PYQ Paper

CSIR NET December 2025 — Part C

All 60 Part C questions we have transcribed from this paper, of the 119 on the site for this sitting — every option and the answer key, with the reasoning for each one.

Part C

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

Q1 Execution slip Likelihood ratio and standard tests

Let $X_{1}, X_{2}, \dots, X_{15}$ be a random sample from an Exponential distribution with the probability density function f(x∣ $\sigma) = (1/\sigma)\exp(-x/\sigma)$ if x>0, 0 elsewhere, where the unknown parameter $\sigma$ is positive. Let ${\bar{X}} = (1/15)\sum_{1}^{15}X_{i}$ . Suppose that $\varphi$ denotes the likelihood ratio test for testing $H_{0}: \sigma\le1$ against $H_{1}: \sigma>1$ at level $\alpha=0.1$ . It is given that $\chi^{2}_{15},_{0}._{1}=22.307, \chi^{2}_{15},_{0}._{9}=8.547, \chi^{2}_{30},_{0}._{1}=40.256, \chi^{2}_{30},_{0}._{9}=20.599$ , where $P(W>\chi^{2}_{n},_{a})=\alpha$ and W~ $\chi^{2}_{n}$ . Then which of the following statements are true?

A.If the observed value of X̄ is 0.6, then $\varphi$ does not reject $H_{0}$ ✓ Correct
B.If the observed value of X̄ is 1.6, then $\varphi$ rejects $H_{0}$ ✓ Correct
C.If the observed value of X̄ is 1.3, then $\varphi$ rejects $H_{0}$
D.If the observed value of X̄ is 1.2, then $\varphi$ does not reject $H_{0}$ ✓ Correct

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Q2 Execution slip Polynomial rings and irreducibility tests

Which of the following statements are true?

A. $x^{2} +$ xy $^{2} - x - y + 1$ is irreducible in $\mathbb{Q}[x,y]$ .✓ Correct
B. $x^{3} + 100x^{2} + 25x + 10$ is irreducible in $\mathbb{Z}[x]$ .✓ Correct
C. $2x^{2} + 3$ xy $+ y^{2} + 3x + 2y + 1$ is irreducible in $\mathbb{Q}[x,y]$ .
D. $x^{4} + 4x^{2} + 3$ is irreducible in $\mathbb{Z}[x]$ .

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Q3 Limit assumed to exist Sufficiency, completeness, UMVUE, Cramér–Rao

Let $X_{1}, X_{2}, \dots, X_{n} (n\ge2)$ be a random sample from an Exponential distribution with the probability density function f(x∣ $\mu,\sigma) = (1/\sigma)\exp((\mu-x)/\sigma)$ if $x>\mu, 0$ otherwise, where parameters $\mu$ and $\sigma$ are unknown and positive. Let ${\bar{X}}_{n}, S_{n}^{2}$ and $X_{1}:_{n}$ denote the sample mean, the sample variance and the sample smallest order statistic, respectively, and let $\theta = \sigma/\mu$ . Then which of the following statements are true?

A. $S_{n}(X_{1}:_{n})$ ⁻ $^{1}$ is a consistent estimator of $\theta$ ✓ Correct
B. $({\bar{X}}_{n} - X_{1}:_{n})(X_{1}:_{n})$ ⁻ $^{1}$ is a consistent estimator of $\theta$ ✓ Correct
C. $({\bar{X}}_{n} - X_{1}:_{n})({\bar{X}}_{n} - S_{n})$ ⁻ $^{1}$ is a consistent estimator of $\theta$ ✓ Correct
D. $S_{n}({\bar{X}}_{n})$ ⁻ $^{1}$ is a consistent estimator of $\theta$

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Q4 Execution slip SRS, stratified and systematic sampling

Consider a dataset of n observations given by A = { $x_{1},x_{2},\dots,x_{n}$ }. Create a new dataset, given by B = { $x_{1},x_{2},\dots,x_{n},-x_{1},-x_{2},\dots,-x_{n}$ }. Which of the following statements are always true? (Variance is calculated with divisor as the number of observations in the corresponding dataset.)

A.Mean of the observations in dataset B is 0✓ Correct
B.Median of the observations in dataset B is 0✓ Correct
C.Variance of the observations in dataset B ≥ Variance of the observations in dataset A✓ Correct
D.Range of the observations in dataset B ≥ Range of the observations in dataset A✓ Correct

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Q5 Standard counterexample Sylow theorems and groups of small order

Let G be a group of order 8. Which of the following statements are necessarily true?

A.If there are no elements of order 4 in G, then G is abelian.✓ Correct
B.If there are exactly two elements of order 4 in G, then G is abelian.
C.If there are exactly six elements of order 4 in G, then G is abelian.
D.There is an element of order 4 in G.

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Q6 Standard counterexample Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

Let $\alpha = \sum_{l=1}^{2025}$ $a_{l}/10$ ˡ with $a_{l} \in$ {0,1,…,9} for all 1≤l≤2025 and $a_{2025}\ne0$ . Let $(\beta_{n})_{n}\ge1$ be a strictly increasing sequence of positive real numbers in (0,1) that converges to $\alpha$ . For each n≥1, write $\beta_{n} = \sum_{l=1}^\infty$ $b_{n,l}/10$ ˡ with $b_{n,l}$ ∈{0,1,…,9}, for the infinite decimal expansion of $\beta_{n}$ . Which of the following statements are necessarily true?

A.There exists a positive integer N such that for all n≥N, $b_{n,l}$ $= a_{l}$ for all 1≤l≤2023.✓ Correct
B.There exists a positive integer N such that for all n≥N, $b_{n,2025}$ $= a_{2025} - 1$ .✓ Correct
C. $\beta_{n}$ is rational for infinitely many n.
D.There exists a positive integer N such that for all n≥N, $b_{n,2024}$ $= a_{2024} - 1$ .

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Q7 Hypothesis dropped Improper integrals and convergence tests

Let F be the set of all functions $f : [0,\infty) \to [0,\infty)$ that are Riemann integrable on [0,t] for all $t\in[0,\infty)$ . Which of the following statements are necessarily true?

A.If f∈F is a continuous function and f(n)=0 for all positive integers n, then $\lim(x\to\infty)\int_{0}$ ˣf(t)dt exists in $\mathbb{R}$ .
B.If f∈F is uniformly continuous and $\lim(x\to\infty)\int_{0}$ ˣf(t)dt exists in $\mathbb{R}$ , then $\lim(x\to\infty)f(x)=0$ .✓ Correct
C.There exists a function f∈F such that $\lim(x\to\infty)\int_{0}$ ˣf(t)dt exists in $\mathbb{R}$ but $\lim(x\to\infty)f(x)\ne0$ .✓ Correct
D.If f∈F is Lebesgue measurable and $\lim(x\to\infty)\int_{0}$ ˣf(t)dt=0, then f=0 almost everywhere on $[0,\infty)$ .✓ Correct

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Q8 Execution slip Sturm–Liouville problems and Green's functions

Consider the ordinary differential equation (ODE) y″ + r(x)y = 0, where $r(x) = x^{3}\sin(1/x)$ for x≠0, and r(0)=0. Then which of the following statements are true?

A.There exists a non-trivial solution $\varphi$ of ODE, and $\alpha,\beta\in\mathbb{R}, \alpha<\beta$ such that $\varphi$ has infinitely many zeros in $[\alpha,\beta]$ .
B.There exists a non-trivial solution $\varphi$ of ODE, and a sequence { $x_{n}$ } such that $x_{n}\to\infty$ and $\varphi(x_{n})=0$ for all $n\in\mathbb{N}$ .✓ Correct
C.If $\varphi_{1},\varphi_{2}$ are two linearly independent solutions of ODE, then their Wronskian is a constant function on $\mathbb{R}$ .✓ Correct
D.There exists a non-trivial solution of ODE which vanishes at most at one point in $(0,\infty)$ .

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Q9 Hypothesis dropped Compactness and Tychonoff

Consider $X = \prod_{c\in(0,\infty)}$ [0,c] together with the product topology, where [0,c] is equipped with the euclidean topology. Which of the following statements are necessarily true?

A.X is metrizable.
B.X is compact.✓ Correct
C.X is Hausdorff.✓ Correct
D.X is connected.✓ Correct

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Q10 Boundary and endpoint MLE and method of moments

Let $X_{1}, X_{2}, \dots, X_{n} (n\ge2)$ be a random sample from Uniform $[-3\theta/2, \theta/2]$ distribution, where $\theta>0$ is an unknown parameter. Let ${\bar{X}}, X_{(1)}$ and $X_{(n)}$ denote respectively the mean, the smallest order statistic and the largest order statistic of the sample. Then which of the following statements are true?

A.The method of moments estimator of $\theta$ is −2X̄✓ Correct
B.The method of moments estimator of $\theta$ is 2X̄
C.The maximum likelihood estimator of $\theta$ is min{ $-2X_{(1)}/3, 2X_{(n)}$ }
D.The maximum likelihood estimator of $\theta$ is max{ $-2X_{(1)}/3, 2X_{(n)}$ }✓ Correct

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Q11 Standard counterexample Continuity, homeomorphism, separation axioms

Let $\tau_E$ denote the euclidean topology on $\mathbb{R}$ . Which of the following statements are necessarily true?

A. $(\mathbb{R}, \tau_E)$ is a normal space.✓ Correct
B.Let $\tau$ be a topology on $\mathbb{R}$ . If the identity function of $\mathbb{R}$ is a continuous map from $(\mathbb{R},\tau_E)$ to $(\mathbb{R},\tau)$ , then $(\mathbb{R},\tau)$ is a regular space.
C.Let $\tau$ be a topology on $\mathbb{R}$ . If the identity function of $\mathbb{R}$ is a continuous map from $(\mathbb{R},\tau)$ to $(\mathbb{R},\tau_E)$ , then $(\mathbb{R},\tau)$ is a Hausdorff space.✓ Correct
D. $\mathbb{R}$ is a regular space in the finite-complement topology.

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Q12 Standard counterexample Fredholm and Volterra equations

Let y(x) be the solution of the integral equation (IE) y(x) = eˣ $+ e + 1 + (1/3)\int_{0}^{1}y(t)$ dt, and R(x,t,1/3) be the resolvent kernel associated to IE. Then which of the following statements are true?

A.R(0,1,1/3) = 3/2✓ Correct
B.R(1/2,1/2,1/3) = 2/3
C.y(1) = 3e+1✓ Correct
D.y(1) = e+3

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Q13 What the inference means Axioms, conditional probability, independence, Bayes

Suppose X∣ $\theta$ ~ Binomial $(7,\theta), 0<\theta<1$ , and the prior distribution of $\theta$ is Beta $(\alpha,\beta)$ where $\alpha>0$ and $\beta>0$ are known. Then which of the following statements MAY NOT be true?

A.Posterior mean of $\theta$ given X=2 is less than the prior mean✓ Correct
B.Posterior mean of $\theta$ given X=3 is greater than the prior mean✓ Correct
C.Posterior mean of $\theta$ given X=4 is not equal to the prior mean✓ Correct
D.Posterior mean of $\theta$ given X=5 is equal to the prior mean✓ Correct

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Q14 Standard counterexample Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let $A \in M_{4}(\mathbb{C})$ be such that $A^{2} = I$ and Trace(A) = 0. Which of the following statements are necessarily true?

A.The set { $B \in M_{4}(\mathbb{C})$ ∣ AB+BA=0} is an 8-dimensional $\mathbb{C}-$ vector space.✓ Correct
B.The set { $B \in M_{4}(\mathbb{C})$ ∣ AB−BA=0} is an 8-dimensional $\mathbb{C}-$ vector space.✓ Correct
C.There exists $B \in M_{4}(\mathbb{C})$ such that AB+BA $=0, B^{2}=I$ and Trace(B)=0.✓ Correct
D.If $B \in M_{4}(\mathbb{C})$ is such that AB+BA=0, then B is diagonalisable.

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Q15 Hypothesis dropped Markov chains: classification of states, stationary distributions

Consider an M/M/2 queuing system with the birth rate $\lambda=4$ per minute, the death rate $\mu=1$ per minute, and the total capacity of 3 customers (including the ones that are being served). Let $p_{0}$ and $p_{3}$ denote the long-run probabilities that the system will be empty (i.e. without customers) and will be blocked (i.e. full), respectively. Which of the following statements are true?

A. $p_{0}+p_{3} = 17/29$ ✓ Correct
B. $p_{0} > p_{3}$
C. $p_{3}/p_{0} = 2$
D. $p_{3}-p_{0} = 15/29$ ✓ Correct

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Q16 Boundary and endpoint Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

For n≥3, consider the space $M_{n}(\mathbb{C})$ of n×n complex matrices endowed with the inner product ⟨A,B⟩ = Trace(A*B). For 0≤k≤n, let $W_{k} =$ { $A \in M_{n}(\mathbb{C})$ : ⟨A,B⟩=0 for all $B \in M_{n}(\mathbb{C})$ with rank k}. Which of the following statements are necessarily true?

A. $W_{0} =$ {0}
B. $W_{1} =$ {0}✓ Correct
C. $W_{2} =$ {0}✓ Correct
D. $W_{n} =$ {0}✓ Correct

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Q17 Boundary and endpoint

There are six persons. On each day of the year 2024 at least one of them borrowed money from another. A person P is labeled troublesome in a calendar month M of 2024 if P borrowed from the same person at least twice during month M. Which of the following statements are necessarily true?

A.In every calendar month of 2024, at least one person was troublesome.
B.In any two consecutive calendar months of 2024, at least one person was troublesome in at least one of the two calendar months.✓ Correct
C.At least one person was troublesome in two calendar months of 2024.✓ Correct
D.At least one person was troublesome in January 2024.✓ Correct

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Q18 Execution slip Bases, dimension, rank–nullity

Let V be a real vector space. Suppose that {u,v,w,x,y} ⊆ V is a spanning set of V and that {u,v,x,y} is linearly independent. Which of the following statements are necessarily true?

A.The dimension of V is 4 or 5.✓ Correct
B.If 2u−3v+5w=0, then {v,w,x,y} is a basis of V.✓ Correct
C.If u−6v+7w=0, then the span of {v,w,x} is a 2-dimensional vector space.
D.If u+w=v+x, then {u,v,w,y} is a basis of V.✓ Correct

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Q19 Execution slip Isoperimetric problems

Suppose y(x) is the extremal of the variational problem $J(y) = \int_{0}^{1} x^{2}(y')^{2}$ dx subject to $y(0)=0, y(1)=1, \int_{0}^{1}y^{2}$ dx = 1/7. Then which of the following statements are true?

A.y′(1/2) = 3/4✓ Correct
B. $y'(1/\sqrt{3}) = 1$ ✓ Correct
C.y′(1/3) = 1/3✓ Correct
D.y′(1/4) = 1/2

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Q20 Execution slip Multivariate normal distribution

Let $X_{1}, X_{2}, \dots, X_{20}$ be a random sample from $N_{12}(\mu,\sum)$ , where $\mu\in\mathbb{R}^{12}$ and $\sum$ is positive definite. Suppose ${\bar{X}} = (1/20)\sum_{i}$ ₌ $_{1}^{20}X_{i}$ and $S = (1/19)\sum_{i}$ ₌ $_{1}^{20}(X_{i}-{\bar{X}})(X_{i}-{\bar{X}})^{T}$ . Then which of the following statements are true?

A.19(X̄ᵀSX $)({\bar{X}}^{T}\sum{\bar{X}})$ ⁻ $^{1}$ ~ $\chi^{2}_{19}$ ✓ Correct
B.E(S⁻ $^{1}) = (19/6)\sum$ ⁻ $^{1}$ ✓ Correct
C.S ~ $W_{12}(19, 19\sum)$
D.trace $(19\sum$ ⁻ $^{1}S)$ ~ $\chi^{2}_{228}$ ✓ Correct

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Q21 Hypothesis dropped Laplace, heat and wave equations: separation of variables

Let u(x,t) be the solution of the initial-boundary value problem $u_{tt} - u_{xx} + u_{t} = 0, 0<x<\pi, t>0, u(0,t)=0, u(\pi,t)=0, t>0, u(x,0)=\sin x, u_{t}(x,0)=0, 0<x<\pi$ . For t≥0, define $E(t) = \int_{0}^\pi (u_{t}^{2}+u_{x}^{2})$ dx. Then which of the following statements are true?

A. $E(t) \ge \pi$ for all t>0
B. $E(t) \le \pi$ for all t>0✓ Correct
C. $E(t) \ge \pi/2$ for all t>0
D. $E(t) \le \pi/2$ for all t>0✓ Correct

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Q22 Numerical convergence Root finding: bisection, Newton–Raphson, fixed point, order of convergence

Let S be the set of all 2×2 matrices A such that the iterative sequence generated by the Gauss-Seidel method converges for every initial guess, when employed to solve the system of equations $A(x_{1},x_{2})^{T}=(1,2)^{T}$ . Then which of the following statements are true?

A.(4,1;2,3) ∈ S✓ Correct
B.(1,2;3,4) ∈ S
C.(1,5;1,10) ∈ S✓ Correct
D.(−5,2;1,−4) ∈ S✓ Correct

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Q23 Standard counterexample Ideals, quotient rings, prime & maximal ideals, CRT

Consider a commutative ring R with unity with at least five elements such that for any two elements a,b∈R, there exists c∈R such that a=bc or b=ac. Which of the following statements are necessarily true?

A.R has a unique maximal ideal.✓ Correct
B.Every finitely generated ideal of R is principal.✓ Correct
C.R is an integral domain.
D.R is a euclidean domain.

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Q24 Standard counterexample Power series and analyticity

Consider the following statements: (P) The initial value problem $y'+y=e^(-x^{2}), y(0)=0$ has a Taylor series solution about the point x=0. (Q) The initial value problem y′+y=r(x), y(0)=0, where $r(x)=e^(-1/x^{2})$ for x≠0 and r(0)=0, has a Taylor series solution about the point x=0. Then which of the following statements are true?

A.(P) is true.✓ Correct
B.(P) is FALSE.
C.(Q) is true.
D.(Q) is FALSE.✓ Correct

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Q25 Hypothesis dropped Inverse and implicit function theorems, extrema

Let $F : \mathbb{R}^{3} \to \mathbb{R}$ be a continuously differentiable function such that $F(0,0,0)=-1, \partial{}F/\partial{}x(0,0,0)=2, \partial{}F/\partial{}y(0,0,0)=0$ , and $\partial{}F/\partial{}z(0,0,0)=3$ . For $\varepsilon>0$ , define $\Omega_\varepsilon=(-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon)\subset\mathbb{R}^{2}$ . Which of the following statements are necessarily true?

A.There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $g:\Omega_\varepsilon\to(-\delta,\delta)$ such that g(0,0)=0 and F(x,y,g(x,y))=−1 for all $(x,y)\in\Omega_\varepsilon$ .✓ Correct
B.There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $h:\Omega_\varepsilon\to(-\delta,\delta)$ such that h(0,0)=0 and F(x,h(x,z),z)=−1 for all $(x,z)\in\Omega_\varepsilon$ .
C.There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $k_{1},k_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $k_{1}(0)=k_{2}(0)=0$ and $F(x,k_{1}(x),k_{2}(x))=-1$ for all $x\in(-\varepsilon,\varepsilon)$ .✓ Correct
D.There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $j_{1},j_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $j_{1}(0)=j_{2}(0)=0$ and $F(j_{1}(z),j_{2}(z),z)=-1$ for all $z\in(-\varepsilon,\varepsilon)$ .✓ Correct

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Q26 Pointwise vs uniform Pointwise vs uniform convergence, M-test, Dini

For each positive integer n, let $f_{n}:[-1,1]\to\mathbb{R}$ be given by $f_{n}(x)=-1$ if $-1\le{}x\le-1/n, f_{n}(x)=$ nx if −1/n<x<1/n, and $f_{n}(x)=1$ if 1/n≤x≤1. On which of the following intervals does the sequence { $f_{n}$ } $_{n}\ge1$ converge uniformly?

A.[0,1]
B.(0,1]
C.(10⁻ $^{2025},1)$ ✓ Correct
D.(−10⁻ $^{2025},10$ ⁻ $^{2025}]$

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Q27 Standard counterexample Open/closed sets, limit points, closure, interior

Let (X,d) be a metric space. For a non-empty subset A of X, and x∈X, define d(x,A) = $inf_{a\in{}A}$ d(x,a). Which of the following statements are necessarily true?

A.For all x,y∈X and every non-empty subset A of X, we have d(x,A)−d(y,A) ≤ d(x,y).✓ Correct
B.For every non-empty subset A of X, the function x↦d(x,A) is uniformly continuous on X.✓ Correct
C.A non-empty subset A of X is closed if and only if d(x,A)>0 for all x in X outside A.✓ Correct
D.If X is compact and $C_{1},\dots,C_{k}$ are non-empty closed sets of X such that $\cap{}C_{i}=\emptyset$ , then the minimum value of $x\mapsto\sum{}d(x,C_{i})$ on X is 0.

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Q28 Limit assumed to exist Completeness, sup/inf, Archimedean property

Consider the following subset of real numbers A = { $(1+(-1)^{n})n - 1/n : n$ is a positive integer}. Which of the following statements are true?

A.A is bounded below but not bounded above.✓ Correct
B.A is bounded above but not bounded below.
C.inf A = −1✓ Correct
D.sup A = 1

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Q29 Boundary and endpoint Improper integrals and convergence tests

Let $f : (0,\infty) \to (0,\infty)$ be the function defined by $f(x) = \int_{0}$ ˣ $\sqrt{t}/(1+t^{2})$ dt, where $\sqrt{t}$ denotes the positive square root for t>0. Which of the following statements are true?

A.f is a uniformly continuous function.✓ Correct
B.f is a bounded function.✓ Correct
C.There exists $x\in(0,\infty)$ such that f(x)=0.
D.The derivative of f is continuous.✓ Correct

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Q30 Boundary and endpoint Continuity, uniform continuity, Lipschitz

Consider the following real-valued functions $F_{1}$ and $F_{2}$ defined on $\mathbb{R}$ , given by $F_{1}(x)=1$ if x≥1, 0 otherwise, and $F_{2}(x)=\int_{0}$ ˣexp(−t)dt if x≥0, 0 otherwise. Define another function $F:\mathbb{R}\to[0,1]$ by $F(x)=(2/3)F_{1}(x)+(1/3)F_{2}(x)$ for all $x\in\mathbb{R}$ . Which of the following statements are true?

A.F is non-decreasing on $\mathbb{R}$ ✓ Correct
B. $\lim(x\to\infty) F(x) = 1$ ✓ Correct
C.F is left-continuous on $\mathbb{R}$
D.F is right-continuous on $\mathbb{R}$ ✓ Correct

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Q31 Execution slip CRD, RBD, LSD essentials

Consider the following $2^{5}-$ factorial design with 8 blocks (Block 1: (1), acd, bce, abde; Block 2: e, acde, bc, abd; and six further blocks). Which of the following statements are true?

A.ABC is confounded with blocks✓ Correct
B.BCD is confounded with blocks✓ Correct
C.CDE is confounded with blocks✓ Correct
D.ABDE is confounded with blocks✓ Correct

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Q32 Pointwise vs uniform Pointwise vs uniform convergence, M-test, Dini

Let $\mathbb{N}$ denote the set of all positive integers. For $n\in\mathbb{N}$ , let $f_{n}:[0,1]\to\mathbb{R}$ be given by $f_{n}(x)=n(1-$ nx) if 0≤x≤1/n, 0 if x>1/n. Which of the following statements are necessarily true?

A.The sequence of functions { $f_{n}$ } $_{n}\ge1$ is uniformly bounded.
B.The sequence of functions { $f_{n}$ } $_{n}\ge1$ does not converge pointwise.✓ Correct
C.The sequence of functions { $f_{n}$ } $_{n}\ge1$ converges uniformly to the constant function 0.
D.The set { $f_{n} : n\in\mathbb{N}$ } is compact in C[0,1], where C[0,1] denotes the space of all real valued continuous functions on [0,1] equipped with the supremum norm.

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Q33 Standard counterexample Liouville, Morera, maximum modulus principle

Let f be an entire function. Consider the function g given by g(z) = f(z) − 1/z for $z\in\mathbb{C}$ {0}. Which of the following statements are necessarily true?

A.The function g has a pole at 0.✓ Correct
B.If $g(\alpha)=0$ , then | $\alpha$ |≠1.
C.The function g has only finitely many zeros.
D. $max_{|z|=1}$ |g(z)| ≥ 1✓ Correct

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Q34 Standard counterexample L^p spaces essentials

Let $\ell^{2}$ denote the vector space of all square summable sequences { $a_{n}$ } $_{n}\ge1$ of real numbers with the inner product ⟨{ $a_{n}$ },{ $b_{n}$ }⟩ $= \sum_{n}$ ₌ $_{1}^\infty a_{n}b_{n}$ . Let H = {{ $a_{n}$ } $\in\ell^{2}$ : | $a_{n}$ |≤1/n for all positive integers n}. Which of the following statements are true?

A.H contains an orthonormal basis of $\ell^{2}$ .
B.H is a linear subspace of $\ell^{2}$ .
C.H is a bounded subset of $\ell^{2}$ .✓ Correct
D.H is a convex subset of $\ell^{2}$ .✓ Correct

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Q35 Moments and tails Standard discrete and continuous distributions

Let { $X_{n} : n\ge1$ } be a sequence of independent and identically distributed random variables, where $X_{1}$ has an Exponential distribution with mean 1. Define $T_{n} = \max$ { $X_{1},X_{2},\dots,X_{n}$ } − ln n, n≥1. Suppose $T_{n} \to$ ᵈ Y as $n\to\infty$ . Then which of the following statements are true?

A.Y has a Double Exponential distribution with location parameter ln(ln 2) and scale parameter 1
B.Median of Y = ln(ln 2)
C.P(Y≤0) = e⁻ $^{1}$ ✓ Correct
D.The derivative of the cumulative distribution function of Y at ln 3 is e^(−1/3)

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Q36 Converse assumed Riemann integration and criteria

Let f:[0,1]→[0,1] be a monotonically increasing function, that is, a≤b implies f(a)≤f(b). For any $\alpha\in(0,1)$ , let $L_\alpha$ ⁺= $lim_{x\to\alpha^{+}}f(x)$ and $L_\alpha$ ⁻= $lim_{x\to\alpha^{-}}f(x)$ denote the right hand and left hand limits respectively, provided they exist. For $\alpha\in(0,1)$ , if $L_\alpha$ ⁺ and $L_\alpha$ ⁻ exist, define $U_\alpha=(L_\alpha$ ⁻, $L_\alpha$ ⁺) if $L_\alpha$ ⁻ $<L_\alpha$ ⁺, and ∅ if $L_\alpha$ ⁺ $\le{}L_\alpha$ ⁻. Which of the following statements are true?

A. $L_\alpha$ ⁺ and $L_\alpha$ ⁻ exist for every $\alpha\in(0,1)$ .✓ Correct
B.If f is surjective, then f is continuous.✓ Correct
C.If f is Riemann integrable, then f is continuous.
D.If the left and right hand limits exist at $\alpha,\beta\in(0,1), \alpha\ne\beta$ , then $U_\alpha\cap{}U_\beta=\emptyset$ .✓ Correct

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Q37 Execution slip Markov chains: classification of states, stationary distributions

Suppose that the transition probability matrix of a homogeneous Markov chain with state space {1,2,3,4} is given by P, with row 1 = (1/4, 3/4, 0, 0), row 2 = (1, 0, 0, 0), row 3 = (1/8, 0, 7/8, 0), row 4 = (0, 0, 1/9, 8/9). Which of the following statements are true?

A.State 2 is a positive recurrent state✓ Correct
B.Mean recurrence time of state 1 is 7/4✓ Correct
C.State 4 is a transient state✓ Correct
D.State 3 is aperiodic and ergodic

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Q38 Dependence misread Random variables, distributions, moments, MGF

Let X and Y be two independent random variables such that the moment generating functions of X and Y are $M_X(t)=e^(3(e^t-1)), t\in\mathbb{R}$ , and $M_Y(t)=((1/2)e^(-3t)+(1/2)e^(3t))^{2}, t\in\mathbb{R}$ , respectively. Then which of the following statements are true?

A.P(XY=0) = (1+e⁻ $^{3})/2$ ✓ Correct
B.E(X+Y) = 3✓ Correct
C.Var(X+Y) = 21✓ Correct
D.Cov(X+Y,X−Y) = 0

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Q39 Invariants don't determine Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let A be a non-zero 3×3 matrix with integer entries. Let $\lambda_{i}\in\mathbb{C}, 1\le{}i\le3$ be all the eigenvalues of A (not necessarily distinct). Which of the following statements are necessarily true?

A.There exists a cubic polynomial $f(X)\in\mathbb{Q}[X]$ such that $f(\lambda_{i})=0$ for all 1≤i≤3.✓ Correct
B.There exists a quadratic polynomial $f(X)\in\mathbb{Q}[X]$ such that $f(\lambda_{i})=0$ for all 1≤i≤3.
C.If $f(X)\in\mathbb{Q}[X]$ is such that $f(\lambda_{i})=0$ for all 1≤i≤3, then f(A)=0.
D.If $f(X)\in\mathbb{Q}[X]$ is a cubic polynomial such that $f(\lambda_{i})=0$ for all 1≤i≤3, then f(A)=0.

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Q40 Execution slip Hamiltonian formalism and conservation laws

A mechanical system is described using generalized position q and generalized momentum p. Let Q and P denote new generalized position and generalized momentum variables respectively, generated by the generating function $F(q,P)=q^{2}e^P$ , and Q,P are canonical coordinates. Let G(p,Q) be a function such that G(2,e)=0, and it generates the same canonical coordinates Q,P. Then which of the following statements are true?

A. $G(p,Q) = -Q[1 + log_{e}(p^{2}/(4Q))]$ ✓ Correct
B.G(p,Q) = −pQ $[1 + log_{e}(Q^{2}/(4p))]$
C.p = 2qe $^P, Q = q^{2}e^P$ ✓ Correct
D.p = −2qe $^P, Q = -q^{2}e^P$

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Q41 Execution slip Conformal maps, Möbius transformations, Schwarz lemma

Let 𝔻={ $z\in\mathbb{C}$ :|z|<1} and f:𝔻→𝔻 be a holomorphic function which satisfies f(−1/2)=0. Which of the following statements are necessarily true?

A.|f(−1/5)| ≤ 1/5
B.|f(−1/5)| ≤ 1/3✓ Correct
C.|f′(−1/2)| ≤ 1/2
D.|f′(−1/2)| ≤ 4/3✓ Correct

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Q42 Execution slip Residue theorem and standard contour integrals

For $t\in[0,2\pi]$ , let $\gamma_{1}(t)=e^($ it) and $\gamma_{2}(t)=1+i+e^($ it). Which of the following statements are true?

A. $\int_\gamma1$ dz/(z sin z) = 0✓ Correct
B. $\int_\gamma1$ dz $/(z \sin z) = 2\pi{}i$
C. $\int_\gamma2$ dz $/(z \sin z) = 2\pi{}i$
D. $\int_\gamma2$ dz/(z sin z) = 0✓ Correct

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Q43 Boundary and endpoint Diagonalisability criteria

Let V be a finite-dimensional $\mathbb{R}-$ vector space and T:V→V a linear operator such that $T^{2}$ is diagonalizable over $\mathbb{R}$ . Which of the following statements are necessarily true?

A.If T is not diagonalizable over $\mathbb{R}$ , then $T^{2}$ has an eigenvalue ≤0.✓ Correct
B.If $T^{2}$ has only negative eigenvalues, then dim V is an even integer.✓ Correct
C.If $T^{2}$ has only non-negative eigenvalues, then T is diagonalizable over $\mathbb{R}$ .
D.For each non-zero v∈V, {v,Tv, $T^{2}v$ } is linearly dependent.

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Q44 What the inference means Neyman–Pearson lemma and UMP tests

Let $X_{1},X_{2},\dots,X_{7}$ be a random sample drawn from a continuous distribution with unknown unique median M. The null hypothesis $H_{0}:M=2$ is tested against the alternative $H_{1}:M>2$ at level of significance 0.05 using the right-tailed test based on the Sign test statistic K, which is the number of observations in the sample greater than 2. If the observed sample is −3,−6,1,9,4,10,12, which of the following statements are true?

A. $H_{0}$ is rejected
B.The p-value of the test is greater than 0.01✓ Correct
C.Under $H_{0}, E(K) = 3.5$ ✓ Correct
D.If $X_{(i)}$ denotes the i-th smallest observation of the sample, then $[X_{(2)},X_{(6)}]$ is a confidence interval for M with confidence coefficient at least 0.95

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Q45 Execution slip Linear ODE, Wronskian, variation of parameters, systems

Suppose $y_{1}(x)$ and $y_{2}(x)$ are two linearly independent solutions of the differential equation $x^{2}y'' + [(1+\sin x)x/2]y' - (3/2)(\cos x)y = 0, x>0$ , satisfying $y_{2}(0)=0$ . Then which of the following statements are true?

A. $\lim(x\to0+) y_{2}(x)/(x^{2}y_{1}(x))$ exists.✓ Correct
B. $\lim(x\to0+) y_{2}(x)/($ xy $_{1}(x))$ exists.✓ Correct
C.lim(x→0+) xy $_{1}(x)/y_{2}(x)$ does NOT exist.✓ Correct
D. $\lim(x\to0+) x^{2}y_{1}(x)/y_{2}(x)$ exists.

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Q46 Dependence misread Axioms, conditional probability, independence, Bayes

Suppose two fair dice are thrown independently at random. Let X and Y be the numbers on the upper face of the first die and that of the second die, respectively. Then which of the following statements are true?

A.P(X−Y=0 ∣ X+Y=2) = P(X−Y=0 ∣ X+Y=12)✓ Correct
B.E((X−Y)/(X+Y)) = 0✓ Correct
C.Cov(X+Y, X−Y) = 0✓ Correct
D.(X+Y) and (X−Y) are independent

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Q47 Invariants don't determine Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let V be a 4-dimensional complex vector space and A a linear operator on V. Which of the following statements are necessarily true?

A.There exist $\lambda\in\mathbb{C}$ and a non-zero v∈V such that Av $=\lambda{}v$ .✓ Correct
B.There exist $\lambda,\mu\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av $=\lambda{}v$ and Aw $=\mu{}w$ .
C.There exist $\lambda,\mu,\delta\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av $=\lambda{}v$ and Aw $=\mu{}v+\delta{}w$ .✓ Correct
D.There exists a three-dimensional subspace W such that Aw∈W for all w∈W.✓ Correct

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Q48 Standard counterexample Galois theory essentials

Let p≥3 be a prime number and $f(x)\in\mathbb{Q}[x]$ an irreducible polynomial of degree p. Suppose that $a_{1},\dots,a_{p}\in\mathbb{C}$ are the roots of f and that $a_{1}\notin\mathbb{R}, a_{2}\notin\mathbb{R}$ , and $a_{i}\in\mathbb{R}$ for all 3≤i≤p. Let $K=\mathbb{Q}(a_{1},\dots,a_{p})$ be the subfield of $\mathbb{C}$ generated by the roots of f. Consider the Galois group G of K over $\mathbb{Q}$ as a subgroup of $S_{p}$ , the group of permutations of { $a_{1},\dots,a_{p}$ }. Which of the following statements are true?

A.The transposition (1 2) belongs to G.✓ Correct
B.|G| is divisible by p.✓ Correct
C.A p-cycle belongs to G.✓ Correct
D. $G = S_{p}$ ✓ Correct

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Q49 Execution slip Linear transformations, matrix representation, change of basis

Let V be a real vector space and L(V) denote the space of linear operators on V. Let T∈L(V) be a non-zero operator such that $T^{2}=T$ . Consider the subspace W of L(V) spanned by { $I, T^{n} : n$ is a positive integer}. Which of the following statements are necessarily true?

A.The set { $S\in{}W : S^{2}=S$ } contains exactly 2 elements.
B. $dim_\mathbb{R}(W) = 2$ .
C.If U∈L(V) is such that $U^{2}=U$ and $(T+U)^{2}=T+U$ , then TU=0.✓ Correct
D.If U∈L(V) is such that $U^{2}=U$ and $(T-U)^{2}=T-U$ , then (TU $)^{2}=$ TU.✓ Correct

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Q50 Property not inherited Normal subgroups, quotients, isomorphism theorems

Let G be a finite non-abelian group. Which of the following statements are necessarily true?

A.If d is a positive integer that divides |G|, then G has a subgroup of order d.
B.The map f : G×G → G given by f(a,b)=ab is not a group homomorphism.✓ Correct
C.Suppose that for every positive integer d that divides |G|, there exists a subgroup of G of order d. Then G has at least three normal subgroups.✓ Correct
D.|G| ≠ 16

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Q51 Execution slip Standard discrete and continuous distributions

Consider a series system comprising four components, having independent and identically distributed lifetimes with hazard rate $\lambda(t)=1/(1+t), t>0$ , and survival function S(t), t>0. If Y denotes the lifetime of the series system, then which of the following statements are true?

A.Cumulative hazard function of each component is H(t)=2ln(1+t), t>0
B. $S(t)=1/(1+t)^{2}, t>0$
C.P(Y<1/2)={S(1/2)} $^{4}$
D.P(Y<1/2)=65/81✓ Correct

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Q52 Limit assumed to exist Partial derivatives, differentiability, chain rule

Let a,b be distinct positive real numbers. Consider the function $f:\mathbb{R}^{2}\to\mathbb{R}$ given by f(x,y)=(ax+by $)^{2}/($ ax $^{2}+$ by $^{2})$ if (x,y)≠(0,0), and 0 if (x,y)=(0,0). Which of the following statements are necessarily true?

A.lim(x,y)→(0,0) f(x,y) does not exist.✓ Correct
B.The partial derivatives of f at (0,0) do not exist.✓ Correct
C.lim(x→0) f(x,0) = lim(y→0) f(0,y).
D.f is differentiable at (0,0).

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Q53 Execution slip Sufficiency, completeness, UMVUE, Cramér–Rao

Let $X_{1},X_{2},\dots,X_{n} (n>5)$ be independent random variables such that $X_{t}=\alpha+\alpha^{2}t+\varepsilon_{t}$ , for t=1,…,n, where $\varepsilon_{1},\varepsilon_{2},\dots,\varepsilon_{n}$ are independent and identically distributed $N(0,\sigma^{2})$ random variables. Here $\alpha\in\mathbb{R}$ and $\sigma>0$ are unknown parameters. Which of the following statements are true?

A. $(X_{1},X_{2},\dots,X_{n})$ is a sufficient statistic for $(\alpha,\sigma)$ ✓ Correct
B. $(\sum{}t X_{t}, \sum{}t$ tX $_{t}, \sum{}t t^{2}X_{t}^{2})$ is a jointly minimal sufficient statistic for $(\alpha,\sigma)$
C. $X_{4}-X_{3}-X_{2}+X_{1}$ is an ancillary statistic
D. $(X_{4}+X_{1}-X_{2}-X_{3})/(X_{5}+X_{2}-X_{3}-X_{4})$ is an ancillary statistic✓ Correct

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Q54 Boundary and endpoint Euler–Lagrange equation and standard functionals

Suppose y(x) is the extremal of the variational problem $J(y) = \int_{0}^{1} ((y')^{2}\sin x + (2cos x)y)$ dx subject to y(0)=0, y(1)=1. Then which of the following statements are true?

A.y(1/2) = 1
B.y′(0) = 1✓ Correct
C.y(1/4) = 2
D.y′(1/2) = 1✓ Correct

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Q55 Hypothesis dropped Gauss–Markov, regression, ANOVA basics

Let $Y=X\beta+\varepsilon$ be a multiple linear regression model with p regressors and an intercept, where $\beta=(\beta_{0},\beta_{1},\dots,\beta_{p})^{T}$ and the random error $\varepsilon$ ~ $N_{n}(0,\sigma^{2}I), \sigma>0$ and n>p+1. The least squares method provides a unique estimator $\beta$ ̂. Let the total sum of squares (corrected), sum of squares due to regression, and sum of squares due to error, based on $\beta$ ̂, be denoted YᵀAY, YᵀBY and YᵀCY respectively, so YᵀAY=YᵀBY+YᵀCY. Which of the following statements are always true?

A.YᵀAY $/\sigma^{2}$ follows a central $\chi^{2}$ distribution with (n−1) degrees of freedom.
B.YᵀBY $/\sigma^{2}$ follows a central $\chi^{2}$ distribution with p degrees of freedom if $\beta_{1}=\beta_{2}=\cdots=\beta_{p}=0$ .✓ Correct
C.YᵀBY/YᵀCY follows a central F distribution with (p,n−p) degrees of freedom.
D.YᵀBY and YᵀCY are independently distributed if and only if $\beta_{1}=\beta_{2}=\cdots=\beta_{p}=0$ .

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Q56 Existence vs uniqueness Fredholm and Volterra equations

Let $\lambda\in\mathbb{R}$ be such that the integral equation $y(x) = \lambda\int$ ₋ $_{1}^{1}(5$ xt $^{3}+4x^{2}t+3$ xt)y(t)dt admits a non-trivial solution y(x) such that y(1)=5/2. Then which of the following statements are true?

A.y(0)+y′(0) = 3/2✓ Correct
B.y(1/2)+y′(1/2) = 7/2✓ Correct
C.y(−1)+y′(−1) = −1✓ Correct
D.y(1/3)+y′(1/3) = 14/9

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Q57 Numerical convergence Root finding: bisection, Newton–Raphson, fixed point, order of convergence

Let { $x_{n}$ } be a convergent iterative sequence generated by Newton-Raphson method for solving the equation sin x−1=0 such that $x_{n}\to\pi/2$ as $n\to\infty$ . For $n\in\mathbb{N}$ , let $e_{n}=x_{n}-\pi/2$ . Let p>0 be such that $\lim(n\to\infty)$ | $e_{n}$ ₊ $_{1}$ |/| $e_{n}$ |^p exists and is non-zero. Then which of the following statements are true?

A.p = 1✓ Correct
B. $\lim(n\to\infty)$ | $e_{n}$ ₊ $_{1}$ |/| $e_{n}$ |^p = 1/2✓ Correct
C.p = 2
D. $\lim(n\to\infty)$ | $e_{n}$ ₊ $_{1}$ |/| $e_{n}$ |^p = 1

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Q58 Boundary and endpoint Laplace, heat and wave equations: separation of variables

Consider the boundary value problem (BVP $) u_{xx}+u_yy=0$ in $\Omega=$ { $(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1$ }, u(x,y)=e^(x+y) on $\partial\Omega=$ { $(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=1$ }. Then which of the following statements are true?

A.There exists a unique solution to BVP.✓ Correct
B.The BVP does NOT have a solution.
C.There exists a solution u to BVP such that u(x,y)=(1+e)/2 for some $(x,y)\in\Omega\cup\partial\Omega$ .✓ Correct
D.There exists a solution u to BVP such that $u(x,y)=(1+e^{3})/2$ for some $(x,y)\in\Omega\cup\partial\Omega$ .

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Q59 What the inference means Gauss–Markov, regression, ANOVA basics

Consider the simple linear regression model $Y_{i}=\beta{}x_{i}+\varepsilon_{i}, i=1,2,\dots,n$ , where $\beta>0, \sum{}x_{i}^{2}>0$ , and the uncorrelated errors $\varepsilon_{i}$ have zero mean and finite variance $\sigma^{2}(>0)$ . Let $\beta$ ̃ $_{1}=\sum{}a_{i}$ $Y_{i}$ , where $a_{i}$ 's minimize $E(\sum{}a_{i} Y_{i}-\beta)^{2}$ with respect to scalars $a_{1},a_{2},\dots,a_{n}$ . Let $\beta$ ̃ $_{2}$ be the ordinary least squares estimator of $\beta$ . Which of the following statements are true?

A. $Var(\beta$ ̃ $_{1}) > Var(\beta$ ̃ $_{2}), E(\beta$ ̃ $_{1}-\beta)^{2} > E(\beta$ ̃ $_{2}-\beta)^{2}$
B. $Var(\beta$ ̃ $_{1}) < Var(\beta$ ̃ $_{2}), E(\beta$ ̃ $_{1}-\beta)^{2} < E(\beta$ ̃ $_{2}-\beta)^{2}$ ✓ Correct
C. $Var(\beta$ ̃ $_{1}) > Var(\beta$ ̃ $_{2}), E(\beta$ ̃ $_{1}) < E(\beta$ ̃ $_{2})$
D. $Var(\beta$ ̃ $_{1}) < Var(\beta$ ̃ $_{2}), E(\beta$ ̃ $_{1}) > E(\beta$ ̃ $_{2})$

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Q60 Standard counterexample Liouville, Morera, maximum modulus principle

Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=\sin^{2}z+\cos^{2}$ |z|. Which of the following statements are true?

A.f is a real valued function.
B.f(z) = 1 for all $z\in\mathbb{C}$ .
C.f is not an entire function.✓ Correct
D.f has finitely many zeros on the imaginary axis.✓ Correct

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