CSIR NET December 2023 Maths — Part C, all 59 questions

Q1 Limit assumed to exist Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

Let x be a real number. Which of the following statements are true?

A.There exists an integer n ≥ 1 such that $n^{2} \sin(1/n) \ge x$ .✓ Correct
B.There exists an integer n ≥ 1 such that n cos(1/n) ≥ x.✓ Correct
C.There exists an integer n ≥ 1 such that n $e^{-n}$ ≥ x.
D.There exists an integer n ≥ 2 such that n (log $n)^{-1}$ ≥ x.✓ Correct

Solution

$n^{2} \sin(1/n)$ ~ n, n cos(1/n) ~ n and n/log n all tend to $+\infty$ , so each eventually exceeds any x. n $e^{-n}$ ≤ 1/e is bounded, so it fails for x > 1/e.

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Q2 Converse assumed Continuity, uniform continuity, Lipschitz

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function such that |f(x) − f(y)| ≥ log(1 + |x − y|) for all $x, y \in \mathbb{R}$ . Which of the following statements are true?

A.f is necessarily one-one.✓ Correct
B.f need not be one-one.
C.f is necessarily onto.✓ Correct
D.f need not be onto.

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Q3 Pointwise vs uniform Continuity, uniform continuity, Lipschitz

Let $f : [0, \infty) \to \mathbb{R}$ be the periodic function of period 1 given by f(x) = 1 − |2x − 1| for x ∈ [0, 1], and define $g : [0, \infty) \to \mathbb{R}$ by $g(x) = f(x^{2})$ . Which of the following statements are true?

A.f is continuous on $[0, \infty)$ .✓ Correct
B.f is uniformly continuous on $[0, \infty)$ .✓ Correct
C.g is continuous on $[0, \infty)$ .✓ Correct
D.g is uniformly continuous on $[0, \infty)$ .

Solution

f is a continuous tent function with f(0) = f(1) = 0, so its periodic extension is continuous; continuous periodic ⇒ uniformly continuous. g = f∘ $x^{2}$ is continuous but its oscillations speed up as $x \to \infty ($ the tents get compressed), so it is not uniformly continuous.

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Q4 Boundary and endpoint Riemann integration and criteria

For a real number $\lambda$ , consider the improper integrals $I_\lambda = \int_{0}^{1}$ dx $/(1 - x)^\lambda$ and $K_\lambda = \int_{1}^\infty$ dx $/x^\lambda$ . Which of the following statements are true?

A.There exists $\lambda$ such that $I_\lambda$ converges, but $K_\lambda$ does not converge.✓ Correct
B.There exists $\lambda$ such that $K_\lambda$ converges, but $I_\lambda$ does not converge.✓ Correct
C.There exists $\lambda$ such that $I_\lambda, K_\lambda$ both converge.
D.There exists $\lambda$ such that neither $I_\lambda$ nor $K_\lambda$ converges.✓ Correct

Solution

$I_\lambda$ converges iff $\lambda < 1; K_\lambda$ converges iff $\lambda > 1$ .

So they never converge together, and at $\lambda = 1$ neither does.

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Q5 Hypothesis dropped Riemann integration and criteria

Suppose that $f : [-1, 1] \to \mathbb{R}$ is continuous. Which of the following imply that f is identically zero on [−1, 1]?

A. $\int$ ₋ $_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0.✓ Correct
B. $\int$ ₋ $_{1}^{1} f(x)p(x)$ dx = 0 for all real polynomials p(x).✓ Correct
C. $\int$ ₋ $_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0 odd.
D. $\int$ ₋ $_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0 even.

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Q6 Boundary and endpoint Pointwise vs uniform convergence, M-test, Dini

Let $(f_{n})$ be the sequence of functions on [0, 1] defined by $f_{n}(x) = x^{n} \log((1 + \sqrt{x})/2)$ . Which of the following statements are true?

A. $(f_{n})$ converges pointwise on [0, 1].✓ Correct
B. $(f_{n})$ converges uniformly on compact subsets of [0, 1) but not on [0, 1).
C. $(f_{n})$ converges uniformly on [0, 1) but not on [0, 1].
D. $(f_{n})$ converges uniformly on [0, 1].✓ Correct

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Q7 open Partial derivatives, differentiability, chain rule

For real numbers a, b, c, d, e, f, consider the function $F : \mathbb{R}^{2} \to \mathbb{R}^{2}$ given by F(x, y) = (ax + by + c, dx + ey + f). Which of the following statements are true?

A.F is continuous.✓ Correct
B.F is uniformly continuous.✓ Correct
C.F is differentiable.✓ Correct
D.F has partial derivatives of all orders.✓ Correct

Solution

An affine map is Lipschitz (hence uniformly continuous), differentiable with constant derivative, and all higher partials are zero.

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Q8 Converse assumed Partial derivatives, differentiability, chain rule

For a differentiable surjective function f : (0,1) → (0,1), consider $F : (0,1)^{2} \to (0,1)^{2}, F(x, y) = (f(x), f(y))$ . If f′(x) ≠ 0 for every x ∈ (0,1), which of the following statements are true?

A.F is injective.✓ Correct
B.f is increasing.
C.For every $(x', y') \in (0,1)^{2}$ , there exists a unique $(x, y) \in (0,1)^{2}$ such that F(x, y) = (x′, y′).✓ Correct
D.The total derivative DF(x, y) is invertible for all $(x, y) \in (0,1)^{2}$ .✓ Correct

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Q9 Finite-dimensional intuition Partial derivatives, differentiability, chain rule

Let $p : \mathbb{R}^{2} \to \mathbb{R}$ be the function p(x, y) = x. Which of the following statements are true?

A.Let $A_{1} =$ { $x^{2} + y^{2} < 1$ }. For each $\gamma \in p(A_{1})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{1})$ .✓ Correct
B.Let $A_{2} =$ { $x^{2} + y^{2} \le 1$ }. For each $\gamma \in p(A_{2})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{2})$ .
C.Let $A_{3} =$ {xy = 0}. For each $\gamma \in p(A_{3})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{3})$ .✓ Correct
D.Let $A_{4} =$ {xy = 1}. For each $\gamma \in p(A_{4})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{4})$ .✓ Correct

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Q10 Hypothesis dropped Measurable sets and functions

Let { $A_{n}$ } $_{n\ge1}$ be a collection of non-empty subsets of $\mathbb{Z}$ such that $A_{n} \cap A_{m} = \emptyset$ for m ≠ n. If $\mathbb{Z} = \cup_{n\ge1}$ $A_{n}$ , then which of the following statements are necessarily true?

A. $A_{n}$ is finite for every integer n ≥ 1.
B. $A_{n}$ is finite for some integer n ≥ 1.
C. $A_{n}$ is infinite for some integer n ≥ 1.
D. $A_{n}$ is countable (finite or infinite) for every integer n ≥ 1.✓ Correct

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Q11 Invariants don't determine Linear transformations, matrix representation, change of basis

Let $T : \mathbb{R}^{5} \to \mathbb{R}^{5}$ be an $\mathbb{R}-$ linear transformation. Suppose that (1, −1, 2, 4, 0), (4, 6, 1, 6, 0) and (5, 5, 3, 9, 0) span the null space of T. Which of the following statements are true?

A.The rank of T is equal to 2.✓ Correct
B.Suppose that for every vector $v \in \mathbb{R}^{5}$ there exists n such that $T^{n}v = 0$ . Then $T^{2}$ must be zero.
C.Suppose that for every vector $v \in \mathbb{R}^{5}$ there exists n such that $T^{n}v = 0$ . Then $T^{3}$ must be zero.✓ Correct
D.(−2, −8, 3, 2, 0) is contained in the null space of T.✓ Correct

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Q12 Finite-dimensional intuition Bases, dimension, rank–nullity

Let 𝔽 be a finite field and V be a finite dimensional non-zero 𝔽-vector space. Which of the following can NEVER be true?

A.V is the union of 2 proper subspaces.✓ Correct
B.V is the union of 3 proper subspaces.
C.V has a unique basis.
D.V has precisely two bases.

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Q13 Execution slip Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Consider A = [[1, 4], [2, 3]]. Suppose $A^{5} - 4A^{4} - 7A^{3} + 11A^{2} - A - 10I =$ aA + bI for some $a, b \in \mathbb{Z}$ . Which of the following statements are true?

A.a + b > 8.
B.a + b < 7.✓ Correct
C.a + b is divisible by 2.✓ Correct
D.a > b.

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Q14 Execution slip Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let X, Y be two n × n real matrices such that XY $= X^{2} + X + I$ . Which of the following statements are necessarily true?

A.X is invertible.✓ Correct
B.X + I is invertible.
C.XY = YX.✓ Correct
D.Y is invertible.

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Q15 open Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let A be an n × n real symmetric matrix. Which of the following statements are necessarily true?

A.A is diagonalizable.✓ Correct
B.If Aᵏ = I for some positive integer k, then $A^{2} = I$ .✓ Correct
C.If Aᵏ = 0 for some positive integer k, then $A^{2} = 0$ .✓ Correct
D.All eigenvalues of A are real.✓ Correct

Solution

Spectral theorem: A = QDQᵀ with real D.

Aᵏ = I forces each real eigenvalue to satisfy $\lambda^{k} = 1$ , so $\lambda = \pm1$ and $A^{2} = I; A^{k} = 0$ forces all $\lambda = 0$ so A = 0.

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Q16 Invariants don't determine Jordan canonical form

Suppose a 7 × 7 block diagonal complex matrix A has blocks (0), (1), [[0, 1], [0, 0]] and the 3×3 block with $2\pi{}i$ on the diagonal, a 1 in position (1,2) and zeros elsewhere, along the diagonal. Which of the following statements are true?

A.The characteristic polynomial of A is $x^{3}(x - 1)(x - 2\pi{}i)^{3}$ .✓ Correct
B.The minimal polynomial of A is $x^{2}(x - 1)(x - 2\pi{}i)^{3}$ .
C.The dimensions of the eigenspaces for $0, 1, 2\pi{}i$ are 2, 1, 3 respectively.
D.The dimensions of the eigenspaces for $0, 1, 2\pi{}i$ are 2, 1, 2 respectively.✓ Correct

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Q17 open Quadratic forms, positive definiteness, Sylvester's law

Let A be a real diagonal matrix with characteristic polynomial $\lambda^{3} - 2\lambda^{2} - \lambda + 2$ . Define a bilinear form ⟨v, w⟩ = vᵗAw on $\mathbb{R}^{3}$ . Which of the following statements are true?

A.A is positive definite.
B. $A^{2}$ is positive definite.✓ Correct
C.There exists a nonzero $v \in \mathbb{R}^{3}$ such that ⟨v, v⟩ = 0.✓ Correct
D.rank A = 2.

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Q18 Standard counterexample Ideals, quotient rings, prime & maximal ideals, CRT

Let $R = \mathbb{Z}[X]/(X^{2} + 1)$ and $\psi : \mathbb{Z}[X] \to R$ be the natural quotient map. Which of the following statements are true?

A.R is isomorphic to a subring of $\mathbb{C}$ .✓ Correct
B.For any prime number $p \in \mathbb{Z}$ , the ideal generated by $\psi(p)$ is a proper ideal of R.✓ Correct
C.R has infinitely many prime ideals.✓ Correct
D.The ideal generated by $\psi(X)$ is a prime ideal in R.

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Q19 open Ideals, quotient rings, prime & maximal ideals, CRT

Let $n \in \mathbb{Z}$ be such that n ≡ 1 (mod 7) and n ≡ 4 (mod 15). Which of the following statements are true?

A.n ≡ 1 (mod 3).✓ Correct
B.n ≡ 1 (mod 35).
C.n ≡ 1 (mod 21).✓ Correct
D.n ≡ 1 (mod 5).

Solution

n ≡ 4 (mod 15) gives n ≡ 1 (mod 3) and n ≡ 4 (mod 5).

With n ≡ 1 (mod 7), CRT gives n ≡ 1 (mod 21); n ≢ 1 (mod 5) kills the mod-35 claim.

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Q20 Base field or ring Polynomial rings and irreducibility tests

Let $f(X) = X^{2} + X + 1$ and $g(X) = X^{2} + X - 2$ be polynomials in $\mathbb{Z}[X]$ . Which of the following statements are true?

A.For all prime numbers p, f(X) mod p is irreducible in $(\mathbb{Z}/p\mathbb{Z})[X]$ .
B.There exists a prime number p such that g(X) mod p is irreducible in $(\mathbb{Z}/p\mathbb{Z})[X]$ .
C.g(X) is irreducible in $\mathbb{Q}[X]$ .
D.f(X) is irreducible in $\mathbb{Q}[X]$ .✓ Correct

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Q21 Execution slip Group actions, class equation, p-groups

Let G be the group (under matrix multiplication) of 2 × 2 invertible matrices with entries from $\mathbb{Z}/9\mathbb{Z}$ . Let a be the order of G. Which of the following statements are true?

A.a is divisible by $3^{4}$ .✓ Correct
B.a is divisible by $2^{4}$ .✓ Correct
C.a is not divisible by 48.
D.a is divisible by $3^{6}$ .

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Q22 open Galois theory essentials

Let $f(X) = X^{3} - 2 \in \mathbb{Q}[X]$ and let $K \subset \mathbb{C}$ be the splitting field of f(X) over $\mathbb{Q}$ . Let $\omega =$ $e^{2\pi{}i/3}$ . Which of the following statements are true?

A.The Galois group of K over $\mathbb{Q}$ is the symmetric group $S_{3}$ .✓ Correct
B.The Galois group of K over $\mathbb{Q}(\omega)$ is the symmetric group $S_{3}$ .
C.The Galois group of K over $\mathbb{Q}$ is $\mathbb{Z}/3\mathbb{Z}$ .
D.The Galois group of K over $\mathbb{Q}(\omega)$ is $\mathbb{Z}/3\mathbb{Z}$ .✓ Correct

Solution

$K = \mathbb{Q}(\sqrt[3]{2}, \omega)$ has degree 6 over $\mathbb{Q}$ with non-abelian Galois group $S_{3}$ ; over $\mathbb{Q}(\omega)$ the degree is 3, so the group is $\mathbb{Z}/3$ .

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Q23 Property not inherited Continuity, homeomorphism, separation axioms

Consider $\mathbb{R}^{2}$ with the Euclidean topology and $\mathbb{Q}^{2} \subset \mathbb{R}^{2}$ with the subspace topology. Which of the following statements are true?

A. $\mathbb{Q}^{2}$ is connected.
B.If A is a non-empty connected subset of $\mathbb{Q}^{2}$ , then A has exactly one element.✓ Correct
C. $\mathbb{Q}^{2}$ is Hausdorff.✓ Correct
D.{ $(x, y) \in \mathbb{Q}^{2}$ | $x^{2} + y^{2} = 1$ } is compact in the subspace topology.

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Q24 Standard counterexample Power series and analyticity

For every n ≥ 1, consider the entire function $p_{n}(z) = \sum_{k=0}^{n}$ zᵏ/k!. Which of the following statements are true?

A.The sequence $(p_{n})$ converges to an entire function uniformly on compact subsets of $\mathbb{C}$ .✓ Correct
B.For all $n \ge 1, p_{n}$ has a zero in the set { $z \in \mathbb{C}$ : |z| ≤ 2023}.
C.There exists a sequence $(z_{n})$ of complex numbers such that lim | $z_{n}$ | $= \infty$ and $p_{n}(z_{n}) = 0$ for all n ≥ 1.✓ Correct
D.Let $S_{n}$ be the set of all zeros of $p_{n}$ . If $a_{n} =$ $min_{z\in{}S_{n}}$ |z|, then $a_{n} \to \infty$ as $n \to \infty$ .✓ Correct

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Q25 Standard counterexample Liouville, Morera, maximum modulus principle

Let X be an uncountable subset of $\mathbb{C}$ and let $f : \mathbb{C} \to \mathbb{C}$ be an entire function. Assume that for every z ∈ X there exists an integer n ≥ 1 such that $f^{(n)}(z)$ = 0. Which of the following statements are necessarily true?

A.f = 0.
B.f is a constant function.
C.There exists a compact subset K of $\mathbb{C}$ such that $f^{-1}(K)$ is not compact.
D.f is a polynomial.✓ Correct

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Q26 Standard counterexample Liouville, Morera, maximum modulus principle

Let $\Omega_{1} =$ { $z \in \mathbb{C}$ : |z| < 1} and $\Omega_{2} = \mathbb{C}$ . Which of the following statements are true?

A.There exists a holomorphic surjective map $f : \Omega_{1} \to \Omega_{2}$ .✓ Correct
B.There exists a holomorphic surjective map $f : \Omega_{2} \to \Omega_{1}$ .
C.There exists a holomorphic injective map $f : \Omega_{1} \to \Omega_{2}$ .✓ Correct
D.There exists a holomorphic injective map $f : \Omega_{2} \to \Omega_{1}$ .

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Q27 Execution slip Residue theorem and standard contour integrals

For an integer k, consider the contour integral $I_{k} = \int_{|z|=1}$ eᶻ/zᵏ dz. Which of the following statements are true?

A. $I_{k} = 0$ for every integer k.
B. $I_{k} \ne 0$ if k ≥ 1.✓ Correct
C.| $I_{k}$ | ≤ | $I_{k}$ ₊ $_{1}$ | for every integer k.
D. $lim_{k\to\infty}$ | $I_{k}$ | $= \infty$ .

Solution

$I_{k} = 2\pi{}i \cdot ($ coefficient of $z^{k-1}$ in eᶻ $) = 2\pi{}i/(k-1)$ ! for k ≥ 1 and 0 for k ≤ 0.

These values decrease to 0, so (3) and (4) fail.

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Q28 Existence vs uniqueness Existence–uniqueness, Picard, Lipschitz

Consider the initial value problem y′ = y + ½| $\sin(y^{2})$ |, x > 0, y(0) = −1. Which of the following statements are true?

A.There exists an $\alpha \in (0, \infty)$ such that $lim_{x\to\alpha^{-}}$ |y(x)| $= \infty$ .
B.y(x) exists on $(0, \infty)$ and it is monotone.✓ Correct
C.y(x) exists on $(0, \infty)$ , but not bounded below.✓ Correct
D.y(x) exists on $(0, \infty)$ , but not bounded above.

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Q29 Existence vs uniqueness Existence–uniqueness, Picard, Lipschitz

Consider the problem $y' = (1 - y^{2})^{10} \cos y, y(0) = 0$ . Let J be the maximal interval of existence and K be the range of the solution. Which of the following statements are true?

A. $J = \mathbb{R}$ ✓ Correct
B.K = (−1, 1)✓ Correct
C.J = (−1, 1)
D.K = [−1, 1]

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Q30 open Existence–uniqueness, Picard, Lipschitz

Consider the initial value problem $x^{2}y'' - 2x^{2}y' + (4x - 2)y = 0, y(0) = 0$ . Suppose $y = \varphi(x)$ is a polynomial solution satisfying $\varphi(1) = 1$ . Which of the following statements are true?

A. $\varphi(4) = 16$ ✓ Correct
B. $\varphi(2) = 2$
C. $\varphi(5) = 25$ ✓ Correct
D. $\varphi(3) = 3$

Solution

Try $y = x^{2}: 2x^{2} - 4x^{3} + 4x^{3} - 2x^{2} = 0$ .

So $\varphi(x) = x^{2}$ , giving $\varphi(4) = 16$ and $\varphi(5) = 25$ .

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Q31 Existence vs uniqueness First-order PDE: Lagrange, Charpit, characteristics

Consider the Cauchy problem $u\cdot{}u_{x} + u_y = 1$ for $(x, y) \in \mathbb{R} \times (0, \infty)$ , with u(x, 0) = kx, $x \in \mathbb{R}$ , for a real parameter k. For which of the following values of k does the problem have a solution defined on $\mathbb{R} \times (0, \infty)$ ?

A.k = 0✓ Correct
B.k = −2
C.k = 4✓ Correct
D.k = 1✓ Correct

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Q32 Existence vs uniqueness Laplace, heat and wave equations: separation of variables

Let B be the open unit disc in $\mathbb{R}^{2}, \partial{}B$ its boundary and ${\bar{B}} = B \cup \partial{}B$ . For $\lambda \in (0, \infty)$ , let $S_\lambda$ be the set of twice continuously differentiable functions in B, continuous on B̄, satisfying $(u_{x})^{2} + \lambda(u_y)^{2} = 1$ in B and u = 0 on $\partial{}B$ . Which of the following statements are true?

A. $S_{1} = \emptyset$ ✓ Correct
B. $S_{2} = \emptyset$ ✓ Correct
C. $S_{1}$ has exactly one element and $S_{2}$ has exactly two elements.
D. $S_{1}$ and $S_{2}$ are both infinite.

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Q33 Execution slip Interpolation and numerical integration with error terms

The coefficient of $x^{3}$ in the interpolating polynomial for the data x = 0, 1, 2, 3, 4 with y = 1, 2, 1, 3, 5 is

A.−1/3
B.−1/2
C.5/6
D.17/6✓ Correct

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Q34 Numerical convergence Numerical ODE: Euler, Runge–Kutta

For the IVP dy/dx $= f(x, y), y(x_{0}) = y_{0}$ , with step size h, let the first iterate of a second-order scheme be $y_{1} = y_{0} +$ Pk $_{1} +$ Qk $_{2}$ , where $k_{1} = h f(x_{0}, y_{0}), k_{2} = h f(x_{0} + \alpha_{0}h, y_{0} + \beta_{0}k_{1})$ and $P, Q, \alpha_{0}, \beta_{0} \in \mathbb{R}$ . Which of the following statements are correct?

A.If $\alpha_{0} = 2$ , then $\beta_{0} = 2, P = 3/4, Q = 1/4$ ✓ Correct
B.If $\beta_{0} = 3$ , then $\alpha_{0} = 3, P = 5/6, Q = 1/6$ ✓ Correct
C.If $\alpha_{0} = 2$ , then $\beta_{0} = 2, P = 1/4, Q = 3/4$
D.If $\beta_{0} = 3$ , then $\alpha_{0} = 3, P = 1/6, Q = 5/6$

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Q35 open Euler–Lagrange equation and standard functionals

Among the curves connecting the points (1, 2) and (2, 8), let $\gamma$ be the curve on which an extremal of $J[y] = \int_{1}^{2} (1 + x^{3}y') y'$ dx can be attained. Which of the following points lie on the curve $\gamma$ ?

A. $(\sqrt{2}, 3)$
B. $(\sqrt{2}, 6)$ ✓ Correct
C. $(\sqrt{3}, 22/3)$ ✓ Correct
D. $(\sqrt{3}, 23/3)$

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Q36 Existence vs uniqueness Euler–Lagrange equation and standard functionals

Define S = { $y \in C^{1}[0, \pi] : y(0) = y(\pi) = 0$ }, ‖f‖ $_\infty = \max$ |f|, $B_{0}(0, \varepsilon) =$ {f ∈ S : ‖f‖ $_\infty < \varepsilon$ } and $B_{1}(0, \varepsilon) =$ {f ∈ S : ‖f‖ $_\infty +$ ‖f′‖ $_\infty < \varepsilon$ }. For the functional $J[y] = \int_{0}^\pi (1 - (y')^{2}) y^{2}$ dx, there exists $\varepsilon > 0$ such that

A.J[y] ≤ J[0] for all $y \in B_{0}(0, \varepsilon)$
B.J[y] ≤ J[0] for all $y \in B_{1}(0, \varepsilon)$
C.J[y] ≥ J[0] for all $y \in B_{0}(0, \varepsilon)$
D.J[y] ≥ J[0] for all $y \in B_{1}(0, \varepsilon)$ ✓ Correct

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Q37 Execution slip Fredholm and Volterra equations

Let y be the solution to the Volterra integral equation y(x) = eˣ $+ \int_{0}$ ˣ $((1 + x^{2})/(1 + t^{2})) y(t)$ dt. Which of the following statements are true?

A. $y(1) = (1 + \pi/4)e$
B. $y(1) = (1 + \pi/2)e$ ✓ Correct
C. $y(\sqrt{3}) = (1 +$ $3\pi/4)e^{\sqrt{3}}$
D. $y(\sqrt{3}) = (1 +$ $4\pi/3)e^{\sqrt{3}}$ ✓ Correct

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Q38 Existence vs uniqueness Fredholm and Volterra equations

Consider the Fredholm integral equation $y(x) - 3\int_{0}^{1} t x y(t)$ dt = f(x), with f continuous on [0, 1]. Which of the following choices for f(x) have the property that the equation admits at least one solution?

A. $f(x) = x^{2} - 1/2$ ✓ Correct
B.f(x) = eˣ
C.f(x) = 2 − 3x✓ Correct
D.f(x) = x − 1

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Q39 Hypothesis dropped Hamiltonian formalism and conservation laws

Let $q_{1}, q_{2}$ be generalized coordinates and $p_{1}, p_{2}$ their conjugate momenta. Let a and b be such that $Q_{1} = q_{1}, P_{1} = a p_{1} + 16 p_{2}, Q_{2} = p_{2}, P_{2} = 2q_{1} + b q_{2}$ is a canonical transformation. Which of the following statements are true?

A. $a^{2} + b^{2} = 2$ ✓ Correct
B.a − b = 2✓ Correct
C.a + b = 2
D.a = 1, b = 1

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Q40 Moments and tails Random variables, distributions, moments, MGF

Suppose U ~ Uniform(0,1) and $X = \tan(\pi(U -$ ½)). Then which of the following statements are true?

A. $E(X^{4}) = 3$
B.P(X ∈ {1, 2, 5}) = 1/2
C.E(eˣ) does not exist✓ Correct
D.P(X ≤ 0) = 1/2✓ Correct

Solution

X is standard Cauchy: symmetric about 0, continuous (so any finite set has probability 0), and with no finite moments — E(eˣ) diverges too.

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Q41 open Random variables, distributions, moments, MGF

Let X be a discrete random variable with support S = {−1, 0, 1} and P(X = 0) = 1/3. Then which of the following statements are true?

A.E(X) ≤ 2/3✓ Correct
B. $E(X^{2}) = 2/3$ ✓ Correct
C.E(|X|) = 2/3✓ Correct
D.Var(X) > 2/3

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

Q42 Boundary and endpoint Random variables, distributions, moments, MGF

Suppose X is a continuous random variable with density $f(x) = (1/\pi)\cdot1/(1 + (x + 1)^{2})$ , and Y = X/|X| for X ≠ 0, Y = 0 for X = 0. Then which of the following statements are true?

A.E(Y) = 0
B.P(Y > 0) < P(Y < 0)✓ Correct
C.P(Y < −1) < P(Y > 1)
D. $E(Y^{2}) = 1$ ✓ Correct

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

Q43 Execution slip Random variables, distributions, moments, MGF

Let X be a discrete random variable with support {0, 1, …, 25} and $P(X = x) = C(25, x)/2^{25}$ . Then which of the following statements are true?

A.The distributions of X − 12.5 and 12.5 − X are identical✓ Correct
B.P(X ≤ 4) = P(X ≥ 22)
C.The coefficient of variation (in percentage) of X is 20✓ Correct
D.P(X ≤ 4.9) = P(X ≥ 20.1)✓ Correct

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

Q44 Dependence misread Joint distributions, transformations, order statistics

Let $X_{i} (i = 1, 2)$ be independent with densities $f_{i}(x) = i$ $e^{-ix}$ for x ≥ 0. Consider a series system of two independent components with lifetimes $X_{1}$ and $X_{2}$ , and let X denote the lifetime of the system. Which of the following statements are true?

A.P(X > 4) = P(X > 1)P(X > 2)
B.P(X > 4 | X > 2) = P(X > 2)✓ Correct
C.E(X) = 1/3✓ Correct
D.6X ~ $\chi^{2}_{3}$

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

Q45 Execution slip Markov chains: classification of states, stationary distributions

Consider an M/M/1 queue with arrival rate $\lambda = 15$ per hour and service rate $\mu = 45$ per hour. Let N(t) be the number of customers in the system at time t, and $T_{1}, T_{2}$ the time a customer spends in the queue and in the system. Which of the following statements are true?

A. $lim_{t\to\infty}$ P(N(t) = 1) = 2/9✓ Correct
B. $P(T_{1} > 0) = 1/3$ ✓ Correct
C. $E(T_{1}) = 1/90$ ✓ Correct
D. $E(T_{2}) = 1/35$

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Q46 Dependence misread Markov chains: classification of states, stationary distributions

Suppose {X(t)} and {Y(t)} are two independent homogeneous Poisson processes with the same rate $\lambda = 2$ . Let $W_{n}$ ˣ and $W_{n}$ ʸ be the waiting times for the n-th arrival in each process. Which of the following statements are true?

A. $P(W_{2}$ ˣ $< W_{3}$ ʸ) = 11/16✓ Correct
B. $P(W_{1}$ ˣ $< W_{1}$ ʸ) = 1/2✓ Correct
C. $P(W_{2}$ ˣ $< W_{3}$ ʸ) = 13/16
D. $P(W_{1}$ ˣ $< W_{1}$ ʸ) = 1/4

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

Q47 Boundary and endpoint MLE and method of moments

Let $X_{1}, \dots, X_{n}$ be a random sample with density f(x| $\theta) =$ $e^{\theta-x}$ for $x \ge \theta$ , where $\theta \in \mathbb{R}$ is unknown. Define X̄ and $X_{(1)} = \min X_{i}$ . Which of the following statements are true?

A.X̄ is the method of moments estimator of $\theta$
B. $X_{(1)}$ is the maximum likelihood estimator of $\theta$ ✓ Correct
C. $X_{(1)} - 1/n$ is the uniformly minimum variance unbiased estimator of $\theta$ ✓ Correct
D. $X_{(1)}$ is a sufficient statistic for $\theta$ ✓ Correct

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Q48 What the inference means Neyman–Pearson lemma and UMP tests

Let $X_{1}, \dots, X_{25}$ be i.i.d. Bernoulli(p), X̄ their mean, $T_{1} = 5({\bar{X}} - 0.5)/\sqrt{{\bar{X}}(1 - {\bar{X}})}$ for 0 < X̄ < 1 (±5 at the endpoints) and $T_{2} = 10({\bar{X}} - 0.5)$ . For $H_{0}: p = 0.5$ vs $H_{1}: p > 0.5$ , test $\psi_{i}$ rejects iff $T_{i} > 2$ . If the observed X̄ ∈ (0.5, 0.75), which of the following statements are true?

A.If $\psi_{1}$ rejects $H_{0}$ , then $\psi_{2}$ also rejects $H_{0}$
B.If $\psi_{1}$ does not reject $H_{0}$ , then $\psi_{2}$ also does not reject $H_{0}$ ✓ Correct
C.If $\psi_{2}$ rejects $H_{0}$ , then $\psi_{1}$ also rejects $H_{0}$ ✓ Correct
D.If $\psi_{2}$ does not reject $H_{0}$ , then $\psi_{1}$ also does not reject $H_{0}$

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

Q49 What the inference means Neyman–Pearson lemma and UMP tests

Let $X_{1}, \dots, X_{n}$ be a random sample from $N(\mu, 1)$ . To test $H_{0}: \mu = \mu_{0}$ against $H_{1}: \mu > \mu_{0}$ , consider (A) reject iff ${\bar{X}}_{n} > c_{1}$ with $P_{\mu_{0}}(X_{n} > c_{1}) = \alpha$ , and (B) reject iff Median{ $X_{1}, \dots, X_{n}$ } $> c_{2}$ with $P_{\mu_{0}}(Median$ $> c_{2}) = \alpha$ . Which of the following statements are true?

A.The test in (A) is the uniformly most powerful test of size $\alpha$ ✓ Correct
B.The test in (B) is the uniformly most powerful test of size $\alpha$
C. $P_\mu({\bar{X}}_{n} > c_{1}) \to 1$ as $n \to \infty$ for all $\mu > \mu_{0}$ ✓ Correct
D. $P_{\mu_{0}}(Median$ { $X_{1}, \dots, X_{n}$ } $> \mu_{0}) = 1/2$ ✓ Correct

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Q50 Boundary and endpoint Likelihood ratio and standard tests

For n ≥ 2, let $X_{1}, \dots, X_{n}$ be from $N(\mu, \sigma^{2})$ with both unknown, X̄ the mean and $S^{2}$ the sample variance. With $z_\alpha$ and $t_{m,\alpha}$ the $(1 - \alpha)-$ th quantiles, which of the following represent 90% confidence intervals for $\mu$ ?

A. $({\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.05}$ , ${\bar{X}} + (S/\sqrt{n})$ $t_{n-1,0.05})$ ✓ Correct
B. $({\bar{X}} - (\sigma/\sqrt{n})$ $z_{0.05}$ , ${\bar{X}} + (\sigma/\sqrt{n})$ $z_{0.05})$
C. $[{\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.9}$ , $\infty)$
D. $(-\infty, {\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.9})$ ✓ Correct

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

Q51 Moments and tails Likelihood ratio and standard tests

Let $X_{1}, \dots, X_{n}$ be a random sample from an unknown absolutely continuous CDF F, and $F_{0} a$ specified absolutely continuous CDF. For $H_{0}: F = F_{0}$ vs $H_{1}: F \ne F_{0}$ , consider $T_{1,n}$ = sup_x | $F_{n}(x) - F_{0}(x)$ | and $T_{2,n}$ = n·sup_x | $F_{n}(x) - F_{0}(x)$ |, where $F_{n}$ is the empirical CDF. Which of the following statements are true?

A. $T_{1,n}$ → 0 in probability as $n \to \infty$ under $H_{0}$ ✓ Correct
B. $T_{2,n}$ → 0 in probability as $n \to \infty$ under $H_{0}$
C.lim P_F( $T_{2,n}$ > 1) = 1 for all F✓ Correct
D. $T_{2,n}$ converges in distribution to a degenerate random variable under $H_{0}$

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Q52 Hypothesis dropped Gauss–Markov, regression, ANOVA basics

Consider the one-way fixed effects ANOVA model $Y_{ij} = \mu + \alpha_{i} + \varepsilon_{ij}, j = 1, \dots, n_{i}; i = 1, \dots, k$ , with uncorrelated errors of mean 0 and variance $\sigma^{2}$ . Let Ȳ $_{i}$ be the i-th group mean. Which of the following statements are true?

A. $(1/\sum{}n_{i}) \sum_{i}\sum_{j} Y_{ij}$ is an unbiased estimator of $\mu$
B. $2\mu + \alpha_{1} + \alpha_{2}$ is an estimable linear parametric function✓ Correct
C. $\mu + \alpha_{1} + \alpha_{2}$ is an estimable linear parametric function
D. $(1/n_{2}) \sum_{j} (Y_{2j} -$ Ȳ $_{2})$ is an unbiased estimator of $\alpha_{2}$

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Q53 Invariants don't determine Gauss–Markov, regression, ANOVA basics

Consider $Y = X\beta + \varepsilon$ with X a fixed n × (p + 1) matrix of rank p + 1 and $\varepsilon$ ~ $N_{n}(0, \sigma^{2}I)$ . If $\beta$ ̂ is the OLS estimator, which of the following statements are true?

A. $(1/\sigma^{2}) Y^{T}X\beta$ ̂ has a central $\chi^{2}_{p+1}$ distribution
B. $(1/\sigma^{2})(Y - X\beta$ ̂ $)^{T}(Y - X\beta$ ̂) has a central $\chi^{2}_{n-p-1}$ distribution✓ Correct
C. $X\beta$ ̂ and $(Y - X\beta$ ̂ $)^{T}(Y - X\beta$ ̂) are independently distributed✓ Correct
D. $(1/\sigma^{2}) \sum(Y_{i} -$ Ȳ $)^{2}$ has a central $\chi^{2}_{n-1}$ distribution

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Q54 Execution slip Multivariate normal distribution

Suppose $A = (a_{ij})$ ~ $W_{3}(5, \sum)$ , where $\sum = [[2, 1, 1], [1, 2, 0], [1, 0, 2]]$ . Then which of the following statements are true?

A. $a_{22}$ ~ $\chi^{2}_{3}$
B.½ $a_{22}$ ~ $\chi^{2}_{5}$ ✓ Correct
C. $(1/33)(a_{11} - 4a_{13} + 4a_{33})$ ~ $\chi^{2}_{3}$
D. $(1/9)(a_{11} - 4a_{13} + 4a_{33})$ ~ $\chi^{2}_{5}$

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Q55 Hypothesis dropped Multivariate normal distribution

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), (X_{3}, Y_{3})$ be i.i.d. bivariate normal with mean (0, 0) and correlation matrix $[[1, \rho], [\rho, 1]]$ , | $\rho$ | < 1. Let $S_\rho = 3E($ sgn $(X_{1} - X_{2})(Y_{1} - Y_{3}))$ . Which of the following statements are true?

A.If $X_{1}$ and $Y_{1}$ are independent, then $S_\rho = 0$ ✓ Correct
B. $S_\rho = (6/\pi) \sin$ ⁻ $^{1}(\rho/2)$ ✓ Correct
C.If $S_\rho = 0$ , then $X_{1}$ and $Y_{1}$ are independent✓ Correct
D.If $X_{1}$ and $Y_{1}$ are independent, then $S_\rho = 1/2$

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Q56 open SRS, stratified and systematic sampling

Consider a population of 3 units having values 2, 4 and 6. A simple random sample (without replacement) of 2 units is drawn. Let M be the sample mean. Which of the following statements are true?

A.E(M) = 4✓ Correct
B. $E(M^{2}) = 17$
C. $E(M^{3}) = 72$ ✓ Correct
D.Var(M) = 1

Solution

M takes the values 3, 4, 5 with probability 1/3 each: $E(M) = 4, E(M^{2}) = 50/3, E(M^{3}) = 72, Var(M) = 2/3$ .

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Q57 Execution slip SRS, stratified and systematic sampling

Two groups $G_{1}$ and $G_{2}$ have 10 and 30 patients with mean diastolic blood pressures 80 and 100 mmHg and variances 4 and 2 mmHg $^{2}$ . Let ${\bar{X}}, S^{2}, C$ and R be the mean, variance (with divisor n), coefficient of variation (%) and range of the combined group. Which of the following statements are true?

A.X̄ = 95✓ Correct
B. $S^{2} = 77$
C.C > 180/19
D.R > 8✓ Correct

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Q58 Standard counterexample Continuity, uniform continuity, Lipschitz

Which of the following statements are true?

A.f(x) = [x] sin(1/x) for x ≠ 0, f(0) = 0 (where [x] is the integer part) has a discontinuity at 0 which is removable.
B. $f : [0,\infty) \to \mathbb{R}, f(x) = \sin(\log x)$ for x ≠ 0, f(0) = 0, has a discontinuity at 0 which is NOT removable.✓ Correct
C.f(x) = $e^{1/x}$ for x < 0, f(x) = $e^{1/(x+1)}$ for x ≥ 0, has a jump discontinuity at 0.✓ Correct
D.If $f, g : [0,1] \to \mathbb{R}$ are of bounded variation, then fg has at most countably many discontinuities.✓ Correct

Solution

(1) As x → 0⁻, [x] = −1 and −sin(1/x) oscillates: no left limit, not removable.

(2) sin(log x) oscillates as x → 0⁺.

(3) Left limit 0, right limit e.

(4) A product of BV functions is BV, and BV functions have only countably many (jump) discontinuities.

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Q59 open Subgroups, cosets, Lagrange, cyclic groups

Which of the following statements are true?

A.Let $G_{1}$ and $G_{2}$ be finite groups such that | $G_{1}$ | and | $G_{2}$ | are coprime. Then any homomorphism from $G_{1}$ to $G_{2}$ is trivial.✓ Correct
B.Let G be a finite group and f : G → G a homomorphism that fixes more than half of the elements of G. Then f(x) = x for all x ∈ G.✓ Correct
C.Let G be a finite group having exactly 3 subgroups. Then G is of order $p^{2}$ for some prime p.✓ Correct
D.Any finite abelian group G has at least d(|G|) subgroups, where d(m) is the number of positive divisors of m.✓ Correct

Solution

(1) the image has order dividing both.

(2) the fixed set is a subgroup of index < 2, hence all of G.

(3) exactly three subgroups forces cyclic of order $p^{2}$ .

(4) an abelian group has a subgroup of every order dividing |G|.

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CSIR NET December 2023 — Part C

Part C