CSIR NET June 2025 Maths — Part C, all 60 questions

Q1 Limit 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.✓ Correct
B. $lim_{x\to\infty}$ f′(x) = 0.✓ Correct
C. $lim_{x\to\infty}$ x f′(x) need not exist.
D. $lim_{x\to\infty}$ x f′(x) = 0.✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q2 Pointwise vs uniform Pointwise vs uniform convergence, M-test, Dini

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q3 Pointwise vs uniform Pointwise vs uniform convergence, M-test, Dini

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q4 Standard 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.✓ Correct
B.If 0 is an eigenvalue of A, then 0 is an eigenvalue of AB.✓ Correct
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.

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q5 Standard counterexample Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q6 Execution slip Bases, dimension, rank–nullity

Let V be the $\mathbb{R}-$ 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}✓ Correct
D.{f(x + 1), f(x − 1), f(x)}✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q7 Standard 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.✓ Correct
B.The number of surjective linear transformations from V to W is 624.✓ Correct
C.The number of one dimensional subspaces of V is 13.✓ Correct
D.The number of linear transformations from V to W is $3^{6}$ .✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q8 Hypothesis 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.✓ Correct
B.Let v ∈ V be such that T*T(v) = 0. Then T(v) = 0.✓ Correct
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.✓ Correct
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.✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q9 Boundary and endpoint Quadratic forms, positive definiteness, Sylvester's law

Let B(v, w) be a nondegenerate symmetric bilinear form on $\mathbb{R}^{2}$ and let q(v) = B(v, v) be the corresponding quadratic form. Suppose there exist vectors $v, w \in \mathbb{R}^{2}$ 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 $\alpha \in \mathbb{R}$ such that $q(\alpha{}v + w) = 0$ .✓ Correct
C.There are infinitely many $\alpha \in \mathbb{R}$ such that $q(\alpha{}v + w) = 0$ .
D.q is equivalent to the quadratic form $Q(x, y) = x^{2} - y^{2}$ for all $(x, y) \in \mathbb{R}^{2}$ .✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q10 Standard 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 $z \in \mathbb{C}$ then f is a constant function.
B.If f(z) = f(z + 1) = f(z + i) for all $z \in \mathbb{C}$ then f is a constant function.✓ Correct
C.If f(1/z) has a removable singularity at 0 then f is a constant function.✓ Correct
D.If f is a non-constant function then f(1/z) has a pole at 0.

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q11 Standard counterexample Conformal maps, Möbius transformations, Schwarz lemma

Let 𝔻^× = { $z \in \mathbb{C} : 0 <$ |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. $lim_{z\to0}$ f(z) does not exist.
B. $lim_{z\to0}$ f(z) exists and has absolute value ≤ 1.✓ Correct
C. $lim_{z\to0}$ f(z) = 0.✓ Correct
D.There exists $\theta \in \mathbb{R}$ such that f(z) = $e^{i\theta}z$ for all z ∈ 𝔻^×.✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q12 Standard counterexample Power series and analyticity

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q13 Standard 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 $\varphi_{1}(x), \varphi_{2}(x)$ be solutions of the ODE, satisfying $\varphi_{1}(0) = 1, d\varphi_{1}/$ dx(0) = 0, and $\varphi_{2}(0) = 0, d\varphi_{2}/$ dx(0) = 1. Then which of the following statements are true?

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q14 Standard counterexample Linear ODE, Wronskian, variation of parameters, systems

For a continuous function q defined on $\mathbb{R}$ , consider the ordinary differential equation (ODE $) d^{2}y/$ dx $^{2} + q(x)y = 0, x \in \mathbb{R}$ . 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✓ Correct
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✓ Correct
D.There exists a q such that xe^x and x(x − 1)e^x are solutions of ODE✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q15 Hypothesis dropped First-order PDE: Lagrange, Charpit, characteristics

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

A.There is NO neighbourhood of the origin on which (CP) has a solution✓ Correct
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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q16 Execution 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 { $(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} < 1$ }, $u(x, y) = 1 + 2x^{2}y^{2}$ on { $(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} = 1$ }. Then which of the following statements are true?

A.The minimum value of u is 1✓ Correct
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✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q17 Execution slip Interpolation and numerical integration with error terms

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

A. $(\alpha, \beta) = ((3 - \sqrt{3})/2, \sqrt{3})$ or $(\alpha, \beta) = ((3 + \sqrt{3})/2, -\sqrt{3})$ ✓ Correct
B. $(\alpha, \beta) = ((3 - \sqrt{2})/2, \sqrt{2})$ or $(\alpha, \beta) = ((3 + \sqrt{2})/2, -\sqrt{2})$
C. $(\alpha, \beta) = ((3 - \sqrt{5})/2, \sqrt{5})$ or $(\alpha, \beta) = ((3 + \sqrt{5})/2, -\sqrt{5})$
D. $(\alpha, \beta) = ((3 - \sqrt{7})/2, \sqrt{7})$ or $(\alpha, \beta) = ((3 + \sqrt{7})/2, -\sqrt{7})$

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q18 Converse assumed Euler–Lagrange equation and standard functionals

For $\alpha \ge 0$ , consider the functional $J_\alpha[y] = \int_{1}^{2} (y')^{2}/x^\alpha$ 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 $J_{3}$ ✓ Correct
B. $y(x) = (1/3)(x^{2} + 2)$ is an extremal for $J_{1}$ ✓ Correct
C.y(x) = x is an extremal for $J_{0}$ ✓ Correct
D. $y(x) = (1/2)(x^{2} - x + 2)$ is an extremal for $J_{1}$

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q19 Boundary and endpoint Euler–Lagrange equation and standard functionals

For $a, b, c \in \mathbb{R}$ , 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✓ Correct
B.If (a, b, c) = (1, 0, 2), then every admissible extremal is a minimizer✓ Correct
C.If (a, b, c) = (2, −1, 1), then every admissible extremal is a minimizer✓ Correct
D.If (a, b, c) = (1, −2, 5), then every admissible extremal is a minimizer✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q20 Boundary 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. $E(X^{2}) = 4/3$ ✓ Correct
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✓ Correct
D.The value of the moment generating function, M_X(t), at t = 1 is $(e^{2} - 1)/5$

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q21 Execution slip Axioms, conditional probability, independence, Bayes

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q22 Execution slip Markov chains: classification of states, stationary distributions

Consider a Markov chain { $X_{n} : n \ge 1$ } 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).✓ Correct
B.State 5 is absorbing and recurrent.✓ Correct
C.All states are aperiodic.✓ Correct
D. $lim_{n}\to\infty p_{55}$ ⁽ $^{n}$ ⁾ = 1.✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q23 Execution 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. $P_{1} = 2/9$ ✓ Correct
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✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q24 Execution 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✓ Correct
C.E(X | X + Y = 3) = 2✓ Correct
D.The value of the characteristic function of X + Y at the point $t = \pi$ is $e^{-12}$ ✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q25 Execution 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) = $e^{-2t}$ + $e^{-t}$ − $e^{-3t}$ , t > 0✓ Correct
B.The expected lifetime of the given system is 1
C. $h(1) = (2e + e^{2} - 3)/(e + e^{2} - 1)$ ✓ Correct
D. $h(3) = (4e^{3} + e^{6} - 5)/(e^{3} + e^{6} - 1)$

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q26 Standard 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.✓ Correct
B.The probability density function of T is f_T(t) = 1 if 0 < t < 1, and 0 otherwise.✓ Correct
C.Var(S) = 2.
D.E(T) = 2/3.

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q27 Boundary 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| $\theta) =$ $e^{-(x-\theta)}$ if $x \ge \theta$ , and 0 otherwise, where $\theta \in \mathbb{R}$ is the unknown parameter. Further, let $Y = \min(X_{1}, X_{2}, \dots, X_{n})$ . Which of the following are confidence intervals for $\theta$ with the confidence coefficient $(1 - \alpha)$ , where $\alpha \in (0, 1)$ ?

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q28 Dependence 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✓ Correct
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✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q29 Execution slip Gauss–Markov, regression, ANOVA basics

An analyst fits a multiple linear regression model $Y_{i} = \beta_{0} + \beta_{1}X_{1i} + \dots + \beta_{4}X_{4i} + \varepsilon_{i}, i = 1, 2, \dots, n$ , 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., $H_{0} : L^{T}\beta = 0$ against $H_A : H_{0}$ is not true, where $\beta^{T} = (\beta_{0}, \beta_{1}, \dots, \beta_{4})$ . Suppose we have n observations on the response Y and each regressor. Further, assume that the errors with or without restrictions are independent $N(0, \sigma^{2})$ 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)✓ Correct
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 $L^{T}\beta = 0$ follows $\sigma^{2}\chi^{2}_{n-3}$ distribution✓ Correct
D.Sum of squares residuals without the restrictions $L^{T}\beta = 0$ follows a non-central $\sigma^{2}\chi^{2}_{n-4}$ distribution

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q30 Boundary 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 $(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho)$ with $\mu_{1} = 5, \mu_{2} = 6, \sigma_{1}^{2} = 4, \sigma_{2}^{2} = 9$ and $\rho = 1/2$ . Then, which of the following statements are true?

A.The distribution of $(1/\sqrt{7})\sum_{i}$ ₌ $_{1}^{10}(X_{i} - Y_{i} + 1)$ is N(0, 10)✓ Correct
B.The distribution of $(1/19)\sum_{i}$ ₌ $_{1}^{10}(X_{i} + Y_{i} - 11)^{2}$ is $\chi^{2}-$ distribution with degrees of freedom 10✓ Correct
C.The distribution of $2\sqrt{2}(X_{1} - 5)/\surd(\sum_{i}$ ₌ $_{3}^{10}(X_{i} - 5)^{2})$ 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.✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q31 Boundary and endpoint Multivariate normal distribution

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q32 Standard counterexample limsup, liminf and subsequential limits

Consider the sequences $(s_{n})_{n}\ge_{1}$ and $(t_{n})_{n}\ge_{1}$ 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. $limsup_{n\to\infty}$ $t_{n} \le$ $limsup_{n\to\infty}$ $s_{n}$ ✓ Correct
B. $limsup_{n\to\infty}$ $t_{n} \le e$ ✓ Correct
C. $liminf_{n\to\infty}$ $s_{n} \ge e^{2}$
D. $liminf_{n\to\infty}$ $t_{n} \ge e$

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q33 Execution slip Riemann integration and criteria

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

A.2/e
B.4/e✓ Correct
C. $log_{e}2 - 1$
D. $2 log_{e}2 - 1$

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q34 Standard 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 $\sqrt{f^{2} + g^{2}}$ is Riemann integrable.✓ Correct
D.The composition f ∘ g is Riemann integrable, if both f and g are continuous.✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q35 Boundary and endpoint Lebesgue integral, MCT, DCT, Fatou

Let $\mu$ denote the Lebesgue measure on $\mathbb{R}$ . Suppose that f is a non-negative Lebesgue measurable function on $\mathbb{R}$ . Let $0 = a_{0} < a_{1} < a_{2} < \cdots$ be an unbounded sequence such that $a_{n+1}$ ≤ ca $_{n}$ for some real number c and for all n ≥ 1. Let A_k = { $x \in \mathbb{R}$ | 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 $\mathbb{R}$ , then $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite.✓ Correct
B.If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite, then f is Lebesgue integrable on $\mathbb{R}$ .
C.If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite, and $f(x) \ge a_{1}$ for all $x \in \mathbb{R}$ , then f is Lebesgue integrable on $\mathbb{R}$ .✓ Correct
D.If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite and f is bounded, then f is Lebesgue integrable on $\mathbb{R}$ .

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q36 Execution slip Partial derivatives, differentiability, chain rule

Let $f : \mathbb{R}^{2} \to \mathbb{R}$ 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 $(x_{1}, x_{2}) \in \mathbb{R}^{2}$ . 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. $\partial{}f/\partial{}x_{1}(1, -1) - \partial{}f/\partial{}x_{2}(1, -1) = 3f(1, -1)$ ✓ Correct
C. $x_{1}^{2}\partial^{2}f/\partial{}x_{1}^{2}(x_{1}, x_{2}) + x_{2}^{2}\partial^{2}f/\partial{}x_{2}^{2}(x_{1}, x_{2}) + 2x_{1}x_{2} \partial^{2}f/\partial{}x_{1}\partial{}x_{2}(x_{1}, x_{2}) = 6f(x_{1}, x_{2})$ ✓ Correct
D. $x_{1}^{2}\partial^{2}f/\partial{}x_{1}^{2}(x_{1}, x_{2}) + x_{2}^{2}\partial^{2}f/\partial{}x_{2}^{2}(x_{1}, x_{2}) + 2x_{1}x_{2} \partial^{2}f/\partial{}x_{1}\partial{}x_{2}(x_{1}, x_{2}) = 9f(x_{1}, x_{2})$

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q37 Hypothesis dropped Linear transformations, matrix representation, change of basis

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q38 Boundary and endpoint Completeness and Baire category

Let C[0, 1] be the $\mathbb{R}-$ vector space of real valued continuous functions equipped with the norm ‖f‖ = $sup_{x\in[0,1]}$ |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 $\alpha \in (0, 1)$ such that for all f, g ∈ C[0, 1], ‖T(f) − T(g)‖ $\le \alpha$ ‖f − g‖.
B.There exists $\alpha \in (0, 1)$ such that for all f, g ∈ C[0, 1], ‖ $T^{2}(f) - T^{2}(g)$ ‖ $\le \alpha$ ‖f − g‖.✓ Correct
C.The set {f ∈ C[0, 1] : T(f) = f} is a singleton set.✓ Correct
D.‖ $T^{n}$ ‖ $\to \infty$ as $n \to \infty$ .

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q39 Invariants don't determine Jordan canonical form

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q40 Invariants don't determine Jordan canonical form

Which of the following matrices are similar over $\mathbb{R}$ 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)✓ Correct
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)✓ Correct
D.The matrix with rows (0, 1, 0, 0), (1, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0)

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q41 Execution slip Residue theorem and standard contour integrals

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q42 Standard 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 $\pi : G \to S_T$ be the map defined by $\pi(g)(g_{1}H) =$ gg $_{1}H$ . For a prime number p, let 𝔽_p denote the field with p elements. In which of the following cases is $\ker \pi$ trivial?

A.G = GL $_{2}($ 𝔽_p) and H is a subgroup of order p.✓ Correct
B.G = SL $_{2}($ 𝔽_p) and H is a subgroup of order p.✓ Correct
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.

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q43 Property 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.✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q44 Property 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 $(G_{2}G_{1})/G_{1} \cong G_{2}/(G_{2} \cap G_{1})$ .✓ Correct
B.If $H_{1}$ and $H_{2}$ are normal subgroups of $G_{1}$ and $G_{2}$ , respectively, then $(G_{1} \times G_{2})/(H_{1} \times H_{2}) \cong (G_{1}/H_{1}) \times (G_{2}/H_{2})$ .✓ Correct
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.

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q45 Finite-dimensional intuition Polynomial rings and irreducibility tests

Let $f : \mathbb{Q}[X] \to \mathbb{Q}[X]$ 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.✓ Correct
B.ker $f^{n+1}$ $= \ker f^{n}$ for some n ≥ 1.✓ Correct
C.If f is onto, then f is one-to-one.✓ Correct
D.If f is one-to-one, then f is onto.

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q46 Standard counterexample Polynomial rings and irreducibility tests

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q47 Hypothesis 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 $\alpha$ be a root of f in 𝔽̄_p. Then 𝔽 $_p(\alpha)$ is the splitting field of f in 𝔽̄_p.
B.Let f, g ∈ 𝔽_p[X] be irreducible polynomials of same degree and $\alpha$ be a root of f in 𝔽̄_p. Then 𝔽 $_p(\alpha)$ is the splitting field of g in 𝔽̄_p.✓ Correct
C.𝔽_p[X] has infinitely many irreducible polynomials.✓ Correct
D.The set {a + b | a, b ∈ 𝔽_p} is contained in { $a^{2} + b^{2}$ | a, b ∈ 𝔽_p}.✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q48 Converse assumed Compactness: open covers, sequential, Heine–Borel

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q49 Standard counterexample Open/closed sets, limit points, closure, interior

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q50 Existence vs uniqueness Existence–uniqueness, Picard, Lipschitz

Let D = { $(x, y) \in \mathbb{R}^{2} : -1 \le x \le 1, -1 \le y \le 1$ }, and $f : D \to \mathbb{R}$ be the function defined by $f(x, y) = 1 + \surd(y$ ₊), 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✓ Correct
C.IVP has at least one solution✓ Correct
D.IVP has NO solution

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q51 Numerical 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 $n \to \infty$ ✓ Correct
C. $0 \le y_{n} \le 1$ for n = 0, 1, 2, …✓ Correct
D.|y(nh $) - y_{n}$ | → 0 as $n \to \infty$ ✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q52 Execution 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✓ Correct
B. $u(2\pi/\sqrt{3}) =$ $(1/6)(e^{4\pi/\sqrt{3}}$ − $e^{-2\pi/\sqrt{3}})$ ✓ Correct
C. $u(\pi/(2\sqrt{3})) =$ $(1/6)(e^{\pi/\sqrt{3}}$ $- \sqrt{3}$ $e^{-\pi/(2\sqrt{3})})$ ✓ Correct
D. $u(\pi/(2\sqrt{3})) =$ $(1/6)(e^{\pi/\sqrt{3}}$ $+ \sqrt{3}$ $e^{-\pi/\sqrt{3}})$

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q53 Execution 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. $f'(\pi) = 3$ ✓ Correct
B. $f(\pi) + f'(\pi) = 4 - \pi$ ✓ Correct
C. $f(\pi) + f'(\pi) = 2 - \pi$
D.f(0) + f′(0) = −4✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q54 Standard 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. $\partial/\partial{}t [f, g] = [\partial{}f/\partial{}t, g] + [f, \partial{}g/\partial{}t]$ ✓ Correct
B.If f is a constant of motion, and f is independent of t, then [H, f] is a constant of motion✓ Correct
C.[[H, f], g] + [[g, H], f] + [[f, g], H] = 0✓ Correct
D.If f and g are constants of motion, then [f, g] is a constant of motion✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q55 Moments 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. $S_{n}/\sqrt{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)✓ Correct
B. $(T_{n} - n)/\sqrt{3n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)
C. $\sqrt{n} S_{n}/T_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)✓ Correct
D. $(T_{n} - n)/\sqrt{V}_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q56 Execution slip Sufficiency, completeness, UMVUE, Cramér–Rao

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q57 Execution slip Sufficiency, completeness, UMVUE, Cramér–Rao

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 2)$ be a random sample from a continuous distribution with the probability density function f(x| $\theta) =$ $(1/2\theta)e^{-|x-\theta|/\theta}$ , $x \in \mathbb{R}$ , where $\theta (> 0)$ 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 $\theta$ ✓ Correct
B. $S_{n}^{2}$ is an unbiased estimator of $\theta^{2}$
C. $(1/3)V_{n}$ is a consistent estimator of $\theta^{2}$ ✓ Correct
D.The statistic $(U_{n}, V_{n})$ is complete

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q58 What the inference means Likelihood ratio and standard tests

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

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

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q59 What 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 $F(x - \mu)$ and $F(x - \theta)$ , respectively, where $F, \mu$ and $\theta$ are all unknown. Further, let $\mu$ be the unique median of $F(x - \mu)$ and $\theta$ be the unique median of $F(x - \theta)$ . Let $R_{i}$ be the rank of $Y_{i}$ in the combined sample, i = 1, 2, …, 9. For testing $H_{0} : \mu = \theta$ against $H_{1} : \mu > \theta$ , 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 $H_{0}, P(R_{1} = 1, R_{9} = 16) = 1/240$ ✓ Correct

Trap Analysis has the working, and why each of the other options was written to tempt you.

Q60 Invariants 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✓ Correct
B.The design is connected✓ Correct
C.The design is balanced✓ Correct
D.The design is orthogonal

Trap Analysis has the working, and why each of the other options was written to tempt you.

CSIR NET June 2025 — Part C

Part C