RANK BOOSTER-1
SINGLE CORRECT OPTION :
3 3
(esin x 1) 4 cos x 1 esin x e x e x x 3 2 x 2 4
1. y , then the value of y ''(0) is
ex
(A) 4 (B) 4 (C) 8 (D) 8
2. p , q , r are prime numbers and , , are positive integers such that L.C.M. of , , is p 3 q 2 r and
greatest common divisor of , , is pqr then the number of possible triplets ( , , ) will be
(A) 6 (B) 12 (C) 72 (D) 96
3. Consider the locus of the complex number z in the Argand plane given by Re( z ) 2 z 7 2i .
Let P z1 .Q( z2 ). Be two complex numbers satisfying the given locus and also satisfying
z 2 i
arg 1 R . The minimum value of PR.QR where R represents the point
z 2 2 i 2
(7, 2) is
(A) 25 (B) 12 (C) 10 (D) 50
x
4. If f ( x) is a 4 degree polynomial satisfying f ( x) for x 0,1, 2,3, 4, then f (6) is
x 1
12
(A) (B)0 (C) 1 (D) 6
7
1 2 x n e.21 x
5. If f ( x) 2
dx and f (0) 2, then f (1) is
x 2x
(A) 0 (B) 3 (C) 1 (D) 3
6. Given y f ( x) is a solution of differential equation xdy xdx ydy ydx, also f (0) 0,
3
f (1) tan , then the area bounded by y f ( x) on x axis and x 0 to x tan is equal to
8 8
1 1 3 1 3 1 2
(A)
2
2 1 (B)
2
2 1 (C)
2
2 1 (D) 2
2 1
7. The sequence of all positive palindromes are written in ascending order
1, 2,3, 4,5, 6, 7,8,9,11, 22,........ then
(A) Then 2018th positive palindrome is 1019101 (B) The rank of 19999991 is 2998
(C) The rank of 2140412 is 3139 (D) All of these
8. In a ABC , AB 85, BC 90 and AC 102. P is an interior point & line segments are drawn
through P parallel to the sides of the triangle. If these three line-segments (as shown in the diagram
5d
by RQ, R ' Q ', R '' Q '' ) are of equal length d , then is equal to
34
(A) 9 (B) 6 (C) 3 (D) 12
1
9. The solution of differential equation 2 xy 4 e y 2 xy 3 y dx x 2 y 4 e y x 2 y 2 3 x dy 0 is
x2 x x2 x x2 x x2 x
(A) x 2e y 3 c (B) x 2 e y 3 c (C) x 2 e y c (D) x 2 e y c
y y y y y y3 y y3
10. A circle C1 with radius 5 touches x axis and another circle C2 with radius 4 touches y axis. The
two circles touches each other externally so that their point of contact lies in the first quadrant, let
the locus of their point of contact be the curve S. Two tangents are drawn to the curve S from point
P(9,5) to meet the curve at Q and R. The area of PQR is,
29 27 13 25
(A) (B) (C) (D)
4 5 2 3
MULTIPLE CORRECT OPTIONS :
11. Let f : R R, f ( x ) 3sin x sin 3 x, g : R R , g ( x ) 3 cos 2 x 4 cos x and
h :[ , 2 ] [ , 2 ], h ( x ) 1 2 cos x. Then the correct expression(s) is/are
64
(A) f ( x).g ( x).h( x)dx (B) g ( x ) h( x )x [ , 2 ]
0
3
56
(C) f ( x).g ( x).h( x)dx (D) g ( x) h( x)x 0,
0
3 2
12. If a set ' A ' is 1, 0, 1, 2 then which of the following is(are) correct?
(A) A square matrix of order 3 is constructed whose elements are from set ' A ' then the probability
27
that matrix is skew matrix is 9
4
(B) A square matrix of order 3 is constructed whose elements are from set ' A ' then probability that
1
matrix is symmetric is
64
3
(C) A function f : A A is defined then probability that function is onto is
32
3
(D) A function f : A A is defined and f is onto then the probability that f (i ) i i A is
8
13. Which of the following options are correct?
2
1 3 2
(A) nC0 nC3 nC6 ..... nC1 nC2 nC4 nC5 ..... nC1 nC2 nC4 nC5 ..... 1
2 4
5 2
(B) If a and b are two positive numbers such that a b 4 then the maximum value of
log 21/5 a 2 log 21/2 b 2 is equal to 4
2 2
2 2
2
(C) Constant term in x 2 2 2 2 2 .....2 is equal to 2
2
25 25
25C 25
C C 25
C C
(D) The coefficient of x 24 in 25 1 x x 22 25 2 x 32 25 3 x 42 25 4 ..... x 252 25 25
C0 C1 C2 C3 C24
is equal to 2925
2
14. The vertices of a triangle ABC are A (2, 0, 2), B ( 1,1,1) and C 1, 2, 4 . The point D and E
divide the sides AB and CA in the ratio 1: 2 respectively. Another point F is taken in space such that
the perpendicular drawn from F to the plane containing ABC , meets the plane at the point of
intersection of the line segment CD and BE. If the distance of F from the plane of triangle ABC is
2 units, then
7
(A) The volume of the tetrahedron ABCF is cubic units
3
7
(B) The volume of the tetrahedron ABCF is cubic units
6
(C) One of the equation of the line AF is r 2iˆ 2kˆ 2kˆ iˆ R
(D) One of the equation of the line AF is r 2iˆ 2kˆ iˆ 7 kˆ
15. The function f ( x) x1/3 x 1
(A) has two inflection points (B) has one point of extremum
3 2 ,
8/3
(C) is non – differentiable (D) Range of f ( x ) is
16. PQ and RS are normal chord of parabola y 2 8 x at P and R on the curve respectively and the
points P, Q, R, S are concyclic. If A is vertex of parabola, then
(A) centroid of APR lies on x axis (B) centroid of APR lies on y axis
(C) PR is parallel to tangent at the vertex (D) PR is parallel to axis of parabola
x 2 9bx 17
17. If f : R [1, 2], f ( x) is onto function and f '( d ) f '(e) 0. Then
ax3 x 2 bx 33
x2 y2 1
(A)eccentricity of curve 2
1 can be
( d e) b 1 2
x2 y2 5
(B)eccentricity of curve 2
1 can be
( d e) b 1 2
x2 y2 3
(C)eccentricity of curve 2
1 can be
( d e) a 1 2
x2 y2 7
(D) eccentricity of curve 2
1 can be
( d e) a 1 2
18. A differentiable function f : R R satisfies the functional equation
f ( x). f ( y) f ( x y) e x f ( y) e y f ( x) xyx, y R. If f '(0) 0 and f (0) 0, then which of the
following statements is/are correct?
x2
f ( x) 1
(A) lim 2
x0 x 2
(B) f '(t ) f (t ) dt 0 x 1
x
(C) F ( x2 ) F ( x1 )x2 x1 , where F ( x ) f '( x ) f ( x )
(D) There exists at least two horizontal tangents to the curve y f ( x) in (1,1).
1 32 1 1 32 (n 1)5
19. Let Sn lim 6 6 ..... and Tn lim 6 6 ..... , then which of the following
n n
n n n n
n n6
is/are true?
1 1 1 1
(A) Sn (B) S n Tn (C) S n Tn (D) Tn
6 3 3 6
3
20. z1 , z2 .....zn be non-zero complex numbers of equal modulus and satisfying the equation
z n z z n 1 1 0, then
n n zj n n zj n n zj n n zj
(A) Re 0 (B) Re 1 (C) Im 0 (D) Im
j 1 k 1 zk j 1 k 1 zk j 1 k 1 zk j 1 k 1 zk
21. A tangent L1 is drawn to the curve x 2 4 y 2 16 at point A in first quadrant whose abscissa is 5.
Another tangent L2 parallel to L1 meets the curve at B. L3 and L4 are normal to the curve at A and B
lines L1 , L2 , L3 , L4 forms a rectangle, then
(A) equation of normal at B is 12 x 10 y 75 0
2400
(B) area of rectangle L1 L2 L3 L4 is
61
32
(C) radius of largest circle inscribed in the rectangle is
61
109
(D) radius of the circle circumscribing the rectangle is
2
4 z (z z )
22. Four points z1 , z2 , z3 , z4 in complex plane such that z1 , z2 1, z3 1 and z3 2 1 4 , then
3 z1 z4 1
z4 can be equal to
1 1
(A) (B) e (C) (D)
e
n 4 n 1
C4 r 1
23. Let f (n) , then
r 1 r
2 2
(A) f (10) (2 40 1) (B) f (20) (280 1)
21 21
2 2
(C) f (50) (2200 1) (D) f (n) (24 n 1)
21 (n 1)
3
24. If 1 2 3 ..... n be the values of ' ' for which equation x 3 x 0 has all integral roots,
if h( x) x3 3x 1 2 ..... n for x [1, ) and g ( x ) be the inverse of h( x) then:
(A) 12 22 ..... n2 a number which is not the perfect square of an integer
531
(B) area bounded by g ( x ) and y axis between limits y 2 to y 5 is
4
3 22
x 3x dx g ( x)dx 64
3
(C)
1 2
3 22
x 3 x dx g ( x)dx 56
3
(D)
1 2
f ( x)
25. If f ( x y ) f ( x ). f ( y ) for all x, y R and f (0) 0. Also F ( x) , then
1 ( f ( x)) 2
2014 2014
(A) F ( x )dx F ( x )dx
2013 0
2014 2013 2014
(B) F ( x) dx F ( x) dx F ( x) dx
2013 0 0
4
2013
F ( x)
(C) dx 2013 (where f ( x ) is non-zero throughout the interval)
2013
f ( x)
2014 2014
(D) (2 F ( x) F ( x))dx 2 F ( x)dx
2014 0
26. If 2a 2 tan10 tan 50, 2b tan 20 tan 50, 2c 2 tan10 tan 70, 2d tan 20 tan 70, then
which of the following is/are correct?
(A) a d b c (B) a b c (C) a b c d (D) a b c d
27. Tangents are drawn to parabola y 2 16 x at the point’s A, B and C such that three tangents from a
triangle PQR. If 1 , 2 and 3 be the inclinations of these tangents with the axis of x such that their
cotangents from an A.P. with common difference 3. Then which of following are correct:
(A) Area of PQR is 432 (B) Area of ABC is 832
(C) Area of PQR is 416 (D) Area of ABC is 864
28. If cos x cos y a
cos 2 x cos 2 y b
cos 3x cos 3 y c , then which of the following is/are true?
2 b2 a2 b 2
(A) cos x cos y 1 (B) cos x.cos y
2 2 4
3
(C) 2a c 3a(1 b) (D) a b c 3abc
x2 y2
29. Let and f ( ) be the eccentricity of the ellipse 2 2
2 2
1, (3b 2 2a 2 ) and
3b 2a 2b a
x2 y2
2 2
2
1, (2b2 a 2 ) respectively, then
2b a b
(A) f ( ) , b R 0
1 2
1/2
1
(B) ffff ( ) d
0
4
1
(C) e ( f ( ) f ''( ))d e 2 3/ 2
c
(1 ) 1 2
(D) f ( ) , b R 0
2 1 2
30. For every integer n, let a and b be real numbers. Let functions f : IR IR be given by
an sin x, for x [2n, 2n 1]
f ( x)
bn cos x, for x (2n 1, 2n)
For all integers n. If f is continuous, then which of the following hold(s) for all n ?
(A) an1 bn1 0 (B) an bn 1 (C) an bn 1 1 (D) an 1 bn 1
5
PARAGRAPH :
Passage-1
Let r is a position vector of a variable point in Cartesian OXY plane such that r . 10 ˆj 8iˆ rˆ 40
8
2
2
and p1 max r 2iˆ 3 ˆj , p2 min r 2iˆ 3 ˆj . A tangent line is drawn to the curve y 2 at
x
the point A with abscissa 2. The drawn line cuts x axis at a point B.
31. p2 is equal to
32. AB OB is
Passage-2
Let 0 and c denotes the determinant of cofactors, then c n1 , where n ( 0) is the order of
bc a 2 ca b 2 ab c 2
33. If a, b, c are the roots of the equation x3 px 2 r 0, then the value of ca b 2 ab c 2 bc a 2 is
ab c 2 bc a 2 ca b 2
(if p 0.9 )
34. If a, b, c are the roots of the equation x3 3x 2 3x 7 0, then the value of
2bc a 2 c2 b2
c2 2ac b 2 a2 is
b2 a2 2ab c 2
Passage-3
A ten-sided regular polygon is inscribed in a circle z 1. A1 A2 ..... A10 to its vertices, G1 is centroid of
A1 A4 A8 , G2 be centroid of A2 A6 A9 and G3 be centroid of A3 A5 A7 . P is centroid of G1G2G3 . If O
is the center of circle.
35. Angle POA1 is equal to
36. 100 OP is equal to
Passage-4
x2 y 2 x2 y 2
Consider two ellipse S1 : 2
1; S 2 : 2 1; 4 a 2 b2 15
a 4 15 b
Let a tangent is drawn to the curve S1 at point P so as to intersect the curve S2 at point Q and R.
Also tangents to the curve S 2 at Q and R intersect at point S (3, 4). Moreover the foot of
4 2
perpendicular from one of the focus of S1 on the tangent at P is 1 ,2 v
5 5
37. Point of contact P is
9 17 9 8 9 8 9 17
(A) , (B) , (C) , (D) ,
5 5 5 5 5 5 5 5
38. If normals to the curve S 2 are drawn at point Q and R so as to intersect at T , then coordinates of T
are
4 13 3 4 3 4 15 10
(A) , (B) , (C) , (D) ,
11 11 11 11 11 11 11 11
6
Passage-5
The polynomial f ( x) x 2007 17 x 2006 1has distinct zeroes r1 , r2 , r3 ....., r2007 . The polynomial P( x)
and g ( x) of degree 2007 has property that
1
P rj 0 j 1, 2,3,....., 2007
rj
1
g 0
r
j
259 P(1)
39. The value of is equal to
17 P(1)
(A) 17 (B) 18 (C) 19 (D) 20
40. The sum of roots of g ( x) 0
(A) 17 (B) 17 (C) 0 (D) 1
NUMERICAL GRID
41. Consider 5 points in a plane. If m denotes the maximum number of points of intersection of the
perpendiculars drawn from each point to the lines joining the other points then ( m 310) is equal to
1
42. The shortest distance between (1 x)2 ( x y )2 ( y z ) 2 z 2 and 4 x 2 y 4 z 7 0 in 3-
4
dimensional coordinates system is equal to _________
43. A 3 3 determinant has entries either 1 or 1. Let S be the set of all determinants such that the
1 1 1
product of elements of any row or column is 1. For example, 1 1 1 is an element of S and
1 1 1
3 2 3
number of elements in S is m. Let P 2 2 3 and trace of the matrix adj ( adj P ) is n then the
0 1 1
m
value of is
n
44. If a , b , c , d , e are positive real numbers such that a b c d e 15 and ab2c3d 4e5 (120)3 .50,
then the value of a 2 b 2 c 2 d 2 e 2 is
(2n 1)(2n 2 2n 1) n
45. If Tn
(n 1)2 n 2 (n 1) 2 (n 2) 2
, n N , n 2. If K lim
n
r 2
Tr then 27k is equal to
46. Let f : R R be a function defined by f ( x) x3 3x 2 9 x 27. If x ( a , b ) ( c , d ) then
f 3 ( x) 3 f 2 ( x) 9 f ( x) 27 f ( x3 4 x 2 3x 19). Then (b a ) ( c d ) is
9
47. In a triangle ABC , side AC 4 units and sin A sin B sin B sin C sin C sin A . If A is the area of
4
A
the triangle ABC , then is equal to
3
48. If p is the least positive integer which satisfies the equation x 2 x 2 9 x 20 x 2 8 x 18 and
q is the minimum value of sin 2 x sin x 1 and is the only positive root of
2021 2022
1 x 2 p 2021
2022
x 1, then the value of 2022
1 p 2 p 1 2017 pq 8q is
7
49. The number of solution(s) of the equation 2 sin 25 x 1 8sin 50 x cos 2 50 x ; x 0, is/are
4 25
50. The equation z100 (15 z 1)100 0 has 100 complex roots zi , zi (i 1, 2,3,.....50), then the value of
1 100 1
is
100 r 1 zr 2
51. Let the vertices A, B , C of triangle ABC are represented by complex numbers p , q , r respectively in
complex plane. If the angles at B and C are each equal to and
2
(q r ) 2 2 3 p q r p , then 2 is (where [.] represents greatest integer function)
52. Let f ( x) be a non-constant thrice differentiable function defined on (, ) such that
f ( x) f (6 x) and f '(0) 0 f '(2) f '(5). If n is the minimum number of roots of
( f ''( x) 2 ) f '( x) f '''( x) 0 in the interval [0, 6], then the value of n is ____________
53. A cylindrical container is to be made from certain solid material with the following constraints. It
has fixed inner volume of Vmm3 , has a 2mm thick solid wall and is open at the top. The bottom of
the container is solid circular disc of thickness 2mm and is of radius equal to the outer radius of the
container. If the volume of the material used to make the container is minimum when the inner
V
radius of the container is 10mm, then the value of is _________
250
54. All the face cards from a pack of 52 plying cards are removed. From the remaining pack half of the
cards are randomly removed without looking at then and then randomly drawn two cards
simultaneously from the remaining. If the probability that, two cards drawn are both aces, is
p( 38C20 )
40
, then the value of p is?
C20 20C2
55. The projection length of a variable vector xiˆ yjˆ zkˆ on the vector p iˆ 2 ˆj 3kˆ is 6. Let be the
minimum projection length of the vector x 2iˆ y 2 ˆj z 2 kˆ on the vector p, then the value of 3 l 2 15
is
56. A class has 3 teachers and 6 students, they are seated in a line of 9 chairs if the probability that there
are exactly two students between any two adjacent teachers is P, then 14P is equal to
57. The minimum area bounded by the function y f ( x) and y x 9( R ) where f satisfies the
relation f ( x y ) f ( x) f ( y ) y f ( x)x, y R and f '(0) 0 is 9 A, value of A is ______
f ( x)
58. If f ( x ) is a polynomial function such that f ( x) f '( x) f ''( x) f '''( x) x3 and g ( x ) dx
x3
and g (1) 1, then g (e) is (where [.] denotes greatest integer function and e is Napier’s constant)
5 17
59. Let A and B is 2 2 matrix whose elements are elements of set
13 29
P{3,5, 7,11,13,17,19, 23, 29}. If all the elements of AB are multiple of 4, then the total number of
possible matrices B with distinct elements.
x
2 1, 0 x 1
60. Let f ( x ) and g ( x) (2 x 1)( x k ) 3, x 0. Then g ( f ( x)) is continuous at x 1
1 ,1 x 2
2
if 12k is equal to _________
8
ANSWER KEY
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
D C A B D A D A A B ABD BCD ABD AC ABCD
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
AC ABC ABC ACD BC ABCD AB AC ABD BCD ABD AD ABC ABC BD
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
3.40 3 0.53 0 2.51 11.11 B C A C 5 2 8 55 3
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
10 4 6 5 226 5 12 4 6 9 0.5 8 0 960 6
