Math Aptitude Indicator

This is a Multiple Choice Test. For each question only one answer is correct.

Please use separate sheets of paper for calculations, and submit that along

with your solutions as it will assist us in evaluating your performance if you get

any questions wrong.

1. The second order derivative d

2f

dx2

of f (x) = x exp (x) is

(A) e

x

x

2 1

(B) e

x

(x 1) (C) f (x) + 2e

x

(D) f (x) 2e

x

(E) f (x) e

x

2. Given y = ln j1 xj, Önd an expression for dy

dx + x

d

2y

dx2

(A) 1

(1 x)

(B)

1

(1 x)

2

(C) x

(1 x)

2

(D) 1x (E) 1

(1 x

2)

2

3. What is the value of the limiting problem lim

h!0

(x + h)

3 x

3

h

(A) 0 (B) 1 (C) 1 (D) 3x

2

(E) None of the given

answers

4. Calculate limx!0

2x + sin x

x (x 1)

(A) 1 (B) 0 (C) 2 (D) 1 (E) 3

5. If

I =

Z 1

0

x exp

x

2

dx

Then I =

(A) 0 (B) 1 (C) 2 (D)

1

2

(E) 1

2

6. Evaluate the integral Z 1

0

dx

x

2 + 3x + 2

(A) ln 2

3

(B) 1 (C) 0 (D

7. What is the n

th order term in the Taylor series expansion of the function

f (x) = exp x about x = 3

(A) e

3

n!

(x + 3)n

(B) e

3

2

n

(x + 3)n

n!

(C) e

2

n+1

(x 3)n+1

(n + 1)!

(D) e

3

2

n

(x + 3)n1

n!

(E) e

3

2

n

(x 3)n1

(n 1)!

8. Given the Taylor series expansion for

1

1 + x

= 1 x + x

2 x

3 + :::: =

X1

n=0

(1)n

x

n

Obtain the series for ln (1 + x)

(A) P1

n=0

n

2

(1)n1

x

n1

(B)

1P

n=0

(1)n+1 (n + 1) x (C) P1

n=0

(1)n x

n+1

n + 1

(D) P1

n=0

n (1)n1 x

n1

n + 1

(E)

1P

n=0

(1)n x

n1

n 1

9. Let X be a continuous random variable with distribution

p (x) =

k

1 x

3

0 x 1

0 otherwise

Evaluate k

(A) 3

4

(B)

4

3

(C) 4

3

(D) 3

2

(E) 2

10. Consider the probability density function (pdf)

p (x) = 3

4

1 x

2

if jxj 1

0 otherwise

which a random variable X follows. Find the probability that 0 X

1

2

(A) 3

32 (B) 9

32 (C) 12

88 (D) 1

8

(E) 11

32

11. Let X be a Normally distributed random variable with mean () = 4 and

standard deviation () = 2. If E[X] denotes the expectation of X, then

the value of E

X2

is

(A) 16 (B) 20 (C) 4 (D) 2 (E) 3

12. A random variable is uniformly distributed over [0; 1] : What is its skew?

(A) 3 (B) None of t

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