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Sample Objective Type Questions–Mathematics
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Sample Objective Type Questions–Mathematics
1. (2log1/3
1
8 )log4 3 equals
(A) 1
(B) 2
(C) 2p2
(D) 4
(E) 8
2. The sum of all odd integers greater
than 100 and less than 1000 is
(A) 247500
(B) 495000
(C) 990000
(D) 1100000
(E) None of the above.
3. The value of a such that the planes
ax + y + z = 0,
x + 3z = 0, and 5y + 6z = 0
have a line in common is
(A) -1/15
(B) -11/15
(C) 11/15
(D) 1/15
(E) 0
4. The set of real x for which
(x + 1)3(x - 2)2(x - 3) < 0 is
(A) (-1, 3)
(B) (-1,-1) [ (3,1)
(C) (-1,1)
(D) (-1, 3)
(E) (-1, 2) [ (2, 3)
5. lim
x!
tan x
sin(-x
2 )
equals
(A) 0
(B) 1/2
(C) 2
(D) -1
(E) -2
6. If the surface area of a cube is
decreased by 19%, the volume of
the cube decreases by
(A) 9%
(B) 19%
(C) 27.1%
(D) 37.1%
(E) 85.2%
7. Each vertex of a cube is assigned
an integer so that no two vertices
connected by an edge of the cube
are assigned the same integer. The
least number of integers required
for such an assignment is
(A) 2
(B) 3
(C) 4
(D) 6
(E) 8
8. Let f and g be functions such that
f(x) = f(-x),
g(x) = -g(-x),
R2
0 f(x) dx = 7,
and R2
0 g(x)dx = 3.
Calculate R2
-2(f(x)+g(x))dx.
(A) 6
(B) 10
(C) 14
(D) 20
(E) Cannot be determined from
the given data.
9. Which is the smallest number with
exactly 12 divisors? (If n is
a positive integer, 1 and n are
counted as divisors of n. So, for
example, 4 has three divisors: 1, 2,
and 4.)
(A) 72
(B) 211
(C) 12
(D) 48
(E) None of the above.
10. If , are the roots of
(x - 2)(x - p3) = p7,
the roots of
(x - )(x - ) + p7 = 0
are
(A) -2 - p3, p7
(B) 2 - p7, p3 - p7
(C) 2+p3, p7
(D) 2+p7, p3 + p7
(E) 2, p3
Answers: 1. C, 2. A, 3. A,
4. E, 5. E, 6.C, 7. A, 8.C,
9.E (The correct number is 60), 10. E
——————————
Long Answer Type Questions
1. If (1+x)n = c0 +c1x+c2x2 +. . .+
cnxn, show that (8 marks)
c20
+2c21
+· · ·+(n+1)c2
n =
(2n - 1)! (n + 2)
n!(n - 1)!
2. Find the centre and radius of the
circle that is the intersection of the
spheres
x2 + y2 + z2 - 8 = 0
and
x2 +y2 +z2 -4x-4y -2z +4 = 0
(10 marks)
3. What is the probability that a
point chosen at random within a
square is closer to the centre than
to the boundary? (12 marks)
4. Characterize all
arithmetic progressions in positive
integers such that, for all n 1,
the sum up to n terms is a perfect
square. (12 marks)
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