This section contains 8 multiple questions. Each
question has four choices (A), (B), (C) and (D) out of
s.c
which ONLY one is correct.
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SECTION - I: (Single Correct Answer Type)
1. Let a1, a2, a3, … be in harmonic progression with a1 = 5 and
(B) 23
(C) 24
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he
(A) 22
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a20 = 25. The least positive integer n for which an < 0 is
(D) 25
The equation of a plane passing through the line of
intersection of the planes x + 2y + 3z = 2 and x – y + z =
2
from the point (3, 1, – 1) is
3
(A) 5x  11y  z  17
(B) 2x  y  3 2  1
(C) x  y  z  3
ma
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(D)
ne
er
ma
t
he
x  2y 1  2
s.c
3 and at a distance
3.
Let PQR be a triangle of area  with
Pio
a  2, b 
7
5
and c  , where a, b and c are the lengths
2
2
of the sides of the triangle opposite to the angles at P, Q
and R respectively.
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2.
3
4
(B)
2
45
4
2
 45 
(D) 

 4 
Pio
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ma
t
he
ma
tic
 3 
(C) 

 4 
om
(A)
2 sin P  sin 2P
equals
2 sin P  sin 2P
s.c
Then
om
s.c
ma
tic
he
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4.
 


If a and b are vectors such that a  b  29 and


a  2i  3j  4k  2i  3j  4k  2i  3j  4k  b,
 
 

 
then a possible value of  a  b  .  7i  2j  3k  is
er

Pio
ne
(A) 0
(B) 3
(D) 8
5.
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s.c
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(C) 4
If P is a 3  3 matrix such that PT = 2P + I, where PT is
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the transpose of P and I is
3  3 the identity matrix, then there exists a column
er
 x  0
matrix X =  y   0 such that
 z  0
Pio
ne
0
(A) PX  0
0
(B) PX = X
(D) PX = – X
Let and   a  and   a  be the roots of the equation

3

1  a  1 x2 
Then
lim
a 0

 
1 a 1 x 
   a  and
5
(A)  and 1
2
6

1  a  1  0 where a  1
lim
a 0
1
(B)  and  1
2
ne
er
ma
t
7
(C)  and 2
2
Pio
  a  are
he
6.
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(C) PX = 2X
(D)
7.
ma
t
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ma
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s.c
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9
 and 3
2
Four fair dice D1, D2, D3 and D4, each having six faces
numbered 1, 2, 3, 4 , 5 and 6, are rolled simultaneously .
er
The probability that D4 shows a number appearing on
one of D1, D2 and D3 is
91
216
Pio
ne
(A)
(B)
108
216
(D)
127
216
om
125
216
8.
The value of the integral
/2
x 
 2
x

ln

 cos x dx is



x

/2
(B)
Pio
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er
ma
t
(C)
2
4
2
2
4
2
he
(A) 0
ma
tic
s.c
(C)
SECTION -II: Paragraph Type
2
(D)
2
relating to three paragraphs with two questions on each
paragraph. Each question has four choices (A), (B), (C)
Paragraph for Questions 49 and 50
s.c
and (D) out of which ONLY ONE is correct.
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This section contains 6 multiple choice questions
A tangent PT is drawn to the circle x2 + y2 = 4 at the


3 , 1 , A straight line L, perpendicular to PT
ma
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point P
2
is a tangent to the circle  x  3  y 2  1.
A possible equation of L is
(A) x  3 y  1
(B) x  3 y  1
x  3 y 1
(D) x  3 y  5
Pio
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er
ma
t
he
9.
(C)
(A) x = 4
om
(C) x + 3 y=
(D) x  2 2 y  6
Pio
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ma
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ma
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4
(B) y = 2
s.c
10. A common tangent of the two circles is
Paragraph for Questions 51 and 52
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2
Let f  x   1  x  sin2 x  x 2 for all x  IR , and let
 2 t  1

gx  
 1nt  f(t) dt for all x  1,   .
t 1

1
11. Consider the statements :

f  x   2x  2 1  x2

ma
tic
P : There exists some x  IR such that
s.c
x
Q : There exists some x  IR such that
2f  x   1  2x 1  x 
Then
is false
Pio
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er
ma
t
(C) P is false and Q is true
are false
(B) P is true and Q
he
(A) both P and Q are true
(D) both P and Q
(A) g is increasing on 1,  
(B) g is decreasing on
s.c
1,  
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12. Which of the following is true?
(C) g is increasing on (1, 2) and decreasing on 2,  
he
ma
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(D) g is decreasing on (1, 2) and increasing on  2,  
ma
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SECTION - I: (Single Correct Answer Type)
Paragraph for Questions 53 and 54
Let an denote the number of all n-digit positive integers
er
formed by the digits 0, 1 or both such that no
consecutive digits are 0. Let bn = the number of such n-
ne
digit integers ending with digit 1 and cn = the number of
such n-digit integers ending with digit 0.
Pio
13. The value of b6 is
(A) 7
(B) 8
(C) 9
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(D) 11
14. Which of the following is correct?
(B) c17  c16  c15
(A) a17  a16  a15
Pio
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a17  c17  b16
he
(C) b17  b16  c16
(D)
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s.c
ma
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SECTION III: Multiple Correct Answer (s) Type
he
This section contains 6 multiple choice questions. Each
question has four choices (A), (B), (C) and (D) out of
ma
t
which ONE or MORE are correct.
15. For every integer n, let an and bn be real numbers. Let
function f: IR  IR be given by
for x  [2n, 2n  1]
, for all int egers n.
for x   2n  1, 2n 
er
 a n  sin  x,
f x  
bn  cos  x,
If f is continuous, then which of the following hold(s) for
ne
all n?
(A) a n1  bn1  0
Pio
(C) a n  bn1  1
(B) a n  bn  1
(D) a n1  bn  1
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s.c
ma
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16. If the straight lines
x 1 y 1 z

 and
2
k
2
x 1 y 1 z

 are coplanar, then the plane(s)
5
2
k
(B) y + z = – 1
Pio
ne
er
ma
t
(A) y + 2z = – 1
he
containing these two lines is(are)
(D) y – 2z = – 1
ma
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s.c
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(C) y – z = – 1
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1, 4 4
17. If the adjoint of a 3  3 matrix P is  2 1 7  , then the
 1 1 3 
ne
possible value(s) of the determinant of P is (are)
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(A) – 2
(B) – 1
(D) 2
18.
Let f :  1, 1  R be such that
I
f  cos 4  
2
   
for

 0,    ,  . Then the
2  sec2 
 4 4 2
3
2
(B) 1 
2
3
Pio
ne
er
ma
t
(C) 1 
3
2
he
1
value(s) of f   is (are)
3
(A) 1 
ma
tic
s.c
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(C) 1
(D)
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2
3
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ma
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s.c
1
19.
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Let X and Y be two events such that
1
1
1
P  X Y   , P  Y X   and P  X  Y   . Which of
2
3
6
the following is (are) correct?
2
3
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(A) P  X  Y  
and Y are independent
ne
(C) X and Y are not independent


Pio
(D) P X c  Y 
1
3
(B) X
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s.c
ma
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he
ma
t
x
2
20. If f  x    et  t  2 t  3 dt for all x  0,   , then
0
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(A) f has a local maximum at x = 2
(B) f is decreasing on (2, 3)
Pio
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(C) there exists some c   0,   such that f’’(c) = 0
Pio
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(D) f has a local minimum at x = 3