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### Transcript of IIT Bombay - Department of Mathematics, IIT Bombay Screening Test for PhD Admissions (2 Dec, 2015)...

• IIT Bombay PhD Maths Screening Test

Contents(Question Paper):

· IIT Bombay PhD Test: Dec, 2015 · IIT Bombay PhD Test: May, 2016 · IIT Bombay PhD Test: Dec, 2016 · IIT Bombay PhD Test: May, 2017 · IIT Bombay PhD Test: Dec, 2017 · IIT Bombay PhD Test: May, 2018 · IIT Bombay PhD Test: Dec, 2018 · IIT Bombay PhD Test: May, 2019

No. of Pages: 38

[ 1 ]

• Department of Mathematics, IIT Bombay

Screening Test for PhD Admissions (2 Dec, 2015)

Time allowed: 2 hours and 30 minutes Maximum Marks: 40

Note: Q.1-12 carry 3 marks each. Q.13 carries 4 marks.

Q.1 Let A be a 5× 5 matrix s.t. A2 = 0. Compute the least upper bound for its rank.

Q.2 Gram-Schmidt process is applied to the ordered basis {i + j + k, i + j, i} in R3. Find

the resulting orthonormal basis.

Q.3 Let A = [aij] be a square matrix of order n whose entries are given as follows. For

1 ≤ i, j ≤ n we have

aij =

 ij if i 6= j,

1 + ij if i = j.

Evaluate the determinant of A.

Q.4 Arrange the following matrices with their ranks in a non-decreasing order.

P =

 1 2 3

4 5 6

7 8 9

 , Q = 

2 22 23

24 25 26

27 28 29

 , R = 

12 22 32

42 52 62

72 82 92

 . Q.5 If a1 ≥ 0, a2 ≥ 0 and an+2 =

√ anan+1, find the limit of the sequence {an}.

Q.6 Let {Pn} be a sequence of polynomials such that for n = 0, 1, 2, ...

P0 = 0 and Pn+1(x) = Pn(x) + x2 − P 2n(x)

2 .

Assuming the fact that {Pn} is convergent point-wise, find the limit function lim n→∞

Pn(x).

Q.7 Find the values of x, (x ∈ R) for which the series ∞∑ n=1

(−1)nx 2 + n

n2

[ 2 ]

• is convergent.

Q.8 Find the range of values of α for which

∫ ∞ 0

sinx

xα is convergent (i.e. is finite).

Q.9 A point X is picked uniformly at random from the perimeter of a unit circle. Find

the expected value of |X|.

Q.10 Suppose the distribution of Y , conditional onX = x, is Normal(x, x2) and the marginal

distribution of X is Uniform(0, 1). Find the Cov(X, Y ).

Q.11 Let X be an observation from the probability density function

f(x|θ) = ( θ

2

)|x| (1− θ)1−|x|, x = −1, 0, 1; 0 ≤ θ ≤ 1.

Find the maximum likelihood estimator (mle) of θ and its expectation.

Q.12 Let U1, U2, . . . , Un be i.i.d. Uniform(0, θ), θ > 0, random variables. Find the uniformly

minimum variance unbiased estimator (UMVUE) of cos θ.

Q.13 Let A =

 1 r r

r 1 r

r r 1

 . Find the range of values of r so that A is positive definite.

[ 3 ]

• Department of Mathematics, IIT Bombay

Screening Test for PhD Admissions (9 May, 2016)

Time allowed: 2 hours and 30 minutes Maximum Marks: 40

Name: Choice: Math Stat

• Write your name in the blank space at the top of this question-paper, and also tick

‘Math’ or ‘Stat’ to indicate your first choice of one of the 2 PhD programs. Do the

• All questions carry 2 marks.

• The answer to each question is a number, function, set, inequality, random variable

a separate work-sheet for doing rough work (this will not be collected).

[ 4 ]

• Q.1 Consider the vector space R2016 over the field R of real numbers. What is the small-

est positive integer k for which the following statement is true: given any k vec-

tors v1, . . . , vk ∈ R2016, there exist real numbers a1, . . . , ak, not all zero, such that

a1v1 + · · ·+ akvk = 0 and a1 + · · ·+ ak = 0.

Q.2 Find all complex triples (a, b, c) such that the following matrix is diagonalizable 1 a b

0 2 c

0 0 1

 . Q.3 For what values of k does the linear system

x− 3z = −3

2x+ ky − z = −2

x+ 2y + kz = 1

in unknowns x, y, z have no solution?

Q.4 Let V and W be subspaces of the vector space R9 over the field R of real numbers

with dim V = 5 and dim W = 6. Then what is the smallest possible dimension of

V ∩W?

Q.5 Consider the inner product

〈(a1, a2), (b1, b2)〉 = 2a1b1 − a1b2 − a2b1 + 5a2b2

on R2. Write down a vector which is orthogonal to (1, 0) and has norm 1.

Q.6 Find all values α ∈ R for which the matrix α 1 −1

1 2 1

−1 1 4



[ 5 ]

• is positive definite.

Q.7 Let A =

a11 a12 a13 a21 a22 a23

 be a real matrix and c1, c2, c3 be the scalars

c1 = det

 a12 a13 a22 a23

  , c2 = det

 a13 a11 a23 a21

  , c3 = det

 a11 a12 a21 a22

  .

Find all triples (c1, c2, c3) so that rank A = 2.

Q.8 Let A be the complex 3× 3 matrix

A =

 2 0 0

a 2 0

b c −1

 . Find all triples (a, b, c) for which the characteristic and minimal polynomials of A are

different.

Q.9 Given � > 0, what is the largest δ which fits the definition of continuity of the function

f(x) =

 x+3

2 if x ≤ 1

7−x 3

if 1 ≤ x

at x = 1, that is, the largest δ > 0 for which the implication |x − 1| < δ =⇒

|f(x)− f(1)| < � holds?

Q.10 A point ω is said to be a fixed point of a function f if f(ω) = ω. Given that the

function

f(x) = x3 + 1

3

has three fixed points α, β, γ in (−2,−1), (0, 1) and (1, 2) respectively, let us define a

sequence of real numbers {xn} as

x1 = γ − 0.01 , xn+1 = f(xn) n = 1, 2, 3, . . . .

[ 6 ]

• Given that the sequence converges, find

lim n→∞

xn.

Q.11 Suppose f is a real valued continuously differentiable function on [0, 1] with f(0) =

f(1) = 0 and ∫ 1 0

f 2(x) dx = 1.

Find the value of ∫ 1

0 xf(x)f

′ (x) dx.

Q.12 Let

yn = n2

n3 + n+ 1 +

n2

n3 + n+ 2 + · · ·+ n

2

n3 + 2n .

Find limn→∞ yn =?

Q.13 Find all values of x > 0 for which the series

1 + x

1! +

22x2

2! +

33x3

3! + · · ·+ n

nxn

n! + · · ·

converges.

Q.14 Find all values of (p, q) for which the integral∫ 1 0

xp logq(1/x) dx

converges.

Q.15 Suppose X, Y, Z are i.i.d. Uniform[0, 1] random variables. What is the probability

Q.16 Suppose X1, X2, . . . , Xn are i.i.d. random variables for which E(X−11 ) < ∞, where E

denotes expectation. Let Si := X1 +X2 + · · ·+Xi. For m < n calculate E(Sm/Sn).

Q.17 Suppose the distribution of Y , conditional on X = x, is Poisson(x) and random

variable X is exponentially distributed with rate parameter 1. Find the correlation

between X and Y .

[ 7 ]

• Q.18 Let U1, U2, . . . , Un be i.i.d. Uniform(0, θ), θ > 0 random variables. Find the uniformly

minimum variance unbiased estimator (UMVUE) of θ2.

Q.19 Let X1, X2, . . . , Xn be i.i.d. random variables with one of two possible probability

density functions f(x|θ). If θ = 0, then f(x|θ) = I(0,1)(x) while if θ = 1, then

f(x|θ) = 1 2 √

x I(0,1)(x). Find the maximum likelihood estimator θ̂ of θ.

Q.20 Suppose that the random variables Y1, . . . , Yn satisfy

Yi = βxi + �i, i = 1, . . . , n,

where x1, . . . , xn are fixed known constants, and �1, . . . , �n are i.i.d. N(0, σ 2), σ2 known.

What is the distribution of MLE of β?

[ 8 ]

• Department of Mathematics, IIT Bombay

Screening Test for PhD Admissions (Dec 1, 2016)

Time allowed: 2 hours and 30 minutes Maximum Marks: 40

Name: Choice: Math Stat

• Write your name in the blank space at the top of this question-paper, and also tick

‘Math’ or ‘Stat’ to indicate your first choice of one