# Sample Objective Type Questions–Mathematics

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

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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)