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7/31/2019 IIT JEE Maths TopicWisePreviousQuestionsWithAnswers
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Topic wise Previous IIT JEE Mathematics Questions
Complex numbers
01. If z =1 and z 1 , then all the values of2
z
1 zlie on
a) a line not passing through the origin b) z = 2
c) the xaxis d) the yaxis Ans. d
02. A particle P starts from the point z0 = 1+2i, where i 1= . It moves first horizontally a way from
origin by 5 units and then vertically a way from origin by 3 units to reach a point z1. From z1 the
particle moves 2 units in the direction of the vector i + j and then it moves though an angle2
in
anticlockwise direction on a circle with centre at origin, to reach a point z2. The point z2 is given by
a) 6 + 7i b) 7 + 6i Ans. d
c) 7 + 6i d) 6 + 7i
Paragraph question
Let A, B, C be three sets of complex numbers as defined as follows:
A = { z : Im (z) 1}
B = {z: z 2 i = 3
C = {z: Re ((1i)z) = 2 }
03. Then number of elements in the set A B C is Ans. b
a) 0 b) 1 c) 2 d)
04. Let z be any point in A B C . Then z + 1 i2 + z 5 i2 lie between
a) 25 and 29 b) 30 and 34 Ans. c
c) 35 and 39 d) 40 and 44
05. Let z be any point in A B C and let w be any point satisfying w 2 i

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a)2
(p q)(2q p)9
b)2
(q p)(2p q)9
c)2
(q 2p)(2q p)9
d)2
(2p q)(2q p)9
07. Let a, b, c, p, q be real numbers. Suppose , are the roots of the equation x2 + 2px+q=0 and1
,
are
the roots of the equation ax2+2bx+c=0, where
2{ 1,0,1} . Ans. b
Statement1 :2 2
(p q)(b ac) 0
Statement 2 b pa or c qa
Progression
08. If x is the fist term of an infinite G.P., whose sum is 10, then Ans. a
a) 0 < x < 10 b) 5 < x < 5 c) 10 < x < 10 d) x 10
Reasoning type
09. Suppose four distinct positive numbers a1, a2, a3, a4 are in G.P. Let b1 = a1, b2 = b1 + a2, b3 = b2 + a3 and
b4 = b3 + a4.
Statement 1 : The numbers b1, b2, b3, b4 are neither in A.P nor in G.P
Statement 2: The numbers b1, b2, b3, b4 are in H.P. Ans. d
Logarithm
10. If x > 1, y < 1, z < 1 are in G.P., then1 1 1
, ,1 logx 1 log y 1 logz+ + +
are in
a) A.P. b) H.P Ans. b
c) G.P d) None of these
11. The number of solutions of log4 (x1) = log2(x 3) Ans. b
a) 3 b) 1 c) 2 d) 0
Permutations and Combinations
12. The letters of the word COCHIN are permuted and all permutations are arranged in an alphabetical
order as in an English dictionary. The number of words that appear before the word COCHIN is
Ans. c

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a) 36 b) 192 c) 96 d) 48
13. Consider all possible permutations of the letters of the word ENDEANOEL. Match the entries in
Column I with the correctly related quantum number(s) in Column II. Indicate your answer by
darkening the appropriate bubbles of the 4 4 matrix given in the ORS.
Column I Column II
a) The number of permutations p) 5!
b) The number of permutations in q) 2 5!
which the letter E occurs in the
first and the lat positions is
c) The number of permutations in r) 7 5!
which none of the letters D, L, N
occurs in the last five positions is
d) The number of permutations in s) 21 5!
which the letters A, E, O occur only
in odd positions is
Ans:
p q r s
a p q r s
b p q r s
c p q r s
d p q r s
Binomial Theorem
14. If n and k are positive integers, show that
k k 1 k 2n n n n 1
2 2 20 k 1 k 1
+
n n 2
2 k 2
+
kn n k n
( 1)k 0 k
=
wheren
k
stands forn
kC .

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15. If n
k
stands forn
rC , the value of
30 30 30 30
0 10 1 11
+
30 30 30 30....
2 12 20 30
+
is Ans. a
a)
30
10
b)
30
15
c)
31
11
d)
60
30
Matrices
16. Let a, b and c be three distinct real numbers and f(x) be a quadratic polynomial satisfying the equation
2
2
2
4a 4a 1
4b 4b 1
4c 4c 1
f( 1)
f (1)
f(2)
=
2
2
2
3a 3a
3b 3b
3c 3c
+
+ +
Let V be the point of local maxima of y = f(x) and A be the point where y=f(x) meets the xaxis and B
be a point on y=f(x) such that AB subtends a right angle at V. Find the area of the region lying
between the curve and chord AB.
Ans. 2125
(unit)3
Passage type question
Let
1 0 0
2 1 03 2 1
and X1, X2, X3 be three column matrices such that
1 2
1 2
AX 0 ,AX 3
0 0
= =
and 3
2
AX 3
1
=
and let X be a 3 3 matrix whose columns are X1, X2, X3.
17. Value of det (x) is Ans. c
a) 2 b) 1 c) 3 d) 0
18. Sum of the elements of X1 is Ans. b
a) 1 b) 0 c) 4 d)
19. If [a] = [3 2 0] X
3
2
0
. Then a equals Ans. a

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a) 5 b) 4 c) 3/2 d) 5/2
Determinants
20. The value of for which the system of equations x y z 4+ + = , x 2y z 4 0 + + = , 2xyz=2 has no
solution is Ans. b
a)  3 b) 2 c) 0 d) 3
21. Reasoning type Ans. b
Consier the system of equations ax+by =0, cx + dy = 0, where a, b, c, d {0, 1} Statement 1 : The
probability that the system of equations has a unique solution is 3/8
Statemet 2 : The probability that the system of equations has a solutions is 1.
INEQUALITIES
22. A straight line through the vertex P of a triangle PQR intersects the side QR at the point S and the
circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then
Ans. b, d
a)1 1 2
PS ST QS SR+
c)1 1 4
PS ST QR+ < d)
1 1 4
PS ST QR+ >
23. Letn
n 2 2k 1
nS
n kn k ==
+ + and Tn =
n 1
2 2k 0
n
n kn k
= + + , for n=1, 2, 3,, Then.
a) Sn
3 3
c) nT
3 3
< d) nT
3 3
>
Ans. a, c
PROBABILITY
24. Let E denotes the complement of an event E. Let E, F, G be pair wise independent events such that
P(G)>0 and P(E F )=0. Then P(E FG) equals Ans. c
a) P(E) + P(F) b) P(E)P(F) c) P(E)P(F) d) P(E)P(F)
25. Reasoning type Ans. b
Consider the system of equations
ax + by = 0, cx + dy = 0, where a,b,c,d {0, 1}

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Statement 1: The probability that the system of equations has a unique solution is3
8.
Statement 2 : The probability that the system of equations has a solution is 1.
ELEMENTARY TRIGONOMETRY
26. If and are acute angles such that sin = , cos = 1/3, then + lies in
a) ] / 3, / 2[ b) ] / 2, 2 /3[
3. ]2 /3, 4 /3[ d) None of these Ans. b
27. Given (0, / 4), and tan1t (tan )= t2 =
cot(tan ) tan3t (cot )
= and cot4t (cot )= then
a) t1 > t2 > t3 > t4 b) t4 > t3 > t1 > t2
c) t3 > t1 > t2 > t4 d) t2 > t3 > t1 > t4 Ans. b
SOLUTION OF TRIANGLES AND APPLICATIONS OF TRIGONOMETRY
28. If a, b, c denote the lengths of the sides of a triangle opposite angles A, B, C of a triangle ABC, then
the correct relation among a, b, c, A, B and C is given by Ans. b
a) (b + c) sin ((B + C)/2) = a cos (A/2)
b) (b c) cos (A/2) = a sin ((B C)/2)
c) (b c0 cos (A/2) = 2a sin ((B C)/20
d) (b c) sin ((B C)/2) = a cos (A/2)
29. Let a, b, c be the sides of a triangle. No two of them are euqlal and R. If the roots of the equations,
x2+2(a+b+c)x+3 (ab+bc+ca)=0 are real, then Ans. a
a) 4 / 3 < b) 5/ 3 >
c) (1/ 3, 5 / 3) d) (4/ 3, 5/ 3)
TRIGONOMETRIC EQUATIONS
30. 0 2 , 22sin 5sin 2 0, + > then the range of is Ans. a
a) (0, / 6) (5 / 5, 2 ) b) (0, 5 / 6) ( ,2 )

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c) (0, / 6) ( , 2 ) d) None of these
31. The number of solutions, of the pair of equations 22sin cos 2 0 = and 22cos 3sin 0 = in the
interval [0, 2 ] is Ans. c
a) 0 b) 1 c) 2 d) 4
INVERSE TRIGONOMETRIC FUNCTIONS
32. If 0 < x < 1, then 21 x+ [{xcos(cot1x)+sin(cot1x)}21]1/2 =1
a)2
x
1 x+b) x
c)2
x 1 x+ d) 21 x+ Ans. c
CARTESIAN SYSTEM OF RECTANGULAR COORDINATES AND
STRAIGHT LINES
33. Reasoning type Ans. c
Lines L1: y x = 0 and L2: 2x + y = 0 intersect the line L3: y + 2 = 0 at P and Q respectively. The
bisector of the acute angle between L1 and L2 intersect L3 = at R.
Statement 1: The ratio PR: RQ equals 2 2 : 5 because
Statement 2: In any triangle bisector of an angle divides the triangle into two similar triangles.
34. Matrix match type
L1: x+3y5=0, L2: 3xky1=0, L3: 5x+2y12=0
Column I Column II
a) L1, L2, L3 concurrent, if p) K =  9
b) One of L1, L2, L3 is parallel to at least q) K = 6/5
one of the other two
c) L1, L
2, L
3form a triangle r) K = 5/6
d) L1, L2, L3 do not form a triangle s) K = 9
Ans:
p q r s
a p q r s

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b p q r s
c p q r s
d p q r s
CIRCLES AND SYSTEM OF CIRCLES
35. Let ABCD be a quadrilateral with area K, with side AB parallel to the side CD and AB = 2CD. Let AD
be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the
sides, then its radius is Ans. b
a) 3 b) 2 c) 3/2 d) 1
36. Point E and F are given by Ans. a
a) ( 3 / 2, 3/ 2), ( 3,0) b) ( 3 / 2, 1/ 2), ( 3,0)
c) ( 3 / 2, 3/ 2), ( 3 / 2, 1/2) d) ( 3 /2, 3 /2 ), ( 3 /2, 1 /2)
PAIR OF STRAIGHT LINES
37. Area of the triangle formed by the angle bisectors of the pair of lines x2
y2
+ 2y 1 =0 and the line
x+y=3 (in square units) is Ans. b
a) 1 b) 2 c) 3 d) 4
38. Let a and b be nonzero real numbers. Then the equation2 2
(ax by c)+ + 2 2(x 5xy 6y ) 0 + =
represents Ans. b
a) four straight lines, when c=0 and a, b are of the same sign.
b) two straight lines and a circle, when a = b, and c is of sign opposite to that of a.
c) Two straight lines and a hyperbola, when a and b are of the same sign and c is sign opposite to that
of a.
d) a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a .
CONIC SECTION (PARABOLA, ELLIPSE, HYPERBOLA)
39. Let P(x1, y1) and Q(x2, y2), y1

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Ans. b, c
40. Consider a branch of the hyperbola 2 2x 2y 2 2x 4 2y 6 0 = with vertex at the point A. Let B
be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A,
then area of the triangle ABC is
a) 1 2 / 3 b) 3 /2 1
c) 1 2 /3+ d) 1 3/ 2 1+ + Ans. b
THREE DIMENSIONAL GEOMETRY
41. Reasoning type
Consider the planes 3x6y2z=15 and 2x+y2z=5
Statement 1: The parametric equations of the line of intersection of the given planes are x=3+14t,
y=1+2t, z=15t.
Statement 2: The vector 14i + 2j + 15k is parallel to the line of intersection of the given planes.
Ans. d
Linked comprehension type
Consider the lines 1x 1 y 2 z 1
L :3 1 2
+ + += = and
2 x 2 y 2 z 3L :1 2 3 + = =
42. The unit vector perpendicular to both L1 and L2 is Ans. b
a)7 7
99
+ +i j kb)
7 5
5 3
+i j k
c)7 5
5 3
+ +i j kd)
7 7
99
i j k
43. The shortest distance between L1 and L2 is Ans. d
a) 0 b) 17 / 3 c) 41/5 3 d) 17 /5 3
44. The distance of the point (1, 1, 1) from the plane passing through the point (1, 2, 1) and whose
normal perpendicular to both the lines L1 and L2 is Ans. c
a) 2 / 75 b) 7 / 75 c) 13/ 75 d) 23/ 75

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VECTOR ALGEBRA
45. The number of distinct real values of for which the vectors 2 , + +i j k 2 , +i j k and2
,+ i j k are coplanar, is Ans. c
a) zero b) one c) two d) three
46. The edges of a parallelepiped are of unit lengths and are parallel to noncoplanar unit vectors a, b, c
such that a. b = b. c=c. a=1/2. Then the volume of the parallelepiped is
a)1
2b)
1
2 2Ans. a
c)3
2d)
1
3
FUNCTIONS
47. Let X and Y are two nonempty sets. Let f: X Y be a function. For A X and B Y, define1
f(A) {f(x):x A}f (B)= = {x X / f (x) B} , then
a) 1f (f (B)) B = b) 1f (f (B)) B
c)1
f (f (A)) A = d) 1f (f (A)) A Ans. b
LIMITS AND CONTINUITY
48. If 2x 0
((a n)nx tan x)sin nx
lim x
= 0, where n is a nonzero real numbers, then a is equal to
a) 0 b)n
n 1+Ans. d
c) n d) n + 1/n
49. For x > 0,1/ x sin x
x 0lim ((sin x) (1/x) )
+ is Ans. c
a) 0 b) 1 c) 1 d) 2
DIFFERENTIATION
50.2
2
d x
dyequals Ans. b
a)
12
2
d y
dx
b)
12
2
d y
dx
3dy
dx

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c)
22
2
d y dy
dx dx
d)
32
2
d y dy
dx dx
51. Let g(x) =4
m
(x 1);
logcos (x 1)
0 < x < 2, m and n are integers, m 0, n > 0, and let p be the left hand
derivative of x 1  at x =1. Ifx 1limg(x) p,
= then Ans. c
a) n = 1, m = 1 b) n =1, m = 1
c) n = 2, m = 2 d) n > 2, m = n
APPLICATIONS OF DERIVATIVES
52. The tangent to the curve y=ex
drawn at the point (c, ec) intersects the line joining the points (c 1, e
c1)
and (c+1, ec+1
) Ans. a
a) on the left of x = c b) on the right of x = c
c) at no point d) at all point
53. Let the function g : ( , ) ( / 2, / 2) be given by g(u) =2 tan1 (eu) / 2, then g is
a) even and is strictly increasing in (0, ) Ans. c
b) odd and is strictly decreasing in ( , )
c) odd and is strictly increasing in ( , )
d) neither even nor odd, but in strictly increasing in ( , )
INDEFINITE INTEGRAION
54. Reasoning type Ans. d
Let F(x) be an indefinite integral of sin2
x.
Statement 1 : The function F(x) satisfies F(x+ ) = F(x) for all real x.
Statement 2 :2 2
sin ( x) sin x + = for all real x.
55. Letx
4x 2x
eI dx,
e e 1=
+ +x
4x 2x
eJ dx
e e 1
=
+ +. Then for an arbitrary constant C, the value of JI
equals Ans. c
a)1
2log
4x 2x
4x x
e e 1
e e 1
+
+ + + C b)
2x x
2x x
1 e e 1log
2 e e 1
+ +
+ + C

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c)2x x
2x x
1 e e 1log
2 e e 1
+
+ + + C d)
4x 2x
4x 2x
1 e e 1log
2 e e 1
+ +
+ + C
DEFINITE INTEGRALS
56.
2sec
2
2 2x / 4
f(t) dt
lim x /16
equals Ans. a
a)8
f(2)
b)2
f(2)
3.2 1
f2
4. 4f(2)
57. Multiple correct answer type
Let f(x) be a non constant twice differentiable function defined on ( , ) such that f(x) = f(1x) and
1f ' 0
4
=
Then Ans. a, b, c, d
a) f(x) vanishes at lest twice on [0, 1] b) f1
2
= 0
c)
1/ 2
1/ 2
1f x
2
+
sin x dx = 0 d)
1/ 2
sin t
0
f(t)e 1
sin t
1/ 2
dt f (1 t)e = dt
DIFFERENTIAL EQUATIONS
58. The differentiation equation
21 ydy
dx y
= determines a family of circles with
a) variable radii and a fixed centre (0, 1)
b) variable radii and a fixed centre (0, 1)
c) fixed radius 1 and a variable centres along the xaxis
d) fixed radius 1 and variable centres along the yaxis. Ans. c
59. Reasoning type
Let a solution y=y(x) of the differential equation2
x x 1 dy  2y y 1 dx=0 satisfy y(2)= 2 / 3 .
Statement 1: y(x) = sec(sec1
x  / 6 )

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Statement 2: y(x) is given by1
y=
2 3
x
2
11
x Ans. c