CSIR NET June 2023 — Part B

All 22 Part B questions we have transcribed from this paper, of the 38 on the site for this sitting — every option and the answer key, with the reasoning for each one.

Part B

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

Q1 Hypothesis dropped Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements

Consider the series $\sum_{n = 1}^{\infty}$ $a_{n}$ , where $a_{n} =$ $(- 1)^{n + 1} (n + 1 - n)$ . Which of the following statements is true?

A.The series is divergent.
B.The series is convergent.
C.The series is conditionally convergent.✓
D.The series is absolutely convergent.

Solution

$n + 1 - n = 1/ (n + 1 + n)$ decreases to 0, so the alternating series converges (Leibniz). The absolute series behaves like $\sum 1/ (2 n)$ , which diverges. Hence conditionally convergent — option 3 is the most precise true statement (official key: 3).

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Q2 Limit assumed to exist limsup, liminf and subsequential limits

Which of the following assertions is correct?

A. $l im s u p_{n} exp (cos ((nπ + (- 1)^{n} \cdot 2 e) / (2 n))) > 1$ .
B. $l i m_{n} exp (l o g_{e} ((n π^{2} + (- 1)^{n} e^{2}) / (7 n)))$ does not exist.
C. $l imin f_{n} exp (sin ((nπ + (- 1)^{n} \cdot 2 e) / (2 n))) < π$ .✓
D. $l i m_{n} exp (tan ((n π^{2} + (- 1)^{n} e^{2}) / (7 n)))$ does not exist.

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

Q3 Pointwise vs uniform Continuity, uniform continuity, Lipschitz

Which one of the following functions is uniformly continuous on the interval (0, 1)?

A.f(x) = sin(1/x)
B.f(x) = $e^{- 1/ x^{2}}$ ✓
C.f(x) = eˣ cos(1/x)
D. $f (x) = cos x \cdot cos (π / x)$

Solution

A continuous function on (0,1) is uniformly continuous iff it extends continuously to [0,1]. $e^{- 1/ x^{2}}$ → 0 as x → 0⁺, so it extends; the other three oscillate without a limit at 0.

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Q4 Invariants don't determine Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let A be a 3×3 real matrix whose characteristic polynomial p(T) is divisible by $T^{2}$ . Which of the following statements is true?

A.The eigenspace of A for the eigenvalue 0 is two-dimensional.
B.All the eigenvalues of A are real.✓
C. $A^{3} = 0$ .
D.A is diagonalizable.

Solution

$p (T) = T^{2} (T - c)$ with c real (degree-3 real polynomial). So all eigenvalues are real. The geometric multiplicity of 0 may be 1 (Jordan block), so (1), (4) fail; A need not be nilpotent (c ≠ 0).

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Q5 Invariants don't determine Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let T be a linear operator on $R^{3}$ . Let $f (X) \in R [X]$ denote its characteristic polynomial. Consider the following statements. (a) Suppose T is non-zero and 0 is an eigenvalue of T. If we write f(X) = X·g(X) in $R [X]$ , then the linear operator g(T) is zero. (b) Suppose 0 is an eigenvalue of T with at least two linearly independent eigenvectors. If we write f(X) = X·g(X) in $R [X]$ , then the linear operator g(T) is zero. Which of the following is true?

A.Both (a) and (b) are true.
B.Both (a) and (b) are false.
C.(a) is true and (b) is false.
D.(a) is false and (b) is true.✓

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

Q6 Execution slip Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

Let $x = (x_{1}, \dots, x_{n})$ and $y = (y_{1}, \dots, y_{n})$ denote vectors in $R^{n}$ for a fixed n ≥ 2. Which of the following defines an inner product on $R^{n}$ ?

A.⟨x, y⟩ $= \sum_{i, j = 1}^{n}$ $x_{i} y_{j}$
B.⟨x, y⟩ $= \sum_{i, j = 1}^{n}$ $(x_{i}^{2} + y_{j}^{2})$
C.⟨x, y⟩ $= \sum_{j = 1}^{n}$ $j^{3} x_{j} y_{j}$ ✓
D.⟨x, y⟩ $= \sum_{j = 1}^{n}$ $x_{j} y_{n}$ ₋ $_{j}$ ₊ $_{1}$

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

Q7 Property not inherited Group actions, class equation, p-groups

Let p be a prime number. Let G be a group such that for each g ∈ G there exists an $n \in N$ such that $g^{p^{n}}$ = 1. Which of the following statements is FALSE?

A.If |G| $= p^{6}$ , then G has a subgroup of index $p^{2}$ .
B.If |G| $= p^{6}$ , then G has at least five normal subgroups.
C.Center of G can be infinite.
D.There exists G with |G| $= p^{6}$ such that G has exactly six normal subgroups.✓

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

Q8 Execution slip Ideals, quotient rings, prime & maximal ideals, CRT

The number of solutions of the equation $x^{2} = 1$ in the ring $Z /105 Z$ is

A.0
B.2
C.4
D.8✓

Solution

By CRT, $Z /105 Z ≅ Z /3 \times Z /5 \times Z /7; x^{2} = 1$ has exactly 2 solutions in each odd prime field, giving $2^{3} = 8$ .

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Q9 open Polynomial rings and irreducibility tests

How many real roots does the polynomial $x^{3} + 3 x - 2023$ have?

A.0
B.1✓
C.2
D.3

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

Q10 Finite-dimensional intuition Measurable sets and functions

Suppose S is an infinite set. Assuming that the axiom of choice holds, which of the following is true?

A.S is in bijection with the set of rational numbers.
B.S is in bijection with the set of real numbers.
C.S is in bijection with S × S.✓
D.S is in bijection with the power set of S.

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

Q11 Execution slip Laurent series, classification of singularities, Casorati–Weierstrass

Let $f (z) = exp (z + 1/ z), z \in C$ ∖{0}. The residue of f at z = 0 is

A. $\sum_{l = 0}^{\infty}$ 1/(l+1)!
B. $\sum_{l = 0}^{\infty}$ 1/(l!(l+1))
C. $\sum_{l = 0}^{\infty}$ 1/(l!(l+1)!)✓
D. $\sum_{l = 0}^{\infty}$ $1/ ((l^{2} + l)!)$

Solution

Multiply the series eᶻ $= \sum z^{m} / m$ ! and $e^{1/ z}$ $= \sum$ $z^{- k} / k$ !; the coefficient of $z^{- 1}$ comes from $k = m + 1 : \sum_{m} 1/ (m! (m + 1)!)$ .

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Q12 Execution slip Power series and analyticity

Consider the function f defined by $f (z) = 1/ (1 - z - z^{2})$ for $z \in C$ such that $1 - z - z^{2} \neq = 0$ . Which of the following statements is true?

A.f is an entire function.
B.f has a simple pole at z = 0.
C.f has a Taylor series expansion $f (z) = \sum a_{n} z^{n}$ , where $a_{0} = 1, a_{1} = 0$ and $a_{n}$ ₊ $_{2} = a_{n} + a_{n}$ ₊ $_{1}$ for n ≥ 0.
D.f has a Taylor series expansion $f (z) = \sum a_{n} z^{n}$ , where $a_{0} = 1, a_{1} = 1$ and $a_{n}$ ₊ $_{2} = a_{n} + a_{n}$ ₊ $_{1}$ for n ≥ 0.✓

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

Q13 Numerical convergence Linear ODE, Wronskian, variation of parameters, systems

Suppose x(t) is the solution of the initial value problem in $R^{2}$ : ẋ = Ax, $x (0) = x_{0}$ , where A = [[5, 4], [1, 2]]. Which of the following statements is true?

A.x(t) is a bounded solution for some $x_{0} \neq = 0$ .
B. $e^{- 6 t}$ |x(t)| → 0 as $t \to \infty$ , for all $x_{0} \neq = 0$ .
C. $e^{- t}$ |x(t)| $\to \infty$ as $t \to \infty$ , for all $x_{0} \neq = 0$ .
D. $e^{- 10 t}$ |x(t)| → 0 as $t \to \infty$ , for all $x_{0} \neq = 0$ .✓

Solution

Eigenvalues of A are 6 and 1 (trace 7, det 6), both positive: every non-zero solution grows like $e^{6 t}$ or $e^{t}$ . So (1) fails, (2) fails along the 6-eigendirection, (3) fails along the 1-eigendirection, and (4) holds since 10 > 6.

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Q14 Execution slip First-order PDE: Lagrange, Charpit, characteristics

Let u(x, y) be the solution of the Cauchy problem $u \cdot u_{x} + u_{y} = 0$ for $x \in R, y > 0$ , with u(x, 0) = x for $x \in R$ . Which of the following is the value of u(2, 3)?

A.2
B.3
C.1/2✓
D.1/3

Solution

Characteristics: dx/dy = u with u constant along them, so $x = x_{0} + u \cdot y$ and $u = x_{0}$ . Hence u = x/(1 + y), and u(2, 3) = 1/2.

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Q15 Hypothesis dropped Euler–Lagrange equation and standard functionals

Consider the variational problem $(P) : J (y) = \int_{0}^{1} [(y^{'})^{2} - y$ |y|·y′ + xy] dx, y(0) = 0, y(1) = 0. Which of the following statements is correct?

A.(P) has no stationary function (extremal).
B.y ≡ 0 is the only stationary function (extremal) for (P).
C.(P) has a unique stationary function (extremal) y not identically equal to 0.✓
D.(P) has infinitely many stationary functions (extremals).

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

Q16 Execution slip Fredholm and Volterra equations

For the unknown $y : [0, 1] \to R$ , consider the boundary value problem y″(x) + 2y(x) = 0 for x ∈ (0,1), y(0) = y(1) = 0. It is given that it corresponds to the integral equation $y (x) = 2 \int_{0}^{1} K (x, t) y (t)$ dt. Which of the following is the kernel K(x, t)?

A.K(x,t) = t(1 − x) for t < x; x(1 − t) for t > x✓
B. $K (x, t) = t^{2} (1 - x)$ for $t < x; x^{2} (1 - t)$ for t > x
C. $K (x, t) = t (1 - x)$ for $t < x; x (1 - t)$ for t > x
D. $K (x, t) = t^{3} (1 - x)$ for $t < x; x^{3} (1 - t)$ for t > x

Solution

The Green's function of −y″ with Dirichlet conditions on [0,1] is G(x,t) = t(1 − x) for t < x and x(1 − t) for t > x; then $y = 2 \int G y$ .

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Q17 Execution slip Standard discrete and continuous distributions

Let $X_{1}, X_{2}, X_{3}, X_{4}$ be i.i.d. Bernoulli(1/3) and $X_{(1)} \leq \dots \leq X_{(4)}$ the order statistics. Which of the following is true?

A. $X_{(1)}$ and $X_{(4)}$ are independent.
B.Expectation of $X_{(2)}$ is 1/2.
C.Variance of $X_{(2)}$ is 8/81.✓
D. $X_{(4)}$ is a degenerate random variable.

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

Q18 open Sufficiency, completeness, UMVUE, Cramér–Rao

Let X be a Poisson random variable with mean $λ$ . Which of the following parametric functions is not estimable?

A. $λ$ ⁻ $^{1}$ ✓
B. $λ$
C. $λ^{2}$
D. $e^{- λ}$

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

Q19 Execution slip Likelihood ratio and standard tests

Let $X_{1}, \dots, X_{7}$ and $Y_{1}, \dots, Y_{9}$ be independent random samples from continuous CDFs F and G. For the Wald–Wolfowitz run test of $H_{0} : F = G$ vs $H_{1} : F \neq = G$ , let R be the total number of runs in the combined ordered arrangement. Which of the following is true?

A. $P_{H_{0}} (R$ = 6) = 28/286, $P_{H_{0}} (R$ = 9) = 28/143.
B. $P_{H_{0}} (R$ = 6) = 21/286, $P_{H_{0}} (R$ = 9) = 15/286.
C. $P_{H_{0}} (R$ = 6) = 21/286, $P_{H_{0}} (R$ = 9) = 28/143.✓
D. $P_{H_{0}} (R$ = 6) = 21/286, $P_{H_{0}} (R$ = 9) = 15/286.

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

Q20 Execution slip Gauss–Markov, regression, ANOVA basics

Consider the simple linear regression model $Y_{i} = β x_{i} + ε_{i}, i = 1, \dots, n$ , where $E (ε_{i}) = 0, C o v (ε_{i}, ε_{k}) = 0$ for i ≠ k and $V a r (ε_{i}) = x_{i}^{2} σ^{2}$ . The best linear unbiased estimator of $β$ is

A. $\sum Y_{i} x_{i} / \sum x_{i}^{2}$
B. $\sum Y_{i} / \sum x_{i}$
C. $(1/ n) \sum Y_{i} / x_{i}$ ✓
D. $(1/ n) \sum Y_{i} x_{i} / x_{i}^{2}$

Solution

Heteroscedastic errors: divide by $x_{i}$ to get $Y_{i} / x_{i} = β + ε_{i} / x_{i}$ with constant variance $σ^{2}$ ; the BLUE is the plain mean of $Y_{i} / x_{i} ($ weighted least squares with weights $1/ x_{i}^{2})$ .

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Q21 open Multivariate normal distribution

Let $X = (X_{1}, X_{2})^{T}$ be bivariate normal with mean (0, 0)ᵀ and covariance $\sum = [[5, - 3], [- 3, 10]]$ . The mean vector and covariance matrix of $Y = (X_{1}, 5 - 2 X_{2})^{T}$ are

A.(0, 5)ᵀ, [[5, −3], [−3, 40]]
B.(0, 5)ᵀ, [[5, −6], [−6, 20]]
C.(0, 5)ᵀ, [[5, 3], [3, 20]]
D.(0, 5)ᵀ, [[5, 6], [6, 40]]✓

Solution

$E [5 - 2 X_{2}] = 5; V a r (5 - 2 X_{2}) = 4 \cdot 10 = 40; C o v (X_{1}, 5 - 2 X_{2}) = - 2 \cdot (- 3) = 6$ .

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Q22 Execution slip Multivariate normal distribution

Suppose $X = (X_{1}, X_{2}, X_{3}, X_{4})^{T}$ ~ $N_{4} (0, I_{2}$ ⊗ $\sum)$ with $\sum = [[2, - 1], [- 1, 2]]$ . Define $Z = [[X_{1}, X_{2}], [X_{3}, X_{4}]]$ and Q = ZᵀZ. If $W_{m} (n, \sum)$ denotes a Wishart distribution of order m with n degrees of freedom, the distribution of $Q_{11} + Q_{12} + Q_{21} + Q_{22}$ is

A. $W_{1} (2, 2)$ ✓
B. $W_{1} (1, 2)$
C. $W_{1} (2, 1)$
D. $2 χ_{4}^{2}$

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