13. (a) 14. (c) (b)15. 16., (d)
s
15. f 1 (x) is equal to
(a) ^2 + ^/4-logTx
16. The set ‘A’ equals to
(a) [5,2]
Paragraph for Question Nos. 15 to 16
7t 
g:|_2,7r
(b) ^2 + 74+log2x (c) ^4 + ^2+ log2x (d) ^4-^/2 + logT^
Let /:[2,oo)-> [l,oo] defined by /(x) = 2* 4x
sin x + 4g(x) =----------- be two invertible functions, then
sinx-2
Function 23
Exercise-4 : Matching Type Problems
Column-1 Column-11
7.7
1.1
1
3
Column-1 Column-ll
3. Given the graph of y = f(x)
1
To 2
Column-I Column-ll
(A) y = /(l-x) (P)
24
1. If x,y,z eR satisfies the system of equations x + [y] + {z} =12.7, [x] + {y} + z =4.1 and 
{x}+y + [z] = 2
(where {■} and [•] denotes the fractional and integral parts respectively), then match the 
following :
(A)
(B)
(C)
(D)
(A)
(B)
(C)
(D)
{*} + {y} =
[z] + W =
x + {z} = 
z + [y]-{x} =
Domain of Jj (x) is
Range of f2 (x) in the domain of/j (x) is
Range of f2W is
Domain of/2(x) is
(P)
(Q)
(R)
(S)
(T)
[-3,-2]
[-4,-2]
(—00, oo)
(-co,-4] o[-3, oo)
[0, 1]
(P)
(Q)
(R)
(S)
(T)
f2W = -2+ 21ogy2 cos tan
4_________________
2. Consider ax4 + (7a-2b)x3 + (12a-14b-c)x2 - (24b + 7c)x +1-12c = 0, has no real roots 
, , . . Jlog(jI+e) (ax4 + (7a - 2b) x3 + (12a - 14b - c) x2 - (24b + 7c) x + 1 - 12c)
and Ji (x) = -------—--------------------------- - ----------- ------- ---------------------------------
Vay-sgn (1 + ac + b2)
7 'I(sin (7c(cos(jr(x + -))))) I • Then match the following :
Advanced Problems in Mathematics for JEE24
(Q)y = /(2x)(B) 1
1 20
(C) y =-2fM (R) 1
1
y = i-/(x)(D) (S)
1 .0
Column-ll
(A) (P)
(B) (Q)
f(x) = ^/in (cos (sin x)(C) (R)
(D) (S)f (x) = tan
Periodic but not odd function(T)
5.
Column-ll
(A) (P) (0,co)
(B) then0, (Q) (-co, 0]
[-1, «>)
-H—
1/2
Range contains no natural 
number
Range contains atleast one 
integer
Many one but not even 
function
Both many one and even 
function
____________Column-I____________ \
If |x2-x|> x2 + x, then complete set of 
values of x is
(C) 
___
If |x + y|>x-y, where 
complete set of values of y is
If log2 x > log2(x2), then complete set of (R) 
values of x is
fix') = - (sin-1 (sin irx))
x2 +1
2 ■ >3
______________ Column-1
/(x) = sin2 2x - 2sin2 x
x +
-1/2 0
25Function
(D) (0,1]
(T) [1, =o)
6.
Column-1 Column-ll
(A) (P)
(Q) [2,oo](B)
function (R)(C) (l,2)o(3,co)
[0,co)(D) (S)Let/(x) =
(-co,-3)o (-2,-l)o (2, oo)(T)
7. Let /(x)
Column-1 Column-ll
1
2
3
4
’ Answers I
2.
3.
4.
5.
6.
7.
A-»R; 
A —> S j 
A-»Q ;
(A)
(B)
(C)
(D)
Domain of /(x) = In tan 
{(x3 - 6x2 + 1 lx - 6) x(ex -1)} is
Then range of function /(g(x)) is 
l + x;0| x|, (where [•] denotes the greatest! (S) 
integer function) then complete set of 
values of x is
(P)
(Q)
(R)
(S)
of
(|x|-i)(^ + ^ + 4)is
x\ =
X + 1 X > 1
Range of/(x) = sin2 — + cos— is
4 4
B—>S; C—>P; D->Q
B->P; C->Q; D-»R
B->R; C-»P; D->S
A-»P, Q, S, T; B-»Q, R; C->P, Q, S; D->P, S
A—> Q; B—>P; C—>S; D—>R
A->R; B->P; C->T; D->S
A —> R ; B —> R j C —■> R j D —> S
Advanced Problems in Mathematics forJEE26
Exercise-5 : Subjective Type Problems
+ sin
y = sin + cos
= 3
-33-x
Z
(0,30] (where [•] denotes greatest integer function)
13. Let 7(x,y) = x2-y 2 and g(x,y) = 2xy.
such that (7(x,y))2 - (g(x,y))2 =1 and /(x,y)-g(x,y) = —
2 4
Find the number of ordered pairs (x,y) ?
4. If the domain of 7(x) = -712-3*
x2 + - 
9j
_____ 1
201VTx
7C/C- (7U)) )={-x}
where f is applied 2013 times and {•} denotes fractional part function.
12. Find the number of elements contained in the range of the function J(x)
1. Let f (x) be a polynomial of degree 6 with leading coefficient 2009. Suppose further, that 
7(1) = 1,7(2) = 3,/(3) = 5,7(4) = 7,f(5) = 9,f'(2) = 2, then the sum of all the digits of7(6)is
2. Let 7(x) = x3 - 3x +1. Find the number of different real solution of the equation 7C7W) =
3. If 7(x + y +1) = Qy/fM + 7/CX))2 V x,y g B and 7(0) = 1, then 7(2) =
% yA
— I is [a, b], then a =
5. The number of elements in the range of the function :
x2 where [•] denotes the greatest integer function is
6. The number of solutions of the equation 7(x-l) + 7(x + l) = sina, 0l1S
7. The number of integers in the range of function 7(x) = [sin x] + [cos x] + [sin x + cos x] is 
(where [•] = denotes greatest integer function)
8. If P(x) is a polynomial of degree 4 such that P(-l) = P(l) = 5 and P(-2) = P(0) = P(2) = 2, then 
find the maximum value of P(x).
9. The number of integral value(s) of k for which the curve y = 4-x 
intersect at 2 distinct points is/are
10. Let the solution set of the equation :
is [a, b). Find the product ab.
(where [•] and {•} denote greatest integer and fractional part function respectively).
11. For all real number x, let 7(x) = 1 Find the number of real roots of the equation
2 - 2x and x + y - k = 0
x
3
Function 27
V x g R, then the smallest integral value of k for which /(x) R; /(x) = — + (m-l)x2 + (m + 5)x + n
20. Let /(x) be continuous function such that /(0) = 1 and /(x) - = ~^x eR, then /(42) =
 (min{/(t): 0 1 is —, where p and q are 
q
25. Let the maximum value of expression y = —
x 
relatively prime natural numbers, then p + q =
26. If /(x) is an even function, then the number of distinct real numbers x 
/(x) = /fcA
I x + 2
G C can be expressed as - where a, b, c and d are 
b d_
(a + b + c + d)
2 '
36. Polynomial P(x) contains only terms of odd degree. When P(x) is divided by (x-3), then 
remainder is 6. If P(x) is divided by (x2 - 9) then remainder is g(x). Find the value of g(2).
37. The equation 2x3 - 3x2 + p = 0 has three real roots. Then find the minimum value of p.
1
Vln cos
31. Let/(x) = ^i^,
cx + d
except --.The unique number which is not in the range of f is 
c
32. It is pouring down rain, and the amount of rain hitting point (x, y) is given by 
f(x, y) =| x3 + 2x2y -5xy2 -6y 3 |. If Mr. ‘A’starts at (0, 0); find number of possible value(s) 
for * m' such that y = mx is a line along which Mr. * A' could walk without any rainof
Advanced Problems in Mathematics for JEE304
2
= 1, where fix') is a positive decreasing7. Consider the ellipse
(d)(c)(a)
? Answers J
9. (d) 10. (c)6. (d)5. (b) 7. (c) 8. (a)4. (c)3. (b)1. (c) 2. (a)
43
5>Ȥ
x2 1 y
f(k2 + 2k+5) /(fc + H) 
function, then the value of k for which major axis coincides with x -axis is :
(a) k 6 (—7, - 5) (b) k 6 (-5, - 3) (c) k e (-3, 2) (d) None of these
x2 y2 /— -A
8. If area of the ellipse — + — = 1 inscribed in a square of side length 5V2 is A, then — equals to:
16 b2 n
(a) 12 (b) 10 (c) 8 (d) 11
9. Any chord of the conic x2 + y2 + xy =1 passing through origin is bisected at a point (p, q), then 
(p + q + 12) equals to :
(a) 13 (b) 14 (c) 11 (d) 12
10. Tangents are drawn from the point (4, 2) to the curve x2 + 9y2 =9, the tangent of angle 
between the tangents :
3^3
5V17
Ellipse 305
Exercise-2 : Comprehension Type Problems
2
11(b) x2+y2 +
(c) x2+y 2
.2
(c)
’ Answers |
2. (a) 3. (c) ±12^.1 5. (a)
Paragraph for Question Nos. 1 to 2
An ellipse has semi-major axis of length 2 and semi-minor axis of length 1. It slides between 
the co-ordinate axes in the first quadrant, while maintaining contact with both x-axis and 
y-axis.
(b) 3
(d) 2^5
1- (b)
3. The eccentricity of the ellipse is equal to :
1 12 1(a) A (b) 4= (c) r (d) A
3 V3 3 2
4. The area of the largest triangle that an incident ray and corresponding reflected ray can enclose 
with the major axis of the ellipse is equal to :
(a) 4V5 (b) P, R; B —> R, S; C —> Pj D —> Q
■ - ....
__
2
3
57C 
T
7C
4
n
2
________________ Column-1________________ \
If the tangent to the ellipse x2 + 4y 2 =16 at the point 
P(4cos(fc 2sin 0, b > 0 (b) a > 0, b 0
4. A circle cuts rectangular hyperbola xy = 1 in the points (xr, y r ), r = 1, 2, 3, 4 then :
(a) 7172/374 =1 (b) x1x2x3x4=1
(C) Xj X2X3X4 =7172/374 =-l (d) 7172/3/4 =°
311Hyperbola
Exercise-3 : Comprehension Type Problems
] Answers 3
1. (b) 2. (c) 3. (a)
Paragraph for Question Nos. 1 to 3
A point P moves such that sum of the slopes of the normals drawn from it to the hyperbola 
xy = 16 is equal to the sum of the ordinates of the feet of the normals. Let ‘P’ lies on the curve 
C, then :
(b) 776^3
(d) None of these
1. The equation of ‘C ’ is :
(a) x2 = 4y (b) x2 = 16y
(c) x2=12y (d) y2=8x
2. If tangents are drawn to the curve C, then the locus of the midpoint of the portion of tangent 
intercepted between the co-ordinate axes, is :
(a) x2 = 4y (b) x2 = 2y
(c) x2 + 2y = 0 (d) x2 + 4y = 0
3. Area of the equilateral triangle, inscribed in the curve C , and having one vertex same as the 
vertex of C is : 
(a) 768V3 
(c) 760^3
312 Advanced Problems in Mathematics for J EE
Exercise-4 : Subjective Type Problems
’ Answers !
2. 4 3. 48
□□□
x2 y2 x2 y21. Let y = mx + c be a common tangent to----- — = 1 and — + — = 1, then find the value of
16 9 4 3
2^2 m + c .
2. The maximum number of normals that can be drawn to an ellipse/hyperbpla passing through a 
given point is :
3. Tangent at P to rectangular hyperbola xy = 2 meets coordinate axes at A and B, then area of 
triangle OAB (where O is origin) is :
I
! 
I 
I
i
22. Compound Angles
23. Trigonometric Equations
24. Solution of Triangles
25. Inverse Trigonometric Functions
!
i
1 A
Trigonometry
■
)[22
Compound Angles
I
Prx.'—jL
Exercise-1: Single Choice Problems
(d)(c)Cb)
22
(d)(c)Cb)(a)
(c)
(d)(c)Cb)(a)
n
6
(d) (a-b)2
Cd)
4
3. If all values of x e (a, b) satisfy the inequality tan x tan 3xfalling on him.
33. Let P(x) be a cubic polynomical with leading co-efficient unity. Let the remainder when P(x) is 
divided by x2 - 5x + 6 equals 2 times the remainder when P(x) is divided by x2 - 5x + 4. If 
P(0) = 100, find the sum of the digits of P(5):
34. Let /(x) = x2 + lOx + 20. Find the number of real solution of the equation /(/(/(/(x)))) = 0 
(lnx)(lnx2) + Inx3 +3
In2 x + Inx2 + 2
prime numbers (not necessarily distinct) then find the value of
interval [0,1] is :
28. Let x1,x2,x3 satisfying the equation x3 -x2+px + y = 0 are in G.P where x1,x2,x3 are 
positive numbers. Then the maximum value of [p] + [y] + 4 is where [•] denotes greatest integer 
function is:
29. Let A = {1,2,3,4} and B = {0,1,2,3,4,5}. If ‘ m’ is the number of strictly increasing function /, 
f: A -> B and n is the number of onto functions g, g: B -> A. Then the last digit of n - m is.
n
30. If £[log2 r] = 2010, where [•] denotes greatest integer function, then the sum.of the digits of n
r=l
is :
x4-x2
6 +2x3 -1
27. The least integral value of m, m eR for which the range of function /(x) = % + m contains the 
x2 +1
Function 29
□□□
26
6
0
5
5
4
7
1
6
9
8
0
18.
25.
32.
1
1
76
7
3
1
6
6
4
2
4
4
8
1
2
5
6
5
3
6
8.
15.
22.
29.
36.
5.
12.
19.
26.
33.
6.
13.
20.
27.
34.
3.
10.
17.
24.
31.
38.
2.
9.
16.
23.
30.
37.
7.
14.
21.
28.
35.
I__
j Answers ;
——r
9
12
7
3
9
2
Limit
Exercise-1: Single Choice Problems
-T' /
X
(c)
(a) 0 (b) 1 (C) 2
, b - lim
Inx
4. If /(x) = cot and g(x) = cos
(a) (c)
is :
6. lim — — = — (where [•] denotes greatest integer function), then p + q (where p, q are relative
(b) 7 (d) 6(c) 5
4 
sinx
1
6
Then a, b, c satisfy : 
(a) a 1
3. Let a = lim 
x->0
5. lim 
x->0
-3 
2(1 +a2)
In (cos 2x) 
3x2
(c) a 
l-3x2,
(d) e-8
sin2 2x 
i----------------
X~*ox(l-ex)
“4
3 
(d)
2
(b)l3 
2(1 +a2)
, .. cos (tan x)-cos x
1. lim -------------------------r
x—>0
(1 + x)*
• e2
(c) e8(b) e-4
(d) b ag(x)-g(a) 2
2. The value of lim Wnx-tanx^-d-^/+x5 equal (0 . 
x-»°7(tan 1 x)7 + (sin-1 x)° +3sin5x
P
\
(a) e4
.. 3fx
x->0
x"’1 +
■ i tt sin —xI 3
(b) ±V3
7. /(x) = lim 
n—>oo
11. lim----
cos
. > 2(a) -
n
4
n
1 = (sin en')e~n + —, then f (0) =
I n2 + l
W)1
(0 "
2
(C) t
—,oo , then function is invertible L3 J
(b)
4
12. Let f be a continuous function on R such that
■=, (where [•] denotes greatest integer function) is :
(3 sin x-sin 3x)
10. lim
2cosx-l
3
, a 2(a) -=V3
(a) it
xn
n2
2
x 
(e(x+3)ln27 j27 _9 
3X - 27
l-cos(x-3) x>3 
(x - 3) tan(x - 3)
If lim /(x) exist, then X = 
x->3
(a) If/: |,oo
(b) /(x) = /(-x) has infinite number of solutions
(c) /(x) =| /(x) | has infinite number of solutions
(d) /(x) is one-one function for all x e R
sin (it cos2 (tan(sin x)))
-------?------- =
Advanced Problems in Mathematics for J EE32
13. lim equals, where {•} is fractional part function and I is an integer, to :
(b) e-2 (d) does not exist(c) I
1/X
is :
(b) e
(b) 1 (d) None of these(c)
(a) 1 (c) 3
2
(a) 2 (c) (d) None of these
(b) 2 (c) 3 (d) -3
-7x)3x is equal to :
3
(c) e11
2
5
15. The value of lim 
x—>0
18. The value of lim 
x->0
lim n
n->co
(l-2x)n £ nCr 
r=0
(d)
-{x}-l
{x}2
17. The number of non-negative integral values of n for which lim
x->0
(cosx-l)(cosx-ex) _ q . .
7*
Cd) 4
.. + nf (x + nT)
+ n2/(x + n2T)/ 
3 
2
e{x}
(a)l
, ' 11
(a) T
limit is : 
(a)Z
n
(d) e3
(a) e-1/3 
19. If lim
x—>co n—>oo
(a) a (b) -a (c) 2a (d) b
20. If f is a positive function such that /(x + T) = /(x) (T > 0), V x e R, then
' f(x + T) + 2/(x + 2T) +........ ■ " "" X
+ T) + 4/ (x + 4T) +
-7x8 +5x6 -21x3 +3x2-7
265l lim-------- —-+3h2_____ 1 =
(/(1-h) _/(i)) sin 5h J
(a) 1
M A
11
21. Let /(x) = 3x10
Cc) e3n
X + x^ 
l-2x
14. lim (ellx
X->co
(c) e’1/6
(a) en
(b) 2 
i
sin x \ -cosx
x J
(b)
-x + l- ax-b) = 0, then for k > 2,(k e N) lim sec^ffclnb) =
(d) e~3n
16. For a certain value of ‘c’, lim [(x5 + 7x4 + 2)c -x] is finite and non-zero. Then the value of
X->CO
Limit 33
(a) 0 (c) -1 (d) -2
x->0
3
(b) ±(a) not exist (c)
is equal to :
(b) e (c) e
cosx is :
(a) 1 (b) 0
, where d, b are non-zero constants is equal to :
(b) bfM
(d) (a + 2b)/(x)
16
22. lim
x->0
25. lim
X—>co
24. lim
it
X->2
x-3 
x+2,
(a) e
26. lim (cosx) 
n
(b) ab
(d) eb,a
cos x - sec x 
x2(x + l)
9
(d) - 
e
28. lim i + 
x-»ol
(d) e5
(c) - 
e
W)±
32
/'(x) is equal to :
(a) a/(x)
(c) (a + b)/(x)
f l-tan^fl - sin x)
1 + tan— (n-2x) 2J
27. If lim {In x} and lim {In x} exists finitely but they are not equal (where {■} denotes fractional 
part function), then :
(a) ‘ c’ can take only rational values
(b) ‘ c’ can take only irrational values
(c) ‘ c’ can take infinite values in which only one is irrational
(d) ‘ c’ can take infinite values in which only one is rational 
1
asinbxy 
cosx J
(a)
(c) eab
(b) -1
23. Let/(x)be a continuous and differentiable function satisfying/(x + y) = /(x)f(y) Vx,y eRif 
/(x) can be expressed as /(x) = 1 + xP(x) + x2Q(x) where lim P(x) = a and lim Q(x) = b, then 
x->0
1
8
Advanced Problems in Mathematics for JEE34
x-»0
(c) -Je + 2(a) (b)
and
is :
is a finite31. The integral value of n so that lim /(x) where /(x) =
x->0
(d) 8(c) 6
, if x * 032. Consider the function J(x) =
, if x = 0
Um [/(x)] =
cos33.
(d) 0
(b) 1
(d) Does not exist
, then lim {/(x)} + Um {/(x)} + 
x->o- x->r
r 3e + -
2
(b) 1 
(2x71-x2)
1 x ——
72
ln(l + sin x) 
x
it 
2(k + 2)
X
is :
30. Let a = lim [ —— 
x—>1 I In x
' i
29. The value of lim (cosx)sin2jf +
;c = Um 
x->0
7t
+ COS--------------
2(k + 2)
(d) 4- + 2 
ve
(sinx-x)f 2sinx-ln| + X
I 11 -x
Um 
Ha)
(b) 272 
n / /
34. lim V sin----- cos—-sin
n’400j^k 2k 2k 
(a) 0 (b) 1
(0 2 (d) 3
35. lim^ [1 + [x]]^, where [•] is greatest integer function, is equal to :
(a) 0
(c) e2
xn
sin2x+2tan 13x+3x2 
In (1 + 3x + sin2 x) + xex
(where {■} denotes fraction part function and [•] denotes greatest integer function)
(a) 0 (b) 1 (c) 2 (d) 3
(2x7l-x2)
1 "ix ——72 J
(c) 472
.. cosUm ----
Ha)’
(a) 72
non-zero number, is :
(a) 2 (b) 4
f 1 max x, —
V x 
. f i nun x, — 
k x 
1
, x" -16x;b= hm-------- -
x-»° 4x + x
1 
xlnx
d = Um-------- ------ -----------, then the matrix [a ^1
x—»—1 3[sin(x +1) — (x +1)] Lc “J
(a) Idempotent (b) Involutary
(c) Non-singular (d) Nilpotent
.3
Limit 35
equals to:
(a) m - n
(c) (d) None of these
= 1, is:
(a) (b) (c) (d)
(a) 1 (b) 2 (c) 3
, then a is equal to :
(0 (d) non existent
is equal to :
(c)
= 1, is :
(a) (b) (c) (d)
n > 1“n
(d) 2(c)
is :,xx->0
(c) Ve + 2
Then the limit of un as n -> 0
5 
2’
5 3 
2’ 2
5 
2’
36. If m and n are positive integers, then lim 
x->0
5
2’
m-n
2mn
x(l + acosx)-bsinx
V
5 3^
2’ 2j
= 1?
2
f 1
43. The value of lim (cosx)sin'
(a)l
5 3
2’ 2
42. Consider the sequence :
-E-- S2r
(d)-l
(b) 1 oo
(I3 -l2) + (23 -22) + ... + (n3 -n2) 
n4 
(b) 1
40. The value of lim cosLsiV>-- 
x->0
(cosx)1/m -(cosx)1/" 
x2
(b) 1-1
n m
5 3
2’ 2
38. What is the value of a + b, if lim —bi(e cosx)
x->o xsin(bx)
sin2x + 2tan-1 3x + 3x2 
In (1 + 3x + sin2 x) + xe
1
4
41. The value of ordered pair (a, b) such that lim + ocosx)—bsinx
x->0 x3
5 3^
2’ 2J
2x +
Advanced Problems in Mathematics for J EE36
(l + sec2nx), the
is equal to :
(d) 2"+1
is :
(c) e
(a) 0
, n e N is equal to :
(a) (d) 0(c) e
(a, b, c, d, e e R -{0}) depends on the sign of:
fM
(c)
50. If /(x) be a cubic polynomialand lim
(c) 3 (d) -2
sinx
equals to :
(c) 2
5
3
49. Let /(x) = lim tan 
n->oo
1 
e
46. If lim 
x->0
e
x—>0
(c) 2n-1
(b) -5
-1
1 - cos— 
n
equals. 
, ' 8 (” 3
Cd) e2
(b) donly
(d) a, b and d only
n2 (2xAand g(x) = lim — In cos — then lim 
n->0 /(x)
x2 + 2x
= i then /(l) can not be equal to :
(d) n + 1 
n
-2g(x) _e 
x5
lim
x->o 2x
(a) 0
(d) 1 
e
(b) e3'2
(b) e3/4
4n2
(c) n
3n3+4
47. The value of lim| — |4n -1 
n->x^nn J
3/4
48. The value of lim
(a) a only
(c) a and d only
(a) 0
9 sinx 
51. lim^—
x->0
I
(b) 2n
1
45. The value of lim (1 + [X]),,,(tanx) 
n 
4
(where [.] denotes greatest integer function).
(a) 0 (b) 1
{(a-n)nx-tanx} sinnx . .--------------- -------------- = 0, n * 0then a is equal to : 
x2
44. For neN , let fn(x) = tan — (1 + secx) (1 + sec2x) (1 + sec4x)
2x
n
37Limit
is equal to :
(d) -±(c)(b) 1(a) -1
54. If fM =
(d) Does not exist(c) -1(b) 1(a) 0
Answers
(b)(c)(b) Cd) (a) 9. 10.(d) Cb) 6. 7. 8.(d)Cb) Cd) 5.4.2. 3.
(a)Cc) (a) 20.Cb) 16. (a) 18. 19.Cb) (d) 17.(a) (a) 14. 15.13.12.
Cd)(d) (c) 30.(a) 29.Cd) (c) 26. 27. 28.Cc) (a) 25.Cc) 23. 24.21. 22.
Cb)(a) 39.(b) 36. Cc)(d) 37. 38.Cc)(c) (a) 34. 35.33.32.31.
(a)Cc)Cd) (a)Cb) 46. 48. 49.(d) CO 47.(d) 45.(a) 42. 43. 44.41.
(c)Cb) (c)(a) 53. 54.51. 52.
K
i
2
53. lim 
x—>0
xcosx 
1 
1
Cb)
(c) I
Cd) |
Cc)
2x sin x x tan x 
x 1
2x 1
Cd) nx"-1
52. If xi,x2,x3). 
(Xj -x4).......
(a) nx1 +b
>11 + sin2
,xn are the roots of x” +ax + b = 0, then the value of (X] -x2)(x1 -x3) 
. (xx -xn) is equal to :
(b) nx"-1 + a
x -^1 -2tanx
sin x +tan x
Cb) 40.
| 50.1I '
, find lim 
x->0 x2
(c) nx" 1
1
2
Advanced Problems in Mathematics for JEE38
Exercise-2 : One or More than One Answer is/are Correct
(d) p = 2,q = 4(c) p = 1, q = 2
(a) lim
(d) f is neither even nor odd function= 2
2 2
(sin x)
cotx
2. lim 2|
(a) lim cosec
(d) lim tan - + x 
x-»0
(b) lim sec 
X—>1
(d) lim 
x-»0
2x -loge(l + x)2 
5x2
-1 + x
7
. x sin —
5
x
; p, q eR then :
.... 2(l-cosx2)(c) hm-------- ------
5x4
3. Let lim (2X + ax +ex)1/x = L
X—>00
which of the following statement(s) is(are) correct ?
(a) if 1 = a (a > 0), then the range of a is [e, a>)
(b) if L = 2e(a > 0), then the range of a is {2e}
(c) if L - e (a > 0), then the range of a is (0, e]
(d) if L = 2a (a > 1), then the range of a is I —, oo I
4. Let tan a • x + sin a • y = a and a cosec a • x + cos a • y = 1 be two variable straight lines, a 
being the parameter. Let P be the point of intersection of the lines. In the limiting position when 
a -> 0, the point P lies on the line :
(a) x = 2 (b) x = -l (c) y + l = 0 (d) y = 2
5. Let f: R -> [-1,1] be defined as f (x) = cosfsin x), then which of the following is(are) correct ?
(a) / is periodic with fundamental period 2n (b) Range of f = [cos 1,1]
(c) lim | /f--xl + /f- + x
"I \ 2 I I 2x—>—k k* / kz .
2
6. Let/(x) = x + Vx2 + 2x and g(x) = a/x2 + 2x -x, then :
(a) limg(x)=l (b) lim /(x) = l (c) lim /(x)=-l (d) limg(x)=-l 
x->co X—>CO X—>—CO X->00
7. Which of the following limits does not exist ?
x
x + 7
(b) lim— 
x->0
1
(c) lim xx 
x-»0+
1. If lim (p tan qx2 - 3 cos2 x + 4)1//(3x 5 = e 
x->0
(a) p = V2,q = -i= (b) p =^=,q=242 
2V2 V2
25x2 + x - 5x is equal to :
7t
8
Limit 39
where [y] denotes largest integer R;fM =
0
where {x} denotes the fractional part of x, then :11. Let/(x) =
x->0+
(d) /(xy)=/(x) + /(y)
1 
fM
8. If /(x) = lim x — + [cosx] 
n->oo I 2
(b) lim /(x) is non-integer 
x->a
(d) /(x) has local minimum at x = a
_ It 
y 2.
then identify the correct statement(s).
(a) lim /(x) = 0
x-»0
1 x =---- ,
2^ 
otherwise
(b) f\- | = 
\xj
(b) lim /(x) = V2 lim /(x) 
x->0“ 
it 
2^2
(b) lim /(x) does not exist
x-»o
(d) lim /(x) /(2x) does not exist 
x->0
(d) lim /(x) = 
x->0*
(c) /(x)=yVxe 0,J (d) /(x) = OVx e| —
- 4
2
then identify the correct statement(s).
(a) lim fM = 0
x->0
(c) lim/(x)/(2x) = 0
x->0
—, then:
12
(b) a = -2
n2 +1 -7n
10. If lim fM = lim [/(x)] ([•] denotes the greatest integer function) and fM is non-constant 
x->a x—>a
continuous function, then :
(a) lim fM is an integer 
x-*a
(c) fM has local maximum at x = a 
cos-1 (1 -{xBsin-1 (1 — {x}) 
72U}(1-{x})
(a) lim fM = —
x->o+ 4
(c) lim fM = -4=
x-»o_ 4V2
12.1flin.(sin(s‘nx);sinJf)
x-»° ox3 + bx5 + c
(a) a = 2 (b) a = -2 (c) c = 0 (d) b e R
13. If fM = lim (n (,x^n -1)) for x > 0, then which of the following is/are true ? 
n->oo
(a) /-1 = 0
\xj
(c) /|T] = -f(x)
\xj
2-3n + l
Advanced Problems in Mathematics for JEE40
3
(c)
such that (sin a + sin 0) + —— = 0 and (sin a + sin 0) . = -1 and
(d) X = 1(b) X = 2 (c)
integer function)
(a) 2 (c) 3
I Answers ?
! (b, c) (a, c, d)2. (a, b, c) (a, c) (b, c) 6. (a, c)3. 4. 5.
(a, d) (a, c, d)8. (b,c) (a, d) (b, d) (a, c)9. 12.10.
(c, d) 14. (c) (a. b) (b)15. 16.
X = lim 
n-»oo
7ta = —
3
14. The value of lim cos 
n->oo
2n
— then :
(a) 1
(b) I
2
| 13.
If lim [/(*)] exists, the possible values a can take is/are (where [•] represents the greatest
2 U (V"
15. If a,0 eJ-^,0
1 + (2sin a) 
(2 sin 0) 20
(a) a = -— 
o
16. Let/(x)=|lx-2l + a2-6a + 9 > x2
+ n2 + 2n)) (whereneN):
1
4 
sin a
sin0
sin a
sin0
41Limit
Exercise-3 : Comprehension Type Problems
A3 Configuration
(d) 11(b) 4 (c) 1
(d)(c)
Let /(x) =
(d) None of theseCc) 5(b) 4
is : ({•} denotes fractional part of function)
(d) None of these(c) 7(b) 5
x 
tanx
1
4
3. lim /([x - tan x]) is : ([•] denotes greatest integer fucntion) 
x->0“
(a) 2
4. lim f 
x->0+
(a) 4
o» -a
7t
(d) Non-existent
Paragraph for Question Nos. 1 to 2
A circular disk of unit radius is filled with a number of smaller 
circular disks arranged in the form of hexagon. Let An denotes a 
stack of disks arranged in the shape of a hexagon having ‘ n’ disks on 
a side. The figure shows the configuration A3. If * A’ be the area of 
large disk, Sn be the number of disks in An configuration and rn be 
the radius of each disk in An configuration, then
(b) 1
cosec
5. lim (f(x)) 
x->2
(a) -
7t
(C) e2'n
1. lim —: 
n->o°
(a) 3
2. lim nrn : 
n->oo
Paragraph for Question Nos. 5 to 6
A certain function /(x) has the property that /(3x) = a/(x) for all positive real values of x 
and /(x) = 1 -| x - 2| for 1 0 x3
(d)
2V3
(c) “7=
2V2
10. The value of a :
11. The value of L:
“4
8. a + b =
*4
m 1
(b) -j=V2
Paragraph for Question Nos. 10 to 11
For the curve sin x + siny =1 lying in the first quadrant there exists a constant a for which
d2vlim xa —4 = L (not zero)
x->0 dx2
Paragraph for Question Nos. 7 to 9
1
3
43Limit
f ■Exercise-4 : Matching Type Problems
Column-1 Column-II
n (P) 2(A)
(Q) 0(B)
(R) 1(C)
(S) 3(D)
= eL, thenL + 2 =
Non-existent(T)
Column-11
(P) 2(A) If/(x) = sin
(Q) 3
and then findg(x} where(B) If /(x) = tan
6h
(R) 4If cos 1 (4x3 - 3x) = a + b cos(C)
(S) -2(D) If /(x) - cos
2
then a + b + 3 =
(T)
§ Answers
1
2
lim 
n->x
(nx), then lim /(x) = 
x—>0+
lim
x->-- x-> 2
1 + ^4
2
f\± + 6h\-f\ ~ 
lim 
h->0
Non 
existent I
2. [•] represents greatest integer function :_____________________________
__________ Column-1________________________\ 
xand lim /(3x-4x3) - a-3 lim /(x),then[a] = 
_ r 
X > 2
xfor-1r
X-> 2
' 2xLet /(x) = lim — tan 
n->x> ft
1. A —> P j B —> Q ; C —> R j D —> S
2. A —> Q ; B —> P j C —> S j D —> Q
cos (tan 1(tanx))
n x —
2
If lim (x)lnsinx
x->0+
x->—i
3x — x3
l-3x2
1
2
Advanced Problems in Mathematics for JEE44
*-Exercise-5 : Subjective Type Problems
,3
. Find the value of kl.is
= 2 In 2 In k In 7, then k =
— x.
7. Find lim where a is root of equation sin x + 1 = x (here [•] represent greatest
Answers ■
2 2. 8 1 03. 4. 5 2 6. 1 7.5.
□□□
equal to unity, then the value of lim 
x->(l/a) (e
1. If lim 
x-»0
6. Find the value of lim | x + —
x
min (sinx,{x})
. ^-1 .
integer and {•} represent fractional part function)
Incotj - -Bx 
14 
tan ox
= 1, then — =
P
gl/x
. (140/ -(35/ -(28/ -(20/ + 7X + 5X + 4X -14. The value of hm ----------- -—----- -—-------—d-----------------------------
x->0 xsin2x
b 1 , ,+ — = - , then b - a -
x2 3
1
2. If lim = 8, lim--------------S^- q---------- = Xand lim (1 + 2/(x)/w = - , then X =
x->0sin2x x->0 2cosx-xex + x3 + x-2 x~>° e
3. If a, pare two distinct real roots of the equation ax3 + x -1 -a = 0, (a * -1,0),none of which is 
(1 + a) x3 - x2 - a al (ka - p)
■1-ax-l)(x-l) a
_ acotx5. If hm--------
x->0 x
Exercise-1 : Single Choice Problems
(a) 4 (b) 3 Cd) 5
= 12
(c) 12 (d) None of these
cosO-sinO
(a) 2 sin 2 (b) 2 cos 2 (d) cos 2
Continuity, Differentiability 
and Differentiation
5. If /(x) is a thrice differentiable function such that, lim 
x-»0
(d) None of these
2. The number of points of non-differentiability for f(x) = max-! ||x|-l|,M is :
H-------------------
1 + (tan 6) 
dythen at 0 = ir/3 is :
1. Let be a differentiable real valued function satisfying ffx + 2y) = f(x) + /(2y) + 
6xy(x + 2y) V x,y e R. Then /"(0), /"(I), /"(2) are in : 
(a) AP (b) GP (c) HP
1_____________
+ (cote)sin°-cot0
then the value of/'"(0) equals to :
(a) 0 (b) 1
6 =_____________ 1_____________
’ y 1 + (tan e)sin°-cos0 + (cot 0)^9-cote
(c) 2
3. Number of points of discontinuity of /(x) = + |j^ in x e [0,10Q] is/are (where [•] denotes
greatest integer function and {•} denotes fractional part function)
(a) 50 (b) 51 (c) 52 (d) 61
4. If /(x) has isolated point of discontinuity at x = a such that | /(x) | is continuous at x = a then :
(a) lim /(x) does not exist (b) lim /(x) + ffa) = 0
x->a x->a
(c) /(a) = 0 (d) None of these
/(4x)-3/(3x) + 3/(2x)-/(x) 
x3
_____________ 1_____________
+ 1 + (tan e)cosa-cot0 + (cot 0)cot°-sinO
(a) 0 (b) 1
(c) V3 (d) None of these
7. Let /'(x) = sin(x2) and y = f(x2 +1) then — at x = 1 is : 
dx
(c) 2 sin 4
Advanced Problems in Mathematics for JEE46
(b)(a)
(d)(c)
is 
(c) -6(a) -4
10. f is a differentiable function such that x = f(t2'),y = and /'(I) * 0 if
t=i
11. Let/(x) =■
(a) 0
12. Ify =1 +
a(a) y (b)
(c) y
13. Iff(x) =
(a) 4 (d) 1(c) 2
14. Let f (x) = then set of points, where /(x) is continuous, is :
(a) (b) a null set
(d) set of all rational numbers
if 
if
1 ----a 
x
y
x^l/x-a
a/x
1 
--Y x
7C 7T
4’4.
a 
1/x-a
1 + sin
1 - tan
V3 + 1
2 
V3-1
2
a
I --- a 
x
(/'(D)2
„ , d2y9. If 2 sm x ■ cosy = 1, then —y at 
dx2
(b) -2
sin2 x 
-sin2 x
(/'(l))2
t p i y
+ 1/x-p + 1/x-y
, P/x ! j/x
1/x-p 1/x-y
(b) 1
P/x____+
--p!x )
X l
(0 2
Y/x2
--pY
J 3
f 7tt A8. If/(x) =|sinx-|cosx||, then /' — =
I 6 )
(d) 4
, dy ., then — is : 
dx
(f'd))2
(e)
3 (/'(I))2
OX + 1
3
2 -
x
—; then /'(0) is equal to : 
x
(b) 3
, x is rational
, x is irrational’
1 11----a x f
a P Y ------ + + —!— 
a-x p-x y-x
P t y
+ 1/x-p + 1/x-y (d) q/
xl^ 1/x-a
Continuity, Differentiability and Differentiation 47
15. The number of values of x in (0,2n) where the function /(x) = is
Cd) 1
is
(c) 6 Cd) 7
(c)
(a) -2 (b) 2 (c) (d) None of these
(a) 1
(b) (c) 0 (d)
22. Let g(x) be the inverse of/(x) such that /'(x) = , then is equal to :
Cb)
23. Let /(x) =
I
tan x + cot x tan x - cot x
nII
d2(yjx)) 
dx2
equal to :
(b) 5 
i d2x 
18. If y = x + ex, then ----[dy2
20. If/(x) = 2 + |x|-|x-l|-|x + l|, then y'f--l + f‘ + f' I — I + f 'I - I is equal to :
continuous but non-derivable:
(a) 3 (b) 4 (c) 0
16. If /(x) =|x — 11 andg(x) = /(/(/(x))), theng'(x) is equal to :
(a) 1 for x > 2 (b) 1 for 2 3
17. If /(x) is a continuous function V x e R and the range of f (x) is (2, 126) and g(x) =
1
1 + x5 
g'(x)
l + (g(x))5
(d) l + (g(x))5
i is: 
x = ln 2 
12 2(a) -1 (b) ~ (c) ±
9 27 27
19. Let /(x) = x3 + 4x2 + 6x and g(x) be its inverse then the value of g'(-4): 
1 
2
./1A
(b) -1 (c) 2 (d) -2
9 I v 7C I21. If fix) = cos(x — 4[x]); 0 — Z
2
(d) lim /(x) = 1
2
29. Let f(x) be a polynomial in x. The second derivative of /(ex) w.r.t. x is :
(a) /"(ex)eJf+/'(ex) (b) /"(ex)e2x+/'(ex)e2x
(c) /"(ex)ex+/'(ex)c2x (d) /"(ex)e2x + ex/'(ex)
it
2
n
X * — r \
4 71 I; x e 0, - .
n 2 Jx = - L
4
_ kA .
’ '2;
Continuity, Differentiability and Differentiation 49
(c) e
33. If/(x) -1, then on the interval [0, n]:
(a) tan(/(x)) and----- are both continuous
(b) tan(/(x)) and----- are both discontinuous
-3
34. Let/(x) =
(d) 4(c) 2
be a continuous function at x = 0. The value of
(d) 2
then b + c = 
(a) 0
x 00
43. If y = tan
(a) (c) (d)^3-j
/2(x)/i(x)
/2(x)/i(x)ex
1
4
1
3x*3(l-
_____ 1
3x1/3 (1 ■
4
fnM
-a1'3
l + x^a1/3 J’
1 
X*3(l + x*3)
x 0
3 
*3(l + x■ ^3-j
O’) fnMfn-lM 
(d) fnWf^M 
dvx > 0, a > 0, then — is :
dx
(a)e “2 (b) e
42. Let /i (x) = ex and fn+1 (x) = e 
4~fnto equals : 
dx
(a) /n(x)
(C)
X^/3
(1 + ax)1/x 
b
(x + c)1/3 -1
(x + l)1/2 -1
(b) 6 
j 2
37. If Jx + y + yjy -x = 5, then —=
dx2
, s 2 ^4 , . 2(a) - (b) — (c) —
5 25 25
38. If /(x) = x3 + x4 + log x and g is the inverse of f, then g'(2) is :
39. The number of points at which the function, 
min{|x|,x2}
min{2x-l, x2}
is not differentiable is :
(a) 0 (b) 1 (d) 3
40. If _f(x) is a function such that /(x) + /"(x) = 0andg(x) = (/(x))2 + (/'(x))2 andg(3) = 8, then 
g(8) =
(a) 0 (b) 3 (c) 5 (d) 8
41. Let f is twice differentiable on R such that /(O) = l,/'(0) = 0 and f"(0) = -1, then for a eR, 
lim f f[-y=
(b) - 
x
(c) e 2 (d) e_2“2
for any n > 1, n g N. Then for any fixed n, the value of
1
25
Continuity, Differentiability and Differentiation 51
x 0
(c) Cd) Not possible
6
7
2
9
is non-differentiable will be :
(a) 1 (b) 2
; if x is rational 
; if x is irrational
discontinuous is : 
(a) 0
x
1 -X
(d) Infinitely many
; x>0
; x co 1 + x2"
Statement-2 : L H. L = R. H. L * /(I).
(a) Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation foi 
Statement-1
(b) Statement-1 is true, Statement-2 is true and Statement-2 is not the correct explanation foi 
Statement-1
(c) Statement-1 is true, Statement-2 is false
(d) Statement-1 is false, Statement-2 is true
55. If f (x) =
(x + 1)7 71Tx
(x2 -x + 1)6
(b) 11
57. If the function /(x) = -4e 2
sin(2x) ... „ .--------- is discontinuous at x = L
Continuity, Differentiability and Differentiation 53
(c) -1 (d) -2
62. Number of points at which the function /(x) = is not
x >1
(d) 3
(d) 2x2y
2 .22
sinx then :
sinx then :
(b) lim /(x) = loge 3*X—>1
(c) lim /(x) = -sinl 
x->r
(c) lim/(x) =-sinl
X->1”
dx
(d) 15 
if -00 R is not identically zero, differentiable function and satisfy the equations 
/(xy) = /(x)/(y)and/(x + z) = /tx) + /(z), then /(5) = 
(a) 3 (b) 5 (c) 10
min. (x, x2)
Cd) lim /(x) does not exist 
X->1”
(d) lim /(x) does not exist 
x-»r
(b) lim /(x) = loge 3
X->1"
(c) 2x(x + 71 -x'
59. Let x = y = —+ - then the value of — -x| 
t3 2t2 t dx I
T . ff > logc(2 + x)-x2n65. Let _f(x) = hm ——---------------
n->» 1 + x2"
(a) /(x) is continuous at x = 1
(a) n2y
64. Ifg(x) =/^x-71 -x
(a) 1-x2
derivable is :
(a) 0 (b) 1
63. Ify =(x + 71 + x2)n, then (1 + x2)—y + x^-is : 
dx2
Cb) -n2y
J
(c) 2
dy . 
_r_ ic • 
dx
(c) -y
and f'M = 1 - x2 then g'(x) equals to :
66. Let /(x + y) = f(xff(y') for all x and y, and /(5) = -2, /'(0) = 3, then /'(5) is equal to :
(a) 3 (b) 1 (c) -6 (d) 6
__ _ cr -x 1- Iogc(2+X)-X2n
67. Let /(x) = hm —------—----
1 + x2"
(a) /(x) is continuous at x = 1
(a) 2 (b) 0
60. If y ~2 = 1 + 2^/2 cos 2x, then :
d2 y—— = y (py +1) (qy -1) then the value of (p + q) equals to : 
dx2
Advanced Problems in Mathematics for JEl54
x * 0 is continuous at x = 0 then, which of the68. If f(x)
x = 0
(c) [k]=-2
(c) [0,4] (d) (-4, 4)
x70. If/(x) = ; then /'(0) is equal to :
x
(a) 4 (d) 1
c and y = ^2'
(d)(c)
— c x * 074. If/(x) =
2
(a) a = b = c (b) a = 2b = 3c (c) a = b = 2c
(a) 0 (b) 1
(a) -101 (b) 48 (d) 190
71. Fort G(0,1); let x
2
equals :
is continuous at x = 0; then : 
x = 0
-3
(d) [k]{k}=y
(b) 3
=72sin-1
+ be
(d) 2a = 2b = c
, d2y f k kA .then —at -, - equals to : 
dx2 7t
X 
i7|7j ’
X 
[idTi ’ 
(a) (-00,00)
|x|>l
, then domain of/'(x) is :
|x|0
Then
(a) g(x) is discontinuous at x = rc
(c) g(x) is differentiable at x = n
83. If/(x) = (4 + x)n, n eN and fr (0) represents the r* derivative of /(x) at x = 0, then the value 
ofyri®
(a) 2n
x + 1 
.(x-1)2
79. If y = f (x) and z = g(x) then —y equals
gr-fr
(g')2
x + l ; x 0
the number of points where g(/(x)) is not differentiable.
(a) 0 (b) 1 (c) 2 (d) None of these
81. Let /(x) = [sin x] + [cosx], x e[0,2n], where [■] denotes the greatest integer function, total
number of points where /(x) is non differentiable is equal to : 
(a) 2 (b) 3 (c) 4
82. Let /(x) = cos x, g(x) =4 min J? 4 “ ’
(b)l
d2y
dz“
(g')3
Advanced Problems in Mathematics for JEE56
6
(c)
(d) Three(c) Two
87. If/(x) =
(a) 4
88. If fM =
(a) 0 (c)