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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 x-axis d) the y-axis 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 ((1-i)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 i|2 + |z 5 i|2 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

Statement-1 :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 (x-1) = 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 x-axis 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 X-1 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 + + = , 2x-y-z=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(cot-1x)+sin(cot-1x)}2-1]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: 3x-ky-1=0, L3: 5x+2y-12=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 non-zero 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 3x-6y-2z=15 and 2x+y-2z=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

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 non-coplanar 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

c-1)

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 tan-1 (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 J-I

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)

Let f(x) be a non constant twice differentiable function defined on ( , ) such that f(x) = f(1-x) 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 x-axis

d) fixed radius 1 and variable centres along the y-axis. 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(sec-1

x - / 6 )

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Statement 2: y(x) is given by1

y=

2 3

x-

2

11

x Ans. c