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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

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    (Provide Your Feedbacks/Comments at maths.whisperer@gmail.com)

    [ 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 ]

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  • 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 ]

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  • 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

    same on the attached answer-sheet.

    • All questions carry 2 marks.

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

    etc. Record only your answers on the answer-sheet as indicated. You are being given

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

    [ 4 ]

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  • 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 ]

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  • 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 ]

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  • 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

    P(XY < Z2)? Write down your answer as a fraction.

    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 ]

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  • 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 ]

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  • 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