CSIR NET June 2025 Mathematical Sciences — Part B & C solved

The Part B and Part C questions transcribed from this paper, worked out in full — not just the answer key, but why each option holds or fails and which trap it tests.

What this paper asked

100 of 118 questions mapped to a syllabus topic. How this compares across every complete paper →

Probability11 · 6 in C
Eigenvalues and Canonical Forms6 · 4 in C
Linear Models and Multivariate6 · 4 in C
The Real Line5 · 1 in C
Vector Spaces and Linear Maps5 · 3 in C
Rings and Fields5 · 3 in C
Ordinary Differential Equations5 · 3 in C
Inner Product Spaces and Forms4 · 2 in C
Groups4 · 3 in C
Partial Differential Equations4 · 2 in C
Limit Theorems and Markov Chains4 · 3 in C
Sequences and Series of Functions3 · 2 in C
Cauchy Theory3 · 1 in C
Numerical Analysis3 · 2 in C
Calculus of Variations3 · 2 in C
Linear Integral Equations3 · 2 in C
Estimation3 · 2 in C
Hypothesis Testing3 · 2 in C
Metric Spaces3 · 3 in C
Continuity and Differentiation2 · 1 in C
Zeros and Mappings2 · 1 in C
Analytic Functions2 · 1 in C
Classical Mechanics2 · 1 in C
Sampling and Design of Experiments2 · 1 in C
Integration2 · 2 in C
Topology1 · 0 in C
Linear Programming1 · 0 in C
Lebesgue Measure and Integration1 · 1 in C
Functions of Several Variables1 · 1 in C
Singularities and Residues1 · 1 in C

118 questionsPart B: 40Part C: 600 free to read

Part B

One correct option. 3 marks, −0.75 for a wrong answer.

Q1Boundary and endpoint Elements of set theory: operations, De Morgan and difference

Let p, q be non-negative integers. Consider the following statements: (A) There is an integer k ≥ 1 such that p + k = q. (B) There is an integer k ≥ 1 such that q + k = p. Which of the following statements is true?

A.There exist non-negative integers p, q such that both (A) and (B) are true.
B.Both (A) and (B) are false if and only if p = q.✓
C.For all non-negative integers p and q, (A) or (B) is true.
D.There exists p ≠ q such that both (A) and (B) are false.

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Q2Standard counterexample Elements of set theory: operations, De Morgan and difference

Let A = { $p / q \in (0, 1) : p \in N, q = 2^{n}$ for some $n \in N \cup$ {0}, gcd(p, q) = 1}, B = { $p / q \in (0, 1) : p \in N, q = 2^{n} 5^{m}$ for some $n, m \in N \cup$ {0}, gcd(p, q) = 1} and C = {p/q ∈ (0, 1) : p/q has terminating decimal expansion} be subsets of (0, 1). Which of the following statements is true?

A.A ⊊ C and B ⊊ C
B.A ⊊ C ⊊ B
C.A ⊊ B ⊊ C
D.A ⊊ B = C✓

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Q3Standard counterexample Elements of set theory: operations, De Morgan and difference

Let A, B be non-empty subsets of $N$ with cardinality |A| ≥ 2. Let $S_{1} =$ {f : A → B | f is one-to-one} and $S_{2} =$ {g : B → A | g is onto}. Which of the following statements is true?

A.If A ⊊ B and B is finite, then there is a one-to-one map from $S_{2}$ to $S_{1}$ .
B.If $B = N$ , then there exists a one-to-one map from $S_{2}$ to B.
C.If $B = N$ and A is finite, then there exists a one-to-one map from B to $S_{1}$ .✓
D.If A is finite, then $S_{2}$ is finite for any B.

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Q4Execution slip Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

Let $f : R$ \ $Q \to R$ \ $Q$ be the function defined as f(x) = (3x + 2)/(4x + 3). Let $x_{1} \in R$ \ $Q$ . For n ≥ 1, define $x_{n}$ ₊ $_{1} = f (x_{n})$ . Suppose that the sequence $(x_{n})_{n} \geq_{1}$ converges to a real number $ℓ$ . Which of the following statements is true?

A.If $ℓ$ is positive, then $ℓ = 3 /2$ .
B.If $ℓ$ is positive, then $ℓ = 1/ 2$ .✓
C.If $ℓ$ is negative, then $ℓ = - 3 /2$ .
D.If $ℓ$ is negative, then $ℓ = - 1/2$ .

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Q5Execution slip Differentiability, mean value theorems, Taylor, L'Hôpital

Let $f (x) = x l o g_{e} (1 + 1/ x)$ for $x \in (0, \infty)$ . Which of the following statements is true?

A.f is unbounded.
B.f is increasing.✓
C. $l i m_{x \to \infty}$ f(x) = 2.
D.f is decreasing.

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Q6Pointwise vs uniform Pointwise vs uniform convergence, M-test, Dini

For each n ≥ 1, let $f_{n} : [0, 1] \to R$ be defined as $f_{n} (x) =$ nx if $x \in [0, 1/ n], f_{n} (x) = 2 -$ nx if x ∈ (1/n, 2/n), and $f_{n} (x) = 0$ if x ∈ [2/n, 1]. Which of the following statements is true?

A. $(f_{n})_{n} \geq_{1}$ converges uniformly on [0, 1] to a continuous function f.
B. $(f_{n})_{n} \geq_{1}$ converges pointwise on [0, 1] to a discontinuous function f.
C. $(f_{n})_{n} \geq_{1}$ converges pointwise on [0, 1] to a continuous function f.✓
D. $(f_{n})_{n} \geq_{1}$ does not converge pointwise on [0, 1].

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Q7Execution slip Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Which of the following polynomials is the characteristic polynomial of a real 2 × 2 matrix A such that trace(A) = 7 and trace $(A^{2}) = 29$ ?

A. $t^{2} + 7 t + 10$
B. $t^{2} - 7 t + 29$
C. $t^{2} - 7 t - 10$
D. $t^{2} - 7 t + 10$ ✓

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Q8Standard counterexample Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Consider the real matrix A whose rows are (29, 0, 55, 17), (1, 28, 46, 26), (17, 13, 33, 38) and (21, 67, 0, 13). What is the largest real eigenvalue of A?

A.101✓
B.67
C.103
D.113

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Q9Boundary and endpoint Bases, dimension, rank–nullity

Let 𝔽 $_{5}$ denote the field with 5 elements. How many 2 × 2 matrices with entries in 𝔽 $_{5}$ have rank one?

A.125
B.144✓
C.145
D.480

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Q10Finite-dimensional intuition Bases, dimension, rank–nullity

Let X be the $R -$ vector space of all twice differentiable real valued functions on [0, 1]. Consider the linear map $φ : X \to R^{3}$ defined by $φ (f) = (f (1), f^{'} (1), f^{''} (1))$ . Which of the following statements is true?

A.The dimension of $X / ker φ$ is 3.✓
B. $ker φ$ is finite dimensional.
C.The dimension of $X / ker φ$ is 1.
D.X is finite dimensional.

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Q11Standard counterexample Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

Let $C [0, π]$ be the real vector space of real-valued continuous functions on the closed interval $[0, π]$ . For positive integers n, define $f_{n} \in C [0, π]$ by $f_{n} (x) = sin ($ nx)/sin x if $x \in (0, π), f_{n} (0) = n$ , and $f_{n} (π) = (- 1)^{n}$ ⁻ $^{1} n$ . Let V be the real subspace of $C [0, π]$ spanned by { $f_{1}, f_{2}, f_{3}$ }. Consider the inner product on V given by ⟨f, g⟩ $= (2/ π) \int_{0}^{π} f (x) g (x) sin^{2} x$ dx. Which of the following statements is true?

A. $f_{4} \in V$
B.{ $f_{1}, f_{2}, f_{3}$ } is an orthonormal basis of V.✓
C.The dimension of V is 2.
D.{ $f_{1}, f_{2}, f_{3}$ } is an orthogonal set but not orthonormal.

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Q12Standard counterexample Quadratic forms, positive definiteness, Sylvester's law

Let V = {ax $^{3} +$ bx $^{2} +$ cx | $a, b, c \in R$ }. For f ∈ V, define $Q (f) = \int$ ₋ $_{1}^{1} (f^{'} (t))^{2}$ dt, where f′ denotes the derivative of f. Which of the following statements is FALSE?

A.Q is a positive definite quadratic form on V.
B.Q takes every positive real value.
C.Q(x) = 2.
D.For all f, g ∈ V, Q(f + g) = Q(f) + Q(g).✓

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Q13Boundary and endpoint Liouville, Morera, maximum modulus principle

Let $f : C \to C$ be a polynomial map. For R > 0, let $γ_{R} : [0, 1] \to C$ be the map t ↦ $R e^{2 π i t}$ . Suppose that there exists $c \in R$ such that $\int_{0}^{1}$ |(f ∘ $γ_{R}) (t) γ_{R}^{'} (t)$ | dt → c as $R \to \infty$ . Which of the following statements is FALSE?

A.The function zf(1/z) → 0 as |z| $\to \infty$ .
B.The function f is constant.
C.c = 0.
D.c > 0.✓

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Q14Boundary and endpoint Liouville, Morera, maximum modulus principle

Let f be an entire function such that $f (C) \subset$ {x + iy | y = x + 1}. Which of the following statements is true?

A.|f(z)| $\to \infty$ as |z| $\to \infty$ .
B.f(z)/z → 0 as |z| $\to \infty$ .✓
C.zf(z) → 0 as |z| $\to \infty$ .
D.f(z) → 0 as |z| $\to \infty$ .

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Q15Invariants don't determine Conformal maps, Möbius transformations, Schwarz lemma

Let X be the image of the interval [0, 1] under the Möbius transformation f(z) = (z − i)/(z + i). Which of the following statements is true?

A.X is the line segment joining −1 and −i.
B.X = { $e^{i θ}$ | $θ \in [π, 3 π /2]$ }.✓
C.X is the line segment joining −1 to 1.
D.X = { $e^{i θ}$ | $θ \in [- π /2, π]$ }.

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Q16Execution slip Power series and analyticity

Which of the following statements is true?

A.There exists an entire function f such that f⁽ $^{n}$ ⁾ $(0) = n! / n^{n}$ for all positive integers n.✓
B.There exists an entire function f such that f⁽ $^{n}$ ⁾ $(0) = n! n^{n}$ for all positive integers n.
C.There exists an entire function f such that f⁽ $^{n}$ ⁾(0) = (n − 1)! for all positive integers n.
D.There exists an entire function f such that f⁽ $^{n}$ ⁾(0) = n!n for all positive integers n.

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Q17Hypothesis dropped Subgroups, cosets, Lagrange, cyclic groups

Which of the following statements is true?

A.p ∤ 1 + (p − 1)! for some odd prime p.
B.p | $(1234)^{p - 1}$ − 1 for all primes p > 700.✓
C.There exist $a \in Z$ and a prime p > 11 such that p ∤ aᵖ − a.
D.p ∤ $(p^{2})! / (p!)^{2}$ for some odd prime p.

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Q18Standard counterexample Ideals, quotient rings, prime & maximal ideals, CRT

Let A be a subring of the field of rationals $Q$ such that for any nonzero rational $r \in Q, r \in A$ or 1/r ∈ A. Which of the following statements is FALSE?

A.The set {a ∈ A : 1/a ∉ A} ∪ {0} is an additive subgroup of $Q$ .
B.A has at most one maximal ideal.
C.If $A \neq = Q$ , then A has infinitely many prime ideals.✓
D.For any nonzero a, b ∈ A, a divides b or b divides a in A.

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Q19Hypothesis dropped Ideals, quotient rings, prime & maximal ideals, CRT

Which of the following statements is true?

A.The ideal $2 Z [i]$ is maximal in $Z [i]$ .
B.The ideal $X C [X, Y]$ is maximal in $C [X, Y]$ .
C.The set of all polynomials in $C [X]$ whose coefficients add up to 0 is a maximal ideal in $C [X]$ .✓
D.The ideal $(2 - 1) Z [2]$ is maximal in $Z [2]$ .

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Q20Hypothesis dropped Continuity, homeomorphism, separation axioms

Let S = {1, 2, 3, 4, 5} be equipped with the topology $τ =$ {∅, {1}, S}. What is the number of homeomorphisms of S onto itself?

A.25
B.120
C.24✓
D.6

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Q21Execution slip Linear ODE, Wronskian, variation of parameters, systems

If $φ (x) = x$ is a solution of the ordinary differential equation (ODE $) d^{2} y /$ dx $^{2} - (2/ x^{2} + 1/ x) (x$ dy/dx $- y) = 0, 0 < x < \infty$ , then the general solution of the ODE is given by

A.(a + $b e^{- 2 x}) x$ , $a, b \in R$
B.(a + $b e^{2 x}) x$ , $a, b \in R$
C.ae^x + bx, $a, b \in R$
D.(a + be $^{x}) x, a, b \in R$ ✓

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Q22Execution slip Sturm–Liouville problems and Green's functions

Let $(λ_{n})_{n} \in N$ be the sequence of eigenvalues of the Sturm-Liouville problem (d/dx)(x dy/dx $) + (λ / x) y = 0, 1 < x <$ $e^{2 π}$ , y(1) = 0, $y (e^{2 π})$ = 0. Then $\sum_{n}$ ₌ $_{1}^{\infty} 1/ λ_{n}$ is equal to

A. $π^{2} /12$
B. $2 π^{2} /3$ ✓
C. $π^{2} /4$
D. $π^{2} /16$

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Q23Boundary and endpoint First-order PDE: Lagrange, Charpit, characteristics

Let u = u(x, y) be the solution to the Cauchy problem $(y + u) \partial u / \partial x + y \partial u / \partial y = x - y, x \in R, y > 0, u (x, 1) = 1 + x, x \in R$ . Then which of the following statements is true?

A.u(1, 1) = 2✓
B.u(2, 2) = 4
C.u(3, 3) = 3/2
D.u(4, 4) = 2/3

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Q24Execution slip Laplace, heat and wave equations: separation of variables

Let u = u(x, t) be the solution of $\partial^{2} u / \partial t^{2} - \partial^{2} u / \partial x^{2} = 0, x \in R, t > 0, u (x, 0) = 1 + x^{2}, x \in R, \partial u / \partial t (x, 0) = x + 1, x \in R$ . Then the value of u(1, 1) is

A.2
B.3
C.4
D.5✓

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Q25Execution slip Interpolation and numerical integration with error terms

If the function $s : [0, 4] \to R$ defined by $s (x) = a (x - 2)^{2} + b (x - 1)^{2}$ for $0 \leq x \leq 1, s (x) = (x - 2)^{2}$ for 1 < x ≤ 3, and $s (x) = 2 c (x - 2)^{2} + (x - 3)^{3}$ for 3 < x ≤ 4, is a cubic spline, then the value of 2a + b + 2c is

A.2
B.3✓
C.4
D.5

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Q26Execution slip Euler–Lagrange equation and standard functionals

Let y(x) be the extremal of the functional $J [y] = \int_{0}^{π /4}$ $((y^{'})^{2} - 4 y^{2} + 2$ xy) dx subject to $y (0) = 0, y (π /4) = 1$ . Then y(x) is equal to

A. $(1 - π /4) sin (2 x) + x$
B. $(1 - π /16) sin (2 x) + x /4$ ✓
C. $(1 + π /4) sin (2 x) - x$
D. $(1 + π /16) sin (2 x) - x /4$

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Q27Standard counterexample Separable kernels and resolvent kernels

If y(x) is the solution of the integral equation $y (x) = x^{2} + 2 \int_{0}^{1}$ xt y(t) dt, then which of the following statements is true?

A.y(0) + y(1) = 1/2
B.y(−1) + y(1) = 1
C.y′(0) + y′(1) = 3/2
D.y′(−1) + y′(1) = 3✓

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Q28Execution slip Lagrangian formalism and generalised coordinates

Suppose a dynamical system has the Lagrangian L = (q̇ $_{1})^{2} + (q$ ̇ $_{2})^{2} + (q_{1})^{2} + q$ ̇ $_{1} q$ ̇ $_{2}$ . If $p_{1}$ and $p_{2}$ are momenta conjugate to $q_{1}$ and $q_{2}$ respectively, then which of the following statements is true?

A.ṗ $_{1} = 2 q_{1}$ , ṗ $_{2} = 0$ ✓
B.ṗ $_{1} = - q_{1}$ , ṗ $_{2} = 0$
C.ṗ $_{1} = - q_{1} /2, p_{2} = q_{2}$
D.ṗ $_{1} = q_{1}, p_{2} = - q_{2}$

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Q29Boundary and endpoint Random variables, distributions, moments, MGF

Suppose that we have a data set consisting of 2n + 1 observations for some $n \in N$ . Value of each observation is either x or x + r, where $x \in N, r \geq 0$ . Then, which of the following statements is always true?

A.The mean and median of the data will be different if and only if r > 0
B.Variance of the data is positive if and only if r > 0
C.Mean and mode of the data will be same if and only if r = 0
D.Median and mode of the data will be same for all values of r ≥ 0✓

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Q30Boundary and endpoint Axioms, conditional probability, independence, Bayes

A biased six-faced die is tossed once. Suppose that the probability of any prime number showing up is twice that of any non-prime number showing up. Then, the probability that an odd number will show up is

A.1/3
B.2/3
C.4/9
D.5/9✓

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Q31Hypothesis dropped Axioms, conditional probability, independence, Bayes

Suppose the distribution of X given $θ$ is normal with mean $θ$ and variance 15. Further, let the prior (improper) distribution of $θ$ be proportional to $1, - \infty < θ < \infty$ . If the observed value of X is 13, then which of the following statements is true?

A.Posterior mean = Maximum likelihood estimate of $θ$ , Posterior variance = Var(X| $θ)$ ✓
B.Posterior mean = Maximum likelihood estimate of $θ$ , Posterior variance < Var(X| $θ)$
C.Posterior mean > Maximum likelihood estimate of $θ$ , Posterior variance = Var(X| $θ)$
D.Posterior mean > Maximum likelihood estimate of $θ$ , Posterior variance < Var(X| $θ)$

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Q32Standard counterexample Markov chains: classification of states, stationary distributions

Let $Z_{1}, Z_{2}, \dots$ be a sequence of independent and identically distributed random variables having discrete uniform distribution over {1, 2, …, 2024}. Let $Y_{n} = \sum_{i}$ ₌ $_{1}^{n} Z_{i}, n \geq 2$ . Further, let $X_{n}$ be the remainder when $Y_{n}$ is divided by 2025. Then, which of the following statements is true?

A. $l i m_{n} \to \infty P (X_{n} = 0) = 1/2026$
B. $l i m_{n} \to \infty P (X_{n} = 0) = 1/2025$ ✓
C. $l i m_{n} \to \infty P (X_{n} = 0) = 1/2024$
D. $l i m_{n} \to \infty P (X_{n} = 0) = 1/2023$

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Q33Moments and tails Standard discrete and continuous distributions

A mobile manufacturing company uses two brands of batteries for its mobiles. The life (in years) of batteries of Brand I follows an exponential distribution with the probability density function f(x) = $e^{- x}$ if x > 0 and 0 otherwise, and that of Brand II follows a gamma distribution with the probability density function g(x) = $(x /4) e^{- x /2}$ if x > 0 and 0 otherwise. The company uses the batteries of Brands I and II in proportion of 20% and 80% respectively, in its mobiles. The probability that a randomly selected mobile has the battery life more that 2 years is

A. $(13/5) e^{- 2}$
B. $(1/5) (e^{- 2}$ + $2 e^{- 1})$
C. $(1/5) (e^{- 2}$ + $8 e^{- 1})$ ✓
D. $(1/5) (4 e^{- 2}$ + $2 e^{- 1})$

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Q34Execution slip MLE and method of moments

Consider a discrete random variable X with the probability mass function $P (X = 0) = θ /3, P (X = 1) = 1 - θ /2, P (X = 2) = θ /6$ , where $θ \in (0, 1)$ is an unknown parameter. In a random sample of size 90 from this distribution, the observed counts for X = 0, 1 and 2 are 20, 60 and 10, respectively. Then, the maximum likelihood estimate of $θ$ is

A.1/3
B.1/2
C.2/3✓
D.3/4

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Q35Execution slip Joint distributions, transformations, order statistics

Let X be a random sample of size 1 from the probability density function f(x| $θ) = (3/ θ^{3}) (θ - x)^{2}$ if $0 < x < θ$ , and 0 otherwise. If $(X / (1 - λ_{1}), X / (1 - λ_{2}))$ is a confidence interval for $θ$ with confidence coefficient $1 - α$ , where $λ_{i} \in (0, 1), i = 1, 2, λ_{1} < λ_{2}$ , and $α \in (0, 1)$ , then which of the following statements is true?

A. $λ_{2}^{2} - λ_{1}^{2} = 1 - α$
B. $λ_{2}^{3} - λ_{1}^{3} = 1 - α$ ✓
C. $λ_{2}^{2} - λ_{1}^{2} = 4 (1 - α)$
D. $λ_{2}^{3} - λ_{1}^{3} = 9 (1 - α)$

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Q36What the inference means Neyman–Pearson lemma and UMP tests

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a continuous distribution with the common probability density function f(x| $θ) = 2^{θ} θ /$ $x^{θ + 1}$ if x > 2, and 0 otherwise, where $θ (> 0)$ is an unknown parameter. Suppose $P (Y > χ_{m, β}^{2})$ $= β$ , where Y ~ $χ_{m}^{2}$ . For testing $H_{0} : θ = 1$ against $H_{1} : θ > 1, a$ uniformly most powerful test of size $α, 0 < α < 1$ , will reject $H_{0}$ if

A. $\sum_{i}$ ₌ $_{1}^{n} ln X_{i} > (1/2) χ_{2 n, α}^{2}$ + n ln 2
B. $\sum_{i}$ ₌ $_{1}^{n} ln X_{i} < (1/2) χ_{2 n, 1 - α}^{2}$ + n ln 2✓
C. $\sum_{i}$ ₌ $_{1}^{n} ln X_{i} > χ_{n, α}^{2}$ + n ln 2
D. $\sum_{i}$ ₌ $_{1}^{n} ln X_{i} < χ_{n, 1 - α}^{2}$ + n ln 2

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Q37Execution slip Gauss–Markov, regression, ANOVA basics

Consider the multiple linear regression model $y_{i} = β_{0} + β_{1} x_{1 i} + \dots + β_{8} x_{8 i} + ε_{i}, i = 1, 2, \dots, 29$ , where $ε_{1}, ε_{2}, \dots, ε_{29}$ are independent and identically normal distributed with mean 0 and variance $σ^{2}$ . Suppose the model is fitted using the method of least squares. If the calculated value of the F-statistic for testing the significance of regression is 2.50, then the possible values of $R^{2}$ and Adjusted $R^{2}$ are respectively

A.0.30 and 0.10
B.0.50 and 0.30✓
C.0.50 and 0.16
D.0.30 and −0.10

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Q38Dependence misread Multivariate normal distribution

Suppose $(X_{1}, X_{2}, X_{3})^{T}$ ~ $N_{3} (0, I_{3})$ , where 0 is the zero mean vector and $I_{3}$ is the 3 × 3 identity matrix, and $(X, Y, Z)^{T} = A (X_{1}, X_{2}, X_{3})^{T}$ where A has rows (3, 0, 0), (2, 2, 0) and (4, 0, 4). Then the partial correlation coefficient $ρ_{Y Z . X}$ is

A.1/2
B.2/3
C.3/4
D.0✓

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Q39Hypothesis dropped SRS, stratified and systematic sampling

Suppose we want to estimate the population mean Ȳ of a variable for a finite population of size 85, with 34 Statisticians and 51 Biologists. We consider the following sampling scheme: a stratified random sample with 2 strata of Statisticians (Stratum-1) and Biologists (Stratum-2), where 12 Statisticians and 15 Biologists are drawn from Stratum-1 and Stratum-2, respectively, using SRSWOR scheme. Denote ȳ_S, ȳ_B, and ȳ as the mean of the variable among the Statistician sample, Biologist sample, and the combined sample, respectively. Which of the following is an unbiased estimator of Ȳ?

A.ȳ
B.(2ȳ_S + 3ȳ_B)/5✓
C.(4ȳ_S + 5ȳ_B)/9
D.(ȳ_S/12 + ȳ_B/15)/(1/12 + 1/15)

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Q40Execution slip Linear programming, simplex and duality

Solve the following linear programming problem: maximize z = x + y subject to 5x + 3y ≤ 30, 2x + 6y ≤ 25, 2x − y ≤ 8, x ≥ 0, y ≥ 0. Then the optimal value of the objective function is

A.45/11
B.74/11
C.85/12✓
D.25/6

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Part C

One or more correct options. 4.75 marks, no negative marking, and credit only for exactly the right set.

Q1Limit assumed to exist Differentiability, mean value theorems, Taylor, L'Hôpital

Let f be a bounded, twice continuously differentiable real-valued function on $(0, \infty)$ such that f″(x) ≥ 0 for all $x \in (0, \infty)$ . Which of the following statements are true?

A.f′(x) ≤ 0 for all x > 0.✓
B. $l i m_{x \to \infty}$ f′(x) = 0.✓
C. $l i m_{x \to \infty}$ x f′(x) need not exist.
D. $l i m_{x \to \infty}$ x f′(x) = 0.✓

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Q2Pointwise vs uniform Pointwise vs uniform convergence, M-test, Dini

Let $(f_{n})_{n} \geq_{1}$ be a sequence of real-valued functions on $R$ . Which of the following statements are true?

A.If each $f_{n}$ is uniformly continuous and $(f_{n})_{n} \geq_{1}$ converges to f uniformly, then f is uniformly continuous.✓
B.If each $f_{n}$ is bounded and $(f_{n})_{n} \geq_{1}$ converges to f pointwise, then f is bounded.
C.If each $f_{n}$ is bounded and continuous, $(f_{n})_{n} \geq_{1}$ converges pointwise to a bounded and continuous function f, then the convergence is uniform.
D.If each $f_{n}$ is differentiable and $(f_{n})_{n} \geq_{1}$ converges to f uniformly, then f is differentiable.

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Q3Pointwise vs uniform Pointwise vs uniform convergence, M-test, Dini

For each n ≥ 1, let $f_{n} : R \to R$ be a function defined by $f_{n} (x) =$ $e^{- n^{2} x^{2}} / n$ . Which of the following statements are true?

A. $(f_{n})_{n} \geq_{1}$ converges uniformly to 0 on $R$ , and $(f_{n}^{'})_{n} \geq_{1}$ converges uniformly to 0 on the interval (−M, M) for some positive real number M.
B. $(f_{n})_{n} \geq_{1}$ converges uniformly to 0 on $R$ , and $(f_{n}^{'})_{n} \geq_{1}$ converges pointwise to 0 on $R$ .✓
C. $(f_{n})_{n} \geq_{1}$ converges uniformly to 0 on $R$ and $(f_{n}^{'})_{n} \geq_{1}$ does not converge pointwise to 0 on $R$ .
D. $(f_{n})_{n} \geq_{1}$ converges pointwise to 0 on $R$ but not uniformly on $R$ .

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Q4Standard counterexample Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let A, B be distinct 2 × 2 real matrices. Which of the following statements are true?

A.If A is invertible, then AB and BA have the same minimal polynomial.✓
B.If 0 is an eigenvalue of A, then 0 is an eigenvalue of AB.✓
C.If 0 is the only eigenvalue of A and of B, then 0 is the only eigenvalue of AB.
D.If AB and BA have the same minimal polynomial, then either A or B is invertible.

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Q5Standard counterexample Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let $S, T : C^{2} \to C^{2}$ be $C -$ linear transformations and I denote the identity transformation on $C^{2}$ . Which of the following statements are necessarily true?

A.(ST − TS $)^{2} = λ I$ for some $λ \in C$ .✓
B.The characteristic polynomial of (ST − TS $)^{2}$ is $(x - λ)^{2}$ for some $λ \in C$ .✓
C.If ST − TS has only one eigenvalue, then ST − TS $= λ I$ for some $λ \in C$ .
D.If ST − TS has only one eigenvalue, then (ST − TS $)^{2}$ is the zero transformation.✓

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Q6Execution slip Bases, dimension, rank–nullity

Let V be the $R -$ vector space of all polynomials with real coefficients. Let $f (x) = x^{2} + x + 1$ . Which of the following subsets of V are linearly independent?

A.{f′(x), f(x) − f(x − 1), 1}
B.{f(x + 1) − f(x), f(x) − f(x − 1), 1}
C.{f(x), f′(x), 1}✓
D.{f(x + 1), f(x − 1), f(x)}✓

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Q7Standard counterexample Bases, dimension, rank–nullity

Consider the field 𝔽 $_{3}$ consisting of 3 elements. Let V be an 𝔽 $_{3} -$ vector space of dimension 3 and W an 𝔽 $_{3} -$ vector space of dimension 2. Which of the following statements are true?

A.The number of two dimensional subspaces of V is 13.✓
B.The number of surjective linear transformations from V to W is 624.✓
C.The number of one dimensional subspaces of V is 13.✓
D.The number of linear transformations from V to W is $3^{6}$ .✓

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Q8Hypothesis dropped Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

Let V be a finite dimensional complex inner product space. For a linear map T : V → V, let T* denote its adjoint. Which of the following statements are true?

A.If trace of TT* is zero, then T = 0.✓
B.Let v ∈ V be such that T*T(v) = 0. Then T(v) = 0.✓
C.Suppose T = T* and N > 1 be an integer. Let v ∈ V be such that $T^{2^{N}} (v)$ = 0. Then T(v) = 0.✓
D.Suppose that TT* = T*T and N > 1 be an integer. Let v ∈ V be such that T^N(v) = 0. Then T(v) = 0.✓

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Q9Boundary and endpoint Quadratic forms, positive definiteness, Sylvester's law

Let B(v, w) be a nondegenerate symmetric bilinear form on $R^{2}$ and let q(v) = B(v, v) be the corresponding quadratic form. Suppose there exist vectors $v, w \in R^{2}$ such that B(v, v) = 0 and B(v, w) ≠ 0. Which of the following statements are necessarily true?

A.B(w, w) = 0
B.There exists an $α \in R$ such that $q (α v + w) = 0$ .✓
C.There are infinitely many $α \in R$ such that $q (α v + w) = 0$ .
D.q is equivalent to the quadratic form $Q (x, y) = x^{2} - y^{2}$ for all $(x, y) \in R^{2}$ .✓

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Q10Standard counterexample Liouville, Morera, maximum modulus principle

Let f be an entire function. Which of the following statements are true?

A.If f(z) = f(z + 1) for all $z \in C$ then f is a constant function.
B.If f(z) = f(z + 1) = f(z + i) for all $z \in C$ then f is a constant function.✓
C.If f(1/z) has a removable singularity at 0 then f is a constant function.✓
D.If f is a non-constant function then f(1/z) has a pole at 0.

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Q11Standard counterexample Conformal maps, Möbius transformations, Schwarz lemma

Let 𝔻^× = { $z \in C : 0 <$ |z| < 1} be the punctured unit disk and f be a bijective holomorphic map of 𝔻^× onto itself. Which of the following statements are true?

A. $l i m_{z \to 0}$ f(z) does not exist.
B. $l i m_{z \to 0}$ f(z) exists and has absolute value ≤ 1.✓
C. $l i m_{z \to 0}$ f(z) = 0.✓
D.There exists $θ \in R$ such that f(z) = $e^{i θ} z$ for all z ∈ 𝔻^×.✓

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Q12Standard counterexample Power series and analyticity

Let f be an entire function which is not a polynomial. Let A = { $α \in C$ | f⁽ $^{n}$ ⁾ $(α) \neq = 0$ for all n ≥ 0}. Which of the following statements are true?

A.A is nonempty.✓
B.A is finite.
C.A is infinite.✓
D.A is uncountable.✓

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Q13Standard counterexample Linear ODE, Wronskian, variation of parameters, systems

Consider the ordinary differential equation (ODE $) d^{2} y /$ dx $^{2} + cos (x)$ dy/dx + sin(x) y = 0. Let $φ_{1} (x), φ_{2} (x)$ be solutions of the ODE, satisfying $φ_{1} (0) = 1, d φ_{1} /$ dx(0) = 0, and $φ_{2} (0) = 0, d φ_{2} /$ dx(0) = 1. Then which of the following statements are true?

A. $φ_{1} (x + 2 π)$ is also a solution of ODE✓
B. $φ_{2} (x + 4 π)$ is also a solution of ODE✓
C.There are NO constants a, b such that $φ_{2} (x + 4 π) = a φ_{1} (x) + b φ_{2} (x)$
D.There exist $a, b \in R$ such that $φ_{1} (x + 2 π) = a φ_{1} (x) + b φ_{2} (x)$ ✓

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Q14Standard counterexample Linear ODE, Wronskian, variation of parameters, systems

For a continuous function q defined on $R$ , consider the ordinary differential equation (ODE $) d^{2} y /$ dx $^{2} + q (x) y = 0, x \in R$ . Then which of the following statements are FALSE?

A.There exists a q such that cos(x) and e^x cos(x) are solutions of ODE✓
B.There exists a q such that sin(x) and cos(x) are solutions of ODE
C.There exists a q such that e^x sin(x) and e^x cos(2x) are solutions of ODE✓
D.There exists a q such that xe^x and x(x − 1)e^x are solutions of ODE✓

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Q15Hypothesis dropped First-order PDE: Lagrange, Charpit, characteristics

Consider the Cauchy problem (CP $) x \partial u / \partial x + \partial u / \partial y = 1, u (0, y) = e^{y}, y \in R$ . Then which of the following statements are true?

A.There is NO neighbourhood of the origin on which (CP) has a solution✓
B.(CP) has a unique solution defined on some neighbourhood of the origin
C.(CP) has a unique solution defined on some neighbourhood of the point (0, 1) in the xy-plane
D.(CP) has an infinite number of solutions, each of which is defined on some neighbourhood of the origin

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Q16Execution slip Laplace, heat and wave equations: separation of variables

Suppose u = u(x, y) is the solution of the boundary value problem $\partial^{2} u / \partial x^{2} + \partial^{2} u / \partial y^{2} + \partial u / \partial x = 0$ in { $(x, y) \in R^{2} : x^{2} + y^{2} < 1$ }, $u (x, y) = 1 + 2 x^{2} y^{2}$ on { $(x, y) \in R^{2} : x^{2} + y^{2} = 1$ }. Then which of the following statements are true?

A.The minimum value of u is 1✓
B.The maximum value of u is 3
C.The minimum value of u is 2
D.The maximum value of u is 3/2✓

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Q17Execution slip Interpolation and numerical integration with error terms

If $α, β \in R$ are such that the equation $\int_{0}^{3} f (x)$ dx $= (3/2) [f (α) + f (α + β)]$ holds for all polynomials f(x) of degree less than or equal to 2, then which of the following statements are true?

A. $(α, β) = ((3 - 3) /2, 3)$ or $(α, β) = ((3 + 3) /2, - 3)$ ✓
B. $(α, β) = ((3 - 2) /2, 2)$ or $(α, β) = ((3 + 2) /2, - 2)$
C. $(α, β) = ((3 - 5) /2, 5)$ or $(α, β) = ((3 + 5) /2, - 5)$
D. $(α, β) = ((3 - 7) /2, 7)$ or $(α, β) = ((3 + 7) /2, - 7)$

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Q18Converse assumed Euler–Lagrange equation and standard functionals

For $α \geq 0$ , consider the functional $J_{α} [y] = \int_{1}^{2} (y^{'})^{2} / x^{α}$ dx defined for all continuously differentiable functions defined on the interval [1, 2] satisfying the conditions y(1) = 1, y(2) = 2. Then which of the following statements are true?

A. $y (x) = (1/15) (x^{4} + 14)$ is an extremal for $J_{3}$ ✓
B. $y (x) = (1/3) (x^{2} + 2)$ is an extremal for $J_{1}$ ✓
C.y(x) = x is an extremal for $J_{0}$ ✓
D. $y (x) = (1/2) (x^{2} - x + 2)$ is an extremal for $J_{1}$

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Q19Boundary and endpoint Euler–Lagrange equation and standard functionals

For $a, b, c \in R$ , consider the variational problem: Minimize $J [y] = \int_{0}^{2} [a (y^{'})^{2} + 2$ byy′ + cy $^{2}]$ dx, subject to y(0) = 10, y(1) = 100. Then which of the following statements are true?

A.If (a, b, c) = (−2, 1, −2), then every admissible extremal is a minimizer✓
B.If (a, b, c) = (1, 0, 2), then every admissible extremal is a minimizer✓
C.If (a, b, c) = (2, −1, 1), then every admissible extremal is a minimizer✓
D.If (a, b, c) = (1, −2, 5), then every admissible extremal is a minimizer✓

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Q20Boundary and endpoint Random variables, distributions, moments, MGF

Let X be a random variable with the cumulative distribution function F_X(x) = 0 if x < 0, F_X(x) = (x + 2)/5 if 0 ≤ x < 2, and F_X(x) = 1 if x ≥ 2. Then, which of the following statements are true?

A. $E (X^{2}) = 4/3$ ✓
B.P(|X − 4/5| ≥ 1) = 19/25
C.The upper bound of P(|X − 4/5| ≥ 1), using Chebyshev's inequality, is 52/75✓
D.The value of the moment generating function, M_X(t), at t = 1 is $(e^{2} - 1) /5$

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Q21Execution slip Axioms, conditional probability, independence, Bayes

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from $N (θ, 1), θ \in R$ . If $θ$ ̂ is the Bayes estimator of $θ$ with respect to some prior $π (θ)$ and loss function $L (θ, d)$ . Then, which of the following statements are true?

A. $θ$ ̂ $= \sum_{i}$ ₌ $_{1}^{n} X_{i} / (n + τ^{2})$ , if the prior is $N (0, 1/ τ^{2}), τ^{2}$ known and $L (θ, d) = (θ - d)^{2}$ ✓
B. $θ$ ̂ $= \sum_{i}$ ₌ $_{1}^{n} X_{i} / (n + τ^{2})$ , if the prior is $N (0, τ^{2}), τ^{2}$ known and $L (θ, d) =$ | $θ - d$ |
C. $θ$ ̂ $= \sum_{i}$ ₌ $_{1}^{n} X_{i} / (n + 1/ τ^{2})$ , if the prior is $N (0, 1/ τ^{2}), τ^{2}$ known and $L (θ, d) =$ | $θ - d$ |
D. $θ$ ̂ $= \sum_{i}$ ₌ $_{1}^{n} X_{i} / n$ , if the prior is the Jeffreys prior and $L (θ, d) = (θ - d)^{2}$ ✓

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Q22Execution slip Markov chains: classification of states, stationary distributions

Consider a Markov chain { $X_{n} : n \geq 1$ } on state space {1, 2, 3, 4, 5} with the transition probability matrix whose rows are (0, 1/2, 1/2, 0, 0), (0, 0, 1, 0, 0), (0, 1/3, 0, 1/3, 1/3), (1, 0, 0, 0, 0) and (0, 0, 0, 0, 1). Then, which of the following statements are true?

A.Stationary distribution is (0, 0, 0, 0, 1).✓
B.State 5 is absorbing and recurrent.✓
C.All states are aperiodic.✓
D. $l i m_{n} \to \infty p_{55}$ ⁽ $^{n}$ ⁾ = 1.✓

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Q23Execution slip Markov chains: classification of states, stationary distributions

Consider the M/M/1 queue in which customers arrive according to a Poisson process with rate 3 and successive service times are independent exponential random variables having mean 1/9. Let $P_{n}$ be the long run probability that there are exactly n customers in the system. Then, which of the following statements are true?

A. $P_{0} = 1/3$
B. $P_{1} = 2/9$ ✓
C.The average number of customers in the system is 1
D.The average amount of time that a customer spends in the system is 1/6✓

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Q24Execution slip Standard discrete and continuous distributions

Let X and Y be independent Poisson random variables with means 4 and 2, respectively. Then, which of the following statements are true?

A.The conditional distribution of X given X + Y = 3 is Binomial(3, 1/3)
B.P(X ≤ 1 | X + Y = 3) = 7/27✓
C.E(X | X + Y = 3) = 2✓
D.The value of the characteristic function of X + Y at the point $t = π$ is $e^{- 12}$ ✓

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Q25Execution slip Standard discrete and continuous distributions

A system has two components $C_{1}$ and $C_{2}$ put in parallel. The components $C_{1}$ and $C_{2}$ have independent lifetimes $X_{1}$ and $X_{2}$ , respectively. The probability distribution of X_j is exponential with mean 1/j, j = 1, 2. Suppose that R(t) and h(t) are the reliability and the hazard rate functions of the system, respectively. Then, which of the following statements are true?

A.R(t) = $e^{- 2 t}$ + $e^{- t}$ − $e^{- 3 t}$ , t > 0✓
B.The expected lifetime of the given system is 1
C. $h (1) = (2 e + e^{2} - 3) / (e + e^{2} - 1)$ ✓
D. $h (3) = (4 e^{3} + e^{6} - 5) / (e^{3} + e^{6} - 1)$

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Q26Standard counterexample Joint distributions, transformations, order statistics

Let X and Y be independent and identically distributed N(0, 1) random variables. Let $S = X^{2} + Y^{2}$ and T = $e^{- (X^{2} + Y^{2}) /2}$ . Then, which of the following statements are true?

A.The probability density function of S is f_S(s) = $(1/2) e^{- s /2}$ if s > 0, and 0 otherwise.✓
B.The probability density function of T is f_T(t) = 1 if 0 < t < 1, and 0 otherwise.✓
C.Var(S) = 2.
D.E(T) = 2/3.

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Q27Boundary and endpoint Joint distributions, transformations, order statistics

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent and identically distributed random variables with the probability density function f(x| $θ) =$ $e^{- (x - θ)}$ if $x \geq θ$ , and 0 otherwise, where $θ \in R$ is the unknown parameter. Further, let $Y = min (X_{1}, X_{2}, \dots, X_{n})$ . Which of the following are confidence intervals for $θ$ with the confidence coefficient $(1 - α)$ , where $α \in (0, 1)$ ?

A. $(Y + (1/ n) ln (1 - α /2), Y - (1/ n) ln (1 - α /2))$
B. $(Y + (1/2 n) ln (α), Y)$
C. $(Y + (2/ n) ln (α /2), Y + (2/ n) ln (1 - α /2))$ ✓
D. $(Y + (1/ n) ln (α /2), Y + (1/ n) ln (1 - α /2))$ ✓

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Q28Dependence misread Gauss–Markov, regression, ANOVA basics

Consider a paired data $(x_{i}, y_{i}); i = 1, 2, 3, 4, 5$ , where $(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = (- 2, - 1, 0, 1, 2)$ and $y_{i} = x_{i}^{2}$ for all i = 1, 2, 3, 4, 5. On this data, a simple linear regression model with an intercept term and a simple linear regression model without an intercept term are fitted using the method of least squares. Which of the following statements are true?

A.The two fitted lines have the same slope✓
B.The two fitted models have the same intercept
C.The model with intercept passes through at least one of the observed data points
D.The model without intercept passes through at lesast one of the observed data points✓

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Q29Execution slip Gauss–Markov, regression, ANOVA basics

An analyst fits a multiple linear regression model $Y_{i} = β_{0} + β_{1} X_{1 i} + \dots + β_{4} X_{4 i} + ε_{i}, i = 1, 2, \dots, n$ , using the method of least squares. However, his coordinator insists that the intercept and two regressors $Z_{1} = X_{1} + X_{3}$ and $Z_{2} = X_{2} - X_{4}$ are enough to represent the model. Suppose the coordinator's claim can be tested in the form of a general linear hypothesis, viz., $H_{0} : L^{T} β = 0$ against $H_{A} : H_{0}$ is not true, where $β^{T} = (β_{0}, β_{1}, \dots, β_{4})$ . Suppose we have n observations on the response Y and each regressor. Further, assume that the errors with or without restrictions are independent $N (0, σ^{2})$ variables. Then, which of the following statements are true?

A.A possible choice of L is the 5 × 2 matrix with rows (0, 0), (1, 0), (0, 1), (−1, 0), (0, 1)✓
B.The test statistic for testing $H_{0}$ against H_A follows an F-distribution with (2, n − 4) degrees of freedom under $H_{0}$
C.Sum of squares residuals under the restrictions $L^{T} β = 0$ follows $σ^{2} χ_{n - 3}^{2}$ distribution✓
D.Sum of squares residuals without the restrictions $L^{T} β = 0$ follows a non-central $σ^{2} χ_{n - 4}^{2}$ distribution

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Q30Boundary and endpoint Multivariate normal distribution

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), \dots, (X_{10}, Y_{10})$ be a random sample from a bivariate normal distribution BVN $(μ_{1}, μ_{2}, σ_{1}^{2}, σ_{2}^{2}, ρ)$ with $μ_{1} = 5, μ_{2} = 6, σ_{1}^{2} = 4, σ_{2}^{2} = 9$ and $ρ = 1/2$ . Then, which of the following statements are true?

A.The distribution of $(1/ 7) \sum_{i}$ ₌ $_{1}^{10} (X_{i} - Y_{i} + 1)$ is N(0, 10)✓
B.The distribution of $(1/19) \sum_{i}$ ₌ $_{1}^{10} (X_{i} + Y_{i} - 11)^{2}$ is $χ^{2} -$ distribution with degrees of freedom 10✓
C.The distribution of $22 (X_{1} - 5) /\sqrt (\sum_{i}$ ₌ $_{3}^{10} (X_{i} - 5)^{2})$ is t-distribution with degrees of freedom 9
D.The distribution of $2 \sum_{i}$ ₌ $_{1}^{3} (Y_{i} - 6)^{2} / \sum_{i}$ ₌ $_{4}^{9} (Y_{i} - 6)^{2}$ is F-distribution with degrees of freedom 3 and 6.✓

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Q31Boundary and endpoint Multivariate normal distribution

Let the random vector $(X_{1}, X_{2}, X_{3})^{T}$ have the positive definite dispersion matrix with rows $(1, ρ, ρ), (ρ, 1, ρ)$ and $(ρ, ρ, 1)$ . Then, which of the following statements are true?

A. $ρ$ may be −0.47✓
B.The first principal component can only explain 32% of the total variation for some $ρ$
C.The second principal component can explain more than 32% of the total variation for any $ρ$
D.The variance of the first principal component is $1 + 2 ρ$ for any $ρ$

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Q32Standard counterexample limsup, liminf and subsequential limits

Consider the sequences $(s_{n})_{n} \geq_{1}$ and $(t_{n})_{n} \geq_{1}$ defined by $s_{n} = \sum_{k = 0}^{n}$ $1/ (k!)^{2}$ and $t_{n} = \sum_{k = 0}^{n}$ C(n, k)(−1)ᵏ/nᵏ. Which of the following statements are true?

A. $l im s u p_{n \to \infty}$ $t_{n} \leq$ $l im s u p_{n \to \infty}$ $s_{n}$ ✓
B. $l im s u p_{n \to \infty}$ $t_{n} \leq e$ ✓
C. $l imin f_{n \to \infty}$ $s_{n} \geq e^{2}$
D. $l imin f_{n \to \infty}$ $t_{n} \geq e$

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Q33Execution slip Riemann integration and criteria

What is the value of the limit $l i m_{n \to \infty}$ (1/n)[(n + 1)(n + 2)⋯(n + n)] $^{1/ n}$ ?

A.2/e
B.4/e✓
C. $l o g_{e} 2 - 1$
D. $2 l o g_{e} 2 - 1$

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Q34Standard counterexample Riemann integration and criteria

Let f and g be real-valued Riemann integrable functions on [a, b] such that g([a, b]) ⊆ [a, b]. Which of the following statements are necessarily true?

A.The composition f ∘ g is Riemann integrable.
B.If g(x) ≠ 0 for each x ∈ [a, b], then f/g is Riemann integrable.
C.The positive square root $f^{2} + g^{2}$ is Riemann integrable.✓
D.The composition f ∘ g is Riemann integrable, if both f and g are continuous.✓

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Q35Boundary and endpoint Lebesgue integral, MCT, DCT, Fatou

Let $μ$ denote the Lebesgue measure on $R$ . Suppose that f is a non-negative Lebesgue measurable function on $R$ . Let $0 = a_{0} < a_{1} < a_{2} < \dots$ be an unbounded sequence such that $a_{n + 1}$ ≤ ca $_{n}$ for some real number c and for all n ≥ 1. Let A_k = { $x \in R$ | a_k ≤ f(x) < $a_{k + 1}$ } for each k ≥ 0. Which of the following statements are true?

A.If f is Lebesgue integrable on $R$ , then $\sum_{k \geq 0}$ $a_{k} μ (A_{k})$ is finite.✓
B.If $\sum_{k \geq 0}$ $a_{k} μ (A_{k})$ is finite, then f is Lebesgue integrable on $R$ .
C.If $\sum_{k \geq 0}$ $a_{k} μ (A_{k})$ is finite, and $f (x) \geq a_{1}$ for all $x \in R$ , then f is Lebesgue integrable on $R$ .✓
D.If $\sum_{k \geq 0}$ $a_{k} μ (A_{k})$ is finite and f is bounded, then f is Lebesgue integrable on $R$ .

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Q36Execution slip Partial derivatives, differentiability, chain rule

Let $f : R^{2} \to R$ be a twice continuously differentiable non-zero function such that f(tx $_{1}$ , tx $_{2}) = t^{3} f (x_{1}, x_{2})$ for all t > 0 and $(x_{1}, x_{2}) \in R^{2}$ . Which of the following statements are necessarily true?

A. $3 \partial f / \partial x_{1} (1, 1) + 3 \partial f / \partial x_{2} (1, 1) = f (1, 1)$
B. $\partial f / \partial x_{1} (1, - 1) - \partial f / \partial x_{2} (1, - 1) = 3 f (1, - 1)$ ✓
C. $x_{1}^{2} \partial^{2} f / \partial x_{1}^{2} (x_{1}, x_{2}) + x_{2}^{2} \partial^{2} f / \partial x_{2}^{2} (x_{1}, x_{2}) + 2 x_{1} x_{2} \partial^{2} f / \partial x_{1} \partial x_{2} (x_{1}, x_{2}) = 6 f (x_{1}, x_{2})$ ✓
D. $x_{1}^{2} \partial^{2} f / \partial x_{1}^{2} (x_{1}, x_{2}) + x_{2}^{2} \partial^{2} f / \partial x_{2}^{2} (x_{1}, x_{2}) + 2 x_{1} x_{2} \partial^{2} f / \partial x_{1} \partial x_{2} (x_{1}, x_{2}) = 9 f (x_{1}, x_{2})$

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Q37Hypothesis dropped Linear transformations, matrix representation, change of basis

Let $M_{2} (R)$ denote the space of real 2 × 2 matrices. Let S be the vector subspace of $M_{2} (R)$ comprising of all symmetric matrices. Let $F : M_{2} (R) \to S$ be the map defined by F(X) = XXᵀ. Let DF $_{A} : M_{2} (R) \to S$ be the derivative of F at $A \in M_{2} (R)$ . Which of the following statements are true?

A.If AAᵀ = I, then DF $_{A} : M_{2} (R) \to S$ is surjective.✓
B.If AAᵀ = I, then DF $_{A} : M_{2} (R) \to S$ need not be surjective.
C.If A is invertible, then DF $_{A} : M_{2} (R) \to S$ is surjective.✓
D.If A is not invertible, then DF $_{A} : M_{2} (R) \to S$ is surjective.

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Q38Boundary and endpoint Completeness and Baire category

Let C[0, 1] be the $R -$ vector space of real valued continuous functions equipped with the norm ‖f‖ = $s u p_{x \in [0, 1]}$ |f(x)|. Let T : C[0, 1] → C[0, 1] be defined as $T (f) (x) = \int_{0}$ ˣ f(t)dt, for x ∈ [0, 1]. Let $T^{n} = T$ ∘ T ∘ ⋯ ∘ T (n times). Which of the following statements are true?

A.There exists $α \in (0, 1)$ such that for all f, g ∈ C[0, 1], ‖T(f) − T(g)‖ $\leq α$ ‖f − g‖.
B.There exists $α \in (0, 1)$ such that for all f, g ∈ C[0, 1], ‖ $T^{2} (f) - T^{2} (g)$ ‖ $\leq α$ ‖f − g‖.✓
C.The set {f ∈ C[0, 1] : T(f) = f} is a singleton set.✓
D.‖ $T^{n}$ ‖ $\to \infty$ as $n \to \infty$ .

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Q39Invariants don't determine Jordan canonical form

Let $T : C^{7} \to C^{7}$ be $a C -$ linear operator with eigenvalues 2, 3 and 5. Consider the subspace W := { $v \in C^{7} : (T - 5 I)^{k} v = 0$ for some integer k > 0} of $C^{7}$ . Suppose that $(T - 2 I)^{2} (T - 3 I)^{2} (T - 5 I)^{2} = 0$ . Which of the following statements are necessarily true?

A.T has at least four linearly independent eigenvectors.✓
B.dim W ≥ 2.
C. $ker ((T - 2 I)^{2025}) = ker ((T - 2 I)^{2026})$ ✓
D.(T − 2I)(T − 3I) is a nilpotent operator.

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Q40Invariants don't determine Jordan canonical form

Which of the following matrices are similar over $R$ to the matrix A whose rows are (−1, 1, 0, 0), (0, −1, 0, 0), (0, 0, 1, 1) and (0, 0, 0, 1)?

A.The matrix with rows (0, 0, 0, −1), (1, 0, 0, 0), (0, 1, 0, 2), (0, 0, 1, 0)✓
B.The matrix with rows (0, 0, 0, 1), (1, 0, 0, 0), (0, 1, 0, −2), (0, 0, 1, 0)
C.The matrix with rows (0, 1, 1, 0), (1, 0, 0, 1), (0, 0, 0, 1), (0, 0, 1, 0)✓
D.The matrix with rows (0, 1, 0, 0), (1, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0)

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Q41Execution slip Residue theorem and standard contour integrals

Let $γ : [0, 1] \to C$ be the function t ↦ $e^{2 π i t}$ and $I = \int_{γ} e^{z}$ $e^{1/ z}$ dz. Which of the following statements are true?

A.I = 0
B. $(1/2 π i) I \in$ { $4 n : n \in Z, n \geq 1$ }
C. $I = 2 π i \sum_{n = 0}^{\infty}$ 1/n!
D. $I = 2 π i \sum_{n = 0}^{\infty}$ 1/(n!(n + 1)!)✓

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Q42Standard counterexample Group actions, class equation, p-groups

Let G be a group, H a subgroup of G, and T = {gH | g ∈ G}, the set of all left cosets of H in G. Let S_T be the set of all permutations of T and $π : G \to S_{T}$ be the map defined by $π (g) (g_{1} H) =$ gg $_{1} H$ . For a prime number p, let 𝔽_p denote the field with p elements. In which of the following cases is $ker π$ trivial?

A.G = GL $_{2} ($ 𝔽_p) and H is a subgroup of order p.✓
B.G = SL $_{2} ($ 𝔽_p) and H is a subgroup of order p.✓
C.p ≡ 3 (mod 4), G = GL $_{2} ($ 𝔽_p)/SL $_{2} ($ 𝔽_p) and H is a subgroup of order 2.
D.p ≡ 1 (mod 4), G = GL $_{2} ($ 𝔽_p)/SL $_{2} ($ 𝔽_p) and H is a subgroup of order 2.

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Q43Property not inherited Normal subgroups, quotients, isomorphism theorems

For a group G, let Aut(G) denote the group (under composition) of all bijective group homomorphisms from G onto itself. Which of the following statements are true?

A.If $G_{1}, G_{2}$ are two groups such that Aut $(G_{1})$ is isomorphic to Aut $(G_{2})$ , then $G_{1}$ is isomorphic to $G_{2}$ .
B.If |G| = 2, then Aut(G × G) is abelian.
C.If G is the group of complex numbers under addition, then Aut(G) is abelian.
D.If G is finite, then Aut(G) is finite.✓

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Q44Property not inherited Normal subgroups, quotients, isomorphism theorems

Let $G_{1}$ and $G_{2}$ be subgroups of a group G. Which of the following statements are true?

A.If $G_{1}$ is normal in G, then $(G_{2} G_{1}) / G_{1} ≅ G_{2} / (G_{2} \cap G_{1})$ .✓
B.If $H_{1}$ and $H_{2}$ are normal subgroups of $G_{1}$ and $G_{2}$ , respectively, then $(G_{1} \times G_{2}) / (H_{1} \times H_{2}) ≅ (G_{1} / H_{1}) \times (G_{2} / H_{2})$ .✓
C.If $G_{1}$ is normal in $G_{2}$ and $G_{2}$ is normal in G, then $G_{1}$ is normal in G.
D.Every subgroup of prime index in G is normal.

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Q45Finite-dimensional intuition Polynomial rings and irreducibility tests

Let $f : Q [X] \to Q [X]$ be a ring homomorphism with f(1) = 1. For n ≥ 1, let $f^{n} = f$ ∘ ⋯ ∘ f (n times). Which of the following statements are true?

A.If f is onto, then so is $f^{n}$ for all n ≥ 1.✓
B.ker $f^{n + 1}$ $= ker f^{n}$ for some n ≥ 1.✓
C.If f is onto, then f is one-to-one.✓
D.If f is one-to-one, then f is onto.

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Q46Standard counterexample Polynomial rings and irreducibility tests

Let $f (X) = X^{5} + X + 1 \in Q [X]$ and $g (X) = X^{5} - X + 1 \in Q [X]$ . Which of the following statements are true?

A.f(X) is irreducible in $Q [X]$ , but g(X) is not.
B.g(X) is irreducible in $Q [X]$ , but f(X) is not.✓
C.Both f(X) and g(X) are irreducible in $Q [X]$ .
D.Neither f(X) nor g(X) is irreducible in $Q [X]$ .

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Q47Hypothesis dropped Field extensions, splitting fields, finite fields

Let p > 2 be a prime number. Let 𝔽_p denote the field with p elements and 𝔽̄_p an algebraic closure of 𝔽_p. Which of the following statements are true?

A.Let f(X) ∈ 𝔽_p[X] and $α$ be a root of f in 𝔽̄_p. Then 𝔽 $_{p} (α)$ is the splitting field of f in 𝔽̄_p.
B.Let f, g ∈ 𝔽_p[X] be irreducible polynomials of same degree and $α$ be a root of f in 𝔽̄_p. Then 𝔽 $_{p} (α)$ is the splitting field of g in 𝔽̄_p.✓
C.𝔽_p[X] has infinitely many irreducible polynomials.✓
D.The set {a + b | a, b ∈ 𝔽_p} is contained in { $a^{2} + b^{2}$ | a, b ∈ 𝔽_p}.✓

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Q48Converse assumed Compactness: open covers, sequential, Heine–Borel

Let $p : R \to R$ be a nonconstant polynomial. Which of the following statements are true?

A.The preimage of a compact set under p is a compact set.✓
B.The preimage of a connected set under p is a connected set.
C.Every point $x \in R$ has an open neighbourhood U_x such that the restriction $p ∣_{U_{x}}$ is a homeomorphism onto an open set in $R$ .
D.The image of a bounded set under p is a bounded set.✓

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Q49Standard counterexample Open/closed sets, limit points, closure, interior

Consider $R$ with the usual topology and S = { $a + b 2$ | $a, b \in Q$ } $\subset R$ with the subspace topology. Which of the following statements are true?

A.S is dense in $R$ .✓
B.S \ $Q$ is dense in $R$ .✓
C.S \ $Q$ is discrete with subspace topology on S.
D.S is connected.

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Q50Existence vs uniqueness Existence–uniqueness, Picard, Lipschitz

Let D = { $(x, y) \in R^{2} : - 1 \leq x \leq 1, - 1 \leq y \leq 1$ }, and $f : D \to R$ be the function defined by $f (x, y) = 1 + \sqrt (y$ ₊), where y₊ = max{y, 0}. Consider the initial value problem (IVP) dy/dx = f(x, y), y(0) = 0. Then which of the following statements are true?

A.f is a Lipschitz continuous function on D
B.f is NOT a Lipschitz continuous function on D✓
C.IVP has at least one solution✓
D.IVP has NO solution

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Q51Numerical convergence Numerical ODE: Euler, Runge–Kutta

Consider the initial value problem (IVP) y′ + y = 0, y(0) = 1. Let $(y_{n})$ be the iterates of forward Euler method, applied to the IVP, with step size h where 0 < h < 1. Then which of the following statements are true?

A.The sequence $(y_{n})$ does NOT converge
B. $y_{n} \to 0$ as $n \to \infty$ ✓
C. $0 \leq y_{n} \leq 1$ for n = 0, 1, 2, …✓
D.|y(nh $) - y_{n}$ | → 0 as $n \to \infty$ ✓

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Q52Execution slip Fredholm and Volterra equations

Let u(x) be the solution to the Volterra integral equation $u (x) = x^{2} + 4 \int_{0}$ ˣ $(t - x)^{2} u (t)$ dt. Then which of the following statements are true?

A.u(0) = 0✓
B. $u (2 π / 3) =$ $(1/6) (e^{4 π / 3}$ − $e^{- 2 π / 3})$ ✓
C. $u (π / (23)) =$ $(1/6) (e^{π / 3}$ $- 3$ $e^{- π / (23)})$ ✓
D. $u (π / (23)) =$ $(1/6) (e^{π / 3}$ $+ 3$ $e^{- π / 3})$

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Q53Execution slip Fredholm and Volterra equations

Let f and K be such that the solution of the initial value problem y″ − 3y′ + 2y = 4sin(x), y(0) = 1, y′(0) = −2 satisfies the Volterra integral equation $y (x) = f (x) + \int_{0}$ ˣ K(x, t)y(t) dt. Then which of the following statements are true?

A. $f^{'} (π) = 3$ ✓
B. $f (π) + f^{'} (π) = 4 - π$ ✓
C. $f (π) + f^{'} (π) = 2 - π$
D.f(0) + f′(0) = −4✓

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Q54Standard counterexample Hamiltonian formalism and conservation laws

Suppose f, g are smooth functions of generalized coordinates $q_{1}, q_{2}, \dots, q_{n}$ , the associated conjugate momenta $p_{1}, p_{2}, \dots, p_{n}$ , and time t. Let [f, g] denote the Poisson bracket of f and g. Suppose H is a Hamiltonian of the system. Then which of the following statements are true?

A. $\partial / \partial t [f, g] = [\partial f / \partial t, g] + [f, \partial g / \partial t]$ ✓
B.If f is a constant of motion, and f is independent of t, then [H, f] is a constant of motion✓
C.[[H, f], g] + [[g, H], f] + [[f, g], H] = 0✓
D.If f and g are constants of motion, then [f, g] is a constant of motion✓

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Q55Moments and tails Modes of convergence, WLLN, SLLN, CLT

Let $X_{1}, X_{2}, \dots$ be a sequence of independent and identically distributed random variables with $E (X_{1}) = 0, E (X_{1}^{2}) = 1, E (X_{1}^{3}) = 0, E (X_{1}^{4}) = 3$ . Let $S_{n} = \sum_{i}$ ₌ $_{1}^{n} X_{i}, T_{n} = \sum_{i}$ ₌ $_{1}^{n} X_{i}^{2}, U_{n} = \sum_{i}$ ₌ $_{1}^{n} X_{i}^{3}$ and $V_{n} = \sum_{i}$ ₌ $_{1}^{n} X_{i}^{4}$ . Then, which of the following statements are true?

A. $S_{n} / n$ converges in distribution to a random variable Z, where Z ~ N(0, 1)✓
B. $(T_{n} - n) / 3 n$ converges in distribution to a random variable Z, where Z ~ N(0, 1)
C. $n S_{n} / T_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)✓
D. $(T_{n} - n) / V_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)

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Q56Execution slip Sufficiency, completeness, UMVUE, Cramér–Rao

Let $X_{1}, X_{2}, \dots, X_{n} (n \geq 3)$ be a random sample from the uniform distribution on the interval $(θ_{1} - θ_{2}, θ_{1} + θ_{2})$ , where $θ_{1} \in R$ and $θ_{2} > 0$ are unknown parameters. Let $X_{(j)}$ be the jᵗʰ order statistic, j = 1, 2, …, n, and let $\overset{ˉ}{X} = (1/ n) \sum_{i}$ ₌ $_{1}^{n} X_{i}$ . Here, $(X_{(1)}, X_{(n)})$ is a complete and sufficient statistic for $(θ_{1}, θ_{2})$ . Then, which of the following statements are true?

A.X̄ is an unbiased estimator of $θ_{1}$ ✓
B. $(\overset{ˉ}{X} - X_{(1)})$ is an unbiased estimator of $θ_{2}$
C. $(X_{(1)} + X_{(n)}) /2$ is the uniformly minimum variance unbiased estimator of $θ_{1}$ ✓
D. $(n + 1) (X_{(n)} - X_{(1)}) / (2 (n - 1))$ is the uniformly minimum variance unbiased estimator of $θ_{2}$ ✓

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Q57Execution slip Sufficiency, completeness, UMVUE, Cramér–Rao

Let $X_{1}, X_{2}, \dots, X_{n} (n \geq 2)$ be a random sample from a continuous distribution with the probability density function f(x| $θ) =$ $(1/2 θ) e^{- ∣ x - θ ∣/ θ}$ , $x \in R$ , where $θ (> 0)$ is an unknown parameter. Let $U_{n} = (1/ n) \sum_{i}$ ₌ $_{1}^{n} X_{i}, V_{n} = (1/ n) \sum_{i}$ ₌ $_{1}^{n} X_{i}^{2}$ , and $S_{n}^{2} = (1/ (n - 1)) \sum_{i}$ ₌ $_{1}^{n} (X_{i} - U_{n})^{2}$ . Then, which of the following statements are true?

A. $U_{n}$ is an unbiased estimator of $θ$ ✓
B. $S_{n}^{2}$ is an unbiased estimator of $θ^{2}$
C. $(1/3) V_{n}$ is a consistent estimator of $θ^{2}$ ✓
D.The statistic $(U_{n}, V_{n})$ is complete

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Q58What the inference means Likelihood ratio and standard tests

Let X be a random sample of size one from the probability density function f(x| $θ) =$ $θ e^{- θ (x - 1)}$ if x > 1, and 0 otherwise, where $θ (> 0)$ is the unknown parameter. Suppose we want to test the null hypothesis $H_{0} : θ = 1$ against the alternative hypothesis $H_{1} : θ \neq = 1$ , based on the observed value x of X. Then, which of the following statements are true?

A.The likelihood function is maximized at $θ = 1/ (x - 1)$ ✓
B.The maximum value of the likelihood function is $e^{- 1} / (x$ − 1)✓
C.The likelihood ratio test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ if (x − $1) e^{- x}$ < k, for some k > 0✓
D.The likelihood ratio test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ if x > c, for some c > 1

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Q59What the inference means Likelihood ratio and standard tests

Let $(X_{1}, X_{2}, \dots, X_{7})$ and $(Y_{1}, Y_{2}, \dots, Y_{9})$ be two independent random samples from the continuous distribution functions $F (x - μ)$ and $F (x - θ)$ , respectively, where $F, μ$ and $θ$ are all unknown. Further, let $μ$ be the unique median of $F (x - μ)$ and $θ$ be the unique median of $F (x - θ)$ . Let $R_{i}$ be the rank of $Y_{i}$ in the combined sample, i = 1, 2, …, 9. For testing $H_{0} : μ = θ$ against $H_{1} : μ > θ$ , the test statistic $T = \sum_{i}$ ₌ $_{1}^{9} R_{i}$ is proposed. Then, which of the following statements are true?

A.The maximum possible value of T is 115
B.Right-tailed test based on T is appropriate for testing $H_{0}$ against $H_{1}$
C.Under $H_{0}, E (T) = 76$
D.Under $H_{0}, P (R_{1} = 1, R_{9} = 16) = 1/240$ ✓

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Q60Invariants don't determine CRD, RBD, LSD essentials

If the incidence matrix of a block design is given by N with rows (1, 1, 1, 0), (1, 1, 0, 1), (1, 0, 1, 1) and (0, 1, 1, 1), then which of the following statements are true?

A.The design is incomplete✓
B.The design is connected✓
C.The design is balanced✓
D.The design is orthogonal

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