Mathematics (2003, Screening Solution)
57.
The centre of the circle inscribed in the square determined by the two pairs of
lines x2  8x + 12 = 0 and y2  14y + 45 = 0, is
(A) (7, 3)
(B) (7, 4)
(C) (4, 7)
(D) (3, 7)
58.
The coefficient of t24 in the expansion of (1 + t2)12 (1 + t12) (1 + t24) is
(A) 12C6 + 3
(B) 12C6
(C) 12C6 + 1
(D) 12C6 + 2
59.
If the angles of a triangle are in the ratio 4 : 1 : 1, then the longest side and the
perimeter are in the ratio
(A)
3:2+ 3
(B) 1 : 6
(C) 1 : 2 + 3
(D) 2 : 3
1
 t 1  t 
m
60.
For natural numbers m and n, if I(m, n) =
n
I m  1, n  1
(A) m  1
2n
n

I  m  1,
(C) m  1 m  1
n  1
0
n
dt
, then I (m ,n) equals
m
I m  1, n  1
(B) n  1
2n
n

I  m  1, n  1
(D) m  1 m  1
61.
The system of equations x + ay = 0, y + az = 0, z + ax = 0 has infinitely many
solutions for
(A) a = 1
(B) a = 0
(C) a = –1
(D) nor real value of a
62.
The slopes of the focal chords of the parabola y2 = 16x which are tangents to the
circle (x – 6)2 + y2 = 2 are
1
(A) 1, –1
(B) 2 , –2
1
(C) – 2 , 2
(D) 2, –2
63.
Let A, B and C be three events and A, B and C be the corresponding
complementary events. If the probabilities of the events B, A  B  C and
3 1
1
,
A  B  C are 4 3 and 3 respectively, then the probability of the event B  C
is
1
1
(A) 9
(B) 12
1
(C) 15
64.
1
(D) 18
The area of the region in the first quadrant that is bounded by the curves y = x ,
x = 2y + 3 and the x-axis is
(A) 2 3
(B) 18
34
(D) 3
(C) 9
65.
x  4 y 2 z k


1
2 lies completely in the plane 2x – 4y + z = 7 for
The line 1
(A) no value of k
(B) k = 7
(C) k = 1
(D) k = – 7
66.
 0 
1 0
 1 1


 and B = 5 1 are two matrices, then A2 = B is true for
If A = 
(A)  = –1
(B)  = 1
(C)  = 4
(D) no real value of 
67.
The minimum value of f (x) = x2 + 2bx + 2c2 is more than the maximum value of g
(x) = –x2– 2cx + b2, x being real, for
(A) |c| > |b| 2
(B) 0 < c < b 2
(C) b 2 < c < 0
68.
69.
(D) no values of b and c
dy
If y (t) is a solution of the differential equation (1 + t) dt – ty = 1; y (0) = –1, then
the value of y (t) at t = 1 is
1
1
(A) – 2
(B) e – 2
1
1
(C) 2
(D) e + 2
Let f (x) be a differential function with f (1) = 4 and f (2) = 6, where f (c) is the
f(2  2h  h2 )  f(2)
2
derivative of f (x) at x = c. Then the limit of f(1  h  h )  f(1) , as h  0,
(A) may not exist
(B) equals 3
3
(C) equals 2
(D) equals – 3
lim
70.
If n is a non-zero integer and
x 0
sin(nx)((a  n)nx  tan(x))
x2
, then a equals
1
(A) n
n 1
(C) n
(B) 0
1
(D) n + n
x2  x  2
71.
2
The range of the function f (x) = x  x  1 , x  (–, ), is
 11
 1,

(A) [1, )
(B)  7 
 7
 1, 
(C)  3 
72.
 7
 1, 
(D)  5 
ˆ ˆ ˆ ˆ
ˆ
The volume of the parallelepiped formed by vectors i  aj  k, j  ak and a ˆi  kˆ , a
> 0 is minimum when the value of a is
(A) 3
1
(B) 3
1
(D) 3
(C) 3
73.
74.
x
The function f : [0, )  [0, ), defined by f (x) = 1  x , is
(A) one-to-one and onto
(B) one-to-one but NOT onto
(C) onto but NOT one-to-one
(D) NEITHER one-to-one NOR onto
If z is a complex number satisfying |z| = 1 and z  –1, then the real part of  =
z 1
z  1 is
1
1
(A) | z  1|
2
2
(C) | z  1|
2
75.
(D) 0
The orthocentre of the triangle with vertices (0, 0), (4, 0) and (3, 4) is
4
5
 4,  3

(A) 
(B) (3, 12)
 5
 3, 
(C)  4 
76.
2
(B) | z  1|
 3
 3, 
(D)  4 
tan2 
2
 
x

x

0,


x2  x , x > 0, is greater than or
for every    2  , the value of
equal to
(B) 2 tan 
(A) 2
5
(C) 2
77.
78.
79.
(D) sec 
x2
 y2  1
27
A tangent is drawn to the ellipse
, at the point (3 3 cos , sin ),

where 0 <  < 2 . The sum of the intercepts of the tangent with the coordinate
axes is least when  equals


(A) 6
(B) 3


(C) 8
(D) 4
Let (0, 0), (21, 0) and (0, 21) be the vertices of a triangle. The number of points
having integer coordinates which are strictly inside the given triangle is
(A) 231
(B) 105
(C) 190
(D) 133
x2 y2

1
5
The area of the quadrilateral formed by the tangents to the ellipse 9
, at
the ends of each of its latus-rectum, is
27
(A) 4
(B) 9
27
(C) 2
(D) 27
x 2 1
80.
The function f (x) =
(A) in (0, )
(C) in (–2, 2)

x2
et
2
dt, x  (– , ), in increasing
(B) in (–, 0)
(D) nowhere

6 is
 1 1
 , 
(B)  2 4 
 1 1
 , 
(D)  2 2 
sin1(2x) 
81.
The natural domain of the function f (x) =
 1 1
 , 
(A)  4 2 
 1 1
 , 
(C)  4 4 
82.
In the interval [0, 1], the mean value theorem is NOT applicable to the function
1
 2  x,


2
 1  x  ,



(A) f (x) =  2
(C) f (x) = x|x|
83.
x
1
2
x
1
2
 sin x
, x0

 x

x0
(B) f(x) = 1,
(D) f (x) = |x|
Two numbers are drawn at random, one after another and without replacement,
from the set {1, 2, 3, 4, 5, 6}. The probability that minimum of the chosen
numbers is smaller than 4 is
1
14
(A) 15
(B) 15
1
4
(C) 5
(D) 5
x2
84.

y2
1
 
 0, 
,    2  , which of
For the family of hyperbolas given by cos  sin 
the following remains unchanged with varying ?
(A) eccentricity
(B) abscissae of the foci
(C) equation of the directrices
(D) abscissae of the vertices
2
2
answers
57.
C
58.
D
59.
A
60.
C
61.
C
62.
A
63.
B
64.
C
65.
B
66.
D
67.
A
68.
A
69.
B
70.
D
71.
C
72.
C
73.
B
74.
D
75.
D
76.
B
77.
A
78.
C
79.
D
80.
B
81.
A
82.
A
83.
D
84.
B