CSIR NET December 2023 — Part B

All 37 Part B questions we have transcribed from this paper, of the 96 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 open Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

Consider the following subset of $R : U =$ { $x \in R : x^{2} - 9 x + 18 \leq 0, x^{2} - 7 x + 12 \leq 0$ }. Which one of the following statements is true?

A.inf U = 5.
B.inf U = 4.
C.inf U = 3.✓
D.inf U = 2.

Solution

$x^{2} - 9 x + 18 \leq 0 \Leftrightarrow x \in [3, 6]; x^{2} - 7 x + 12 \leq 0 \Leftrightarrow x \in [3, 4]$ . So U = [3, 4] and inf U = 3.

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Q2 Limit assumed to exist limsup, liminf and subsequential limits

Consider the sequence $(a_{n})_{n \geq 1}$ , where $a_{n} = cos ((- 1)^{n} nπ /2 + nπ /3)$ . Which one of the following statements is true?

A. $lim sup a_{n} = 3 /2$ .
B. $lim sup a_{2 n} = 1$ .✓
C. $lim sup a_{2 n} = 1/2$ .
D. $lim sup a_{3 n} = 0$ .

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

Q3 open Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements

Consider the following infinite series: $(a) \sum_{n \geq 1}$ $sin (nπ /2) / n, (b) \sum_{n \geq 1}$ $lo g (1 + 1/ n^{2})$ . Which one of the following statements is true?

A.(a) is convergent, but (b) is not convergent.
B.(a) is not convergent, but (b) is convergent.
C.Both (a) and (b) are convergent.✓
D.Neither (a) nor (b) is convergent.

Solution

(a): the non-zero terms are $(- 1)^{k} / 2 k + 1$ , an alternating series with terms decreasing to 0 (Leibniz $) . (b) : lo g (1 + 1/ n^{2})$ ~ $1/ n^{2}$ , comparison with a convergent p-series.

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Q4 Execution slip Differentiability, mean value theorems, Taylor, L'Hôpital

Let f(x) be a cubic polynomial with real coefficients. Suppose that f(x) has exactly one real root and that this root is simple. Which one of the following statements holds for ALL antiderivatives F(x) of f(x)?

A.F(x) has exactly one real root.
B.F(x) has exactly four real roots.
C.F(x) has at most two real roots.✓
D.F(x) has at most one real root.

Solution

F′ = f changes sign exactly once, so F is monotone on each side of one point: at most two real roots (and the constant of integration can realise 0, 1 or 2).

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Q5 open Differentiability, mean value theorems, Taylor, L'Hôpital

Let $f : R \to R$ be a differentiable function such that f and its derivative f′ have no common zeros in [0, 1]. Which one of the following statements is true?

A.f never vanishes in [0, 1].
B.f has at most finitely many zeros in [0, 1].✓
C.f has infinitely many zeros in [0, 1].
D.f(1/2) = 0.

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

Q6 Execution slip Differentiability, mean value theorems, Taylor, L'Hôpital

Let $f : R \to R$ be defined by $f (x) = (1 - x)^{2} sin (x^{2})$ for x ∈ (0, 1) and f(x) = 0 otherwise, and let f′ be its derivative. Let S = { $c \in R : f^{'} (x) \leq c f (x)$ for all $x \in R$ }. Which one of the following is true?

A.S = ∅✓
B.S ≠ ∅ and S is a proper subset of $(1, \infty)$
C. $(2, \infty)$ is a proper subset of S
D.S ∩ (0, 1) ≠ ∅

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

Q7 open Measurable sets and functions

Let X be a non-empty finite set and Y = { $f^{- 1} (0)$ : f is a real-valued function on X}. Which one of the following statements is true?

A.Y is an infinite set.
B.Y has $2^{∣ X ∣}$ elements.✓
C.There is a bijective function from X to Y.
D.There is a surjective function from X to Y.

Solution

Every subset of X is the zero set of its indicator-complement (f = 1 off the subset, 0 on it), so Y is the power set of X, with $2^{∣ X ∣}$ > |X| elements.

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Q8 Execution slip Linear transformations, matrix representation, change of basis

Let A be an n × n matrix with complex entries. If n ≥ 4, which one of the following statements is true?

A.A does not have any non-zero invariant subspace in $C^{n}$ .
B.A has an invariant subspace in $C^{n}$ of dimension n − 3.✓
C.All eigenvalues of A are real.
D. $A^{2}$ does not have any invariant subspace in $C^{n}$ of dimension n − 1.

Solution

Over $C$ every matrix is triangularisable (Schur), so the span of the first k basis vectors of a triangularising basis is an invariant subspace of every dimension k — in particular n − 3 ≥ 1.

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Q9 Converse assumed Linear transformations, matrix representation, change of basis

Which one of the following statements is FALSE?

A.The product of two 2×2 real matrices of rank 2 is of rank 2.
B.The product of two 3×3 real matrices of rank 2 is of rank at most 2.
C.The product of two 3×3 real matrices of rank 2 is of rank at least 2.✓
D.The product of two 2×2 real matrices of rank 1 can be the zero matrix.

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

Q10 Finite-dimensional intuition Linear transformations, matrix representation, change of basis

Let $I_{n}$ denote the n × n identity matrix. Which one of the following statements is true?

A.If A is a real 3×2 matrix and B a real 2×3 matrix with BA $= I_{2}$ , then AB $= I_{3}$ .
B.Let A = [[3, 3], [1, 2]]. Then there is a matrix B with integer entries such that AB $= I_{2}$ .
C.Let A = [[3, 1], [1, 2]] with entries in $Z /6 Z$ . Then there is a matrix B with entries in $Z /6 Z$ such that AB $= I_{2}$ .✓
D.If A is a real non-zero 3×3 diagonal matrix, then there is a real matrix B such that AB $= I_{3}$ .

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

Q11 Execution slip Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let $A = (a_{ij})$ be the n × n real matrix with $a_{ij} =$ ij for all 1 ≤ i, j ≤ n. If n ≥ 3, which one of the following is an eigenvalue of A?

A.1
B.n
C.n(n+1)/2
D.n(n+1)(2n+1)/6✓

Solution

A = vvᵀ with v = (1, 2, …, n) has rank 1; its only non-zero eigenvalue is $v^{T} v = \sum i^{2} = n (n + 1) (2 n + 1) /6$ .

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Q12 Converse assumed Quadratic forms, positive definiteness, Sylvester's law

Let (−, −) be a symmetric bilinear form on $R^{2}$ such that there exist nonzero $v, w \in R^{2}$ with (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2×2 real symmetric matrix representing this form in the standard basis. Which one of the following statements is true?

A. $A^{2} = 0$ .
B.rank A = 1.
C.rank A = 0.
D.There exists $u \in R^{2}, u \neq = 0$ such that (u, u) = 0.✓

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

Q13 Execution slip Quadratic forms, positive definiteness, Sylvester's law

For $a \in R$ , let A_a = [[2, −1, 0], [−1, 2, −1], [0, −1, a]]. Which one of the following statements is true?

A.A_a is positive definite for all a < 3.
B.A_a is positive definite for all a > 3.✓
C.A_a is positive definite for all a ≥ −2.
D.A_a is positive definite only for finitely many values of a.

Solution

Leading principal minors: 2, 3 and det A_a = 3a − 2. Positive definite iff a > 2/3, which contains all a > 3.

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Q14 Property not inherited Subgroups, cosets, Lagrange, cyclic groups

Let G be any finite group. Which one of the following is necessarily true?

A.G is a union of proper subgroups.
B.G is a union of proper subgroups if |G| has at least two distinct prime divisors.
C.If G is abelian, then G is a union of proper subgroups.
D.G is a union of proper subgroups if and only if G is not cyclic.✓

Solution

A finite group is the union of its proper subgroups iff it is not cyclic: if it is cyclic a generator lies in no proper subgroup; if not cyclic every element generates a proper subgroup.

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Q15 Execution slip Ideals, quotient rings, prime & maximal ideals, CRT

Which one of the following is equal to $1^{37} + 2^{37} + 3^{37} + \dots + 8 8^{37}$ in $Z /89 Z$ ?

A.88
B.−88
C.−2
D.0✓

Solution

89 is prime and $(Z /89) *$ is cyclic of order 88. For a generator $g, \sum_{k}$ $k^{37} = \sum_{j}$ $g^{37 j}$ is a geometric sum equal to 0 because 88 ∤ 37 (so $g^{37} \neq = 1)$ .

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Q16 Finite-dimensional intuition Field extensions, splitting fields, finite fields

Consider the field $C$ with the Euclidean topology. Let K be a proper subfield of $C$ that is not contained in $R$ . Which one of the following statements is necessarily true?

A.K is dense in $C$ .✓
B.K is an algebraic extension of $Q$ .
C. $C$ is an algebraic extension of K.
D.The smallest closed subset of $C$ containing K is NOT a field.

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

Q17 Execution slip Cauchy–Riemann equations, harmonic functions

Let $f : C \to C$ be a real-differentiable function and define u(x, y) = Re f(x + iy), v(x, y) = Im f(x + iy). Let $\nabla u = (u_{x}, u_{y})$ denote the gradient. Which one of the following is necessarily true?

A.For $c_{1}, c_{2} \in C$ , the level curves $u = c_{1}$ and $v = c_{2}$ are orthogonal wherever they intersect.
B. $\nabla u \cdot \nabla v = 0$ at every point.
C.If f is an entire function, then $\nabla u \cdot \nabla v = 0$ at every point.✓
D.If $\nabla u \cdot \nabla v = 0$ at every point, then f is an entire function.

Solution

For entire f the Cauchy–Riemann equations give $\nabla u \cdot \nabla v = u_{x} v_{x} + u_{y} v_{y} = u_{x} (- u_{y}) + u_{y} u_{x} = 0$ . Without holomorphy there is no such relation, and orthogonal gradients alone do not force CR (e.g. f = z̄ has $\nabla u \cdot \nabla v = 0)$ .

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Q18 open Cauchy–Riemann equations, harmonic functions

Let ℍ = { $z \in C : I m z > 0$ } and f(z) = $e^{i z}$ . Which one of the following statements is true?

A.f(ℍ $) = C$ ∖{0}.
B.f(ℍ) ∩ ℍ is countable.
C.f(ℍ) is bounded.✓
D.f(ℍ) is a convex subset of $C$ .

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Q19 Execution slip Argument principle, Rouché's theorem, open mapping

How many roots does the polynomial $z^{100} - 50 z^{30} + 40 z^{10} + 6 z + 1$ have in the open disc { $z \in C$ : |z| < 1}?

A.100
B.50
C.30✓
D.0

Solution

On |z| = 1, | $- 50 z^{30}$ | = 50 > 1 + 40 + 6 + 1 = 48 ≥ | $z^{100} + 40 z^{10} + 6 z + 1$ |, so by Rouché the polynomial has as many zeros inside as $- 50 z^{30}$ : thirty.

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Q20 Execution slip Argument principle, Rouché's theorem, open mapping

Let f be a meromorphic function on an open set containing the unit circle C and its interior. Suppose f has no zeros and no poles on C, and let n_p and $n_{0}$ denote the number of poles and zeros of f inside C. Which one of the following is true?

A. $(1/2 π i) \int_{C} ($ zf)′/(zf) dz $= n_{0} - n_{p} + 1$ .✓
B. $(1/2 π i) \int_{C} ($ zf)′/(zf) dz $= n_{0} - n_{p} - 1$ .
C. $(1/2 π i) \int_{C} ($ zf)′/(zf) dz $= n_{0} - n_{p}$ .
D. $(1/2 π i) \int_{C} ($ zf)′/(zf) dz $= n_{p} - n_{0}$ .

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Q21 open Sturm–Liouville problems and Green's functions

The smallest real number $λ$ for which the problem $- y^{''} + 3 y = λ y, y (0) = 0, y (π) = 0$ has a non-trivial solution is

A.3
B.2
C.1
D.4✓

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Q22 open Classification and canonical forms

The partial differential equation $x^{2} u_{xx} - 2$ xy $u_{x}$ ᵧ $- 3 y^{2} u_{y} y + u_{x} - u_{y} = 0$ is

A.elliptic in {(x, y) : y > 0}
B.parabolic in {(x, y) : x > 0, y > 0}
C.hyperbolic in {(x, y) : xy ≠ 0}✓
D.parabolic in {(x, y) : xy ≠ 0}

Solution

$B^{2} - 4$ AC $= 4 x^{2} y^{2} + 12 x^{2} y^{2} = 16 x^{2} y^{2} > 0$ wherever xy ≠ 0: hyperbolic.

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Q23 Limit assumed to exist Laplace, heat and wave equations: separation of variables

Consider the Cauchy problem for the wave equation $u_{t} t - 4 u_{xx} = 0$ on $R \times (0, \infty)$ , with u(x, 0) = $e^{- 1/ x^{2}}$ for x ≠ 0 (and 0 at x = 0), and u_t(x, 0) = x $e^{- x^{2}}$ . Which one of the following is true?

A. $l i m_{t \to \infty}$ u(5, t) = 1✓
B. $l i m_{t \to \infty}$ u(5, t) = 2
C. $l i m_{t \to \infty}$ u(5, t) = 1/2
D. $l i m_{t \to \infty}$ u(5, t) = 0

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Q24 open Numerical ODE: Euler, Runge–Kutta

Using Euler's method with step size 0.05, the approximate value of the solution of dy/dx $= 3 x + 2 y + 1, y (1) = 1$ , at x = 1.1 (rounded to two decimal places) is

A.1.50
B.1.65
C.1.25✓
D.1.15

Solution

$y_{1} = 1 + 0.05 6 \approx 1.1225; y_{2} = 1.1225 + 0.05 3.15 + 2.245 + 1 \approx 1.1225 + 0.05 \cdot 2.529 \approx 1.249 \approx 1.25$ .

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Q25 open Isoperimetric problems

The cardinality of the set of extremals of $J [y] = \int_{0}^{1} (y^{'})^{2}$ dx subject to $y (0) = 1, y (1) = 6, \int_{0}^{1} y$ dx = 3 is

A.0
B.1✓
C.2
D.countably infinite

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Q26 open Fredholm and Volterra equations

The value of $λ$ for which the integral equation $y (x) = λ \int_{0}^{1} x^{2}$ $e^{x + t}$ y(t) dt has a non-zero solution is

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

Solution

Separable kernel: $y = c x^{2} e$ ˣ with $c = λ c \int_{0}^{1} t^{2}$ $e^{2 t}$ dt $= λ c (e^{2} - 1) /4$ , so $λ = 4/ (e^{2} - 1)$ .

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Q27 open Lagrangian formalism and generalised coordinates

Let g denote the acceleration due to gravity and a > 0. A particle of mass m glides without friction on the cycloid $x = a (θ - sin θ), y = a (1 + cos θ), 0 \leq θ \leq 2 π$ . The equation of motion of the particle is

A. $(1 - cos θ) θ$ ̈ + ½ $sin θ (θ$ ̇ $)^{2} - (g /2 a) sin θ = 0$ ✓
B. $(1 - 2 cos θ) θ$ ̈ $+ sin θ (θ$ ̇ $)^{2} - (g / a) sin θ = 0$
C. $m (1 - 2 cos θ) θ$ ̈ $+ sin θ (θ$ ̇ $)^{2} + (g / a) sin θ = 0$
D. $m (1 - 2 cos θ) θ$ ̈ $+ (m /2) sin θ (θ$ ̇ $)^{2} - (g / a) sin θ = 0$

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Q28 open Random variables, distributions, moments, MGF

Suppose X ~ Poisson(3/4). Then which of the following statements is true?

A.P(X > 9) ≥ 11/12
B.P(X < 9) ≥ 11/12✓
C. $E (X - 3/4)^{2} \geq 11/12$
D.(11/9)X ~ Poisson(11/12)

Solution

Markov: P(X ≥ 9) ≤ E[X]/9 = 1/12, so P(X < 9) ≥ 11/12. The variance is 3/4 < 11/12, and a scaled Poisson is not Poisson.

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Q29 Moments and tails Random variables, distributions, moments, MGF

Let (X, Y) have joint moment generating function $M (t_{1}, t_{2}) = (3/4 +$ $e^{t_{1}} /4)^{2}$ (5/6 + $e^{t_{2}} /6)^{3}$ . Then P(X + 2Y > 1) equals

A.1581/3456✓
B.1875/3456
C.125/3456
D.3331/3456

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Q30 Moments and tails Modes of convergence, WLLN, SLLN, CLT

Let $X_{1}, X_{2}, \dots$ be i.i.d. with CDF F(x) = 0 for x < 5 and 1 − $e^{5 - x}$ for x ≥ 5. Define $Y_{n} = min$ { $X_{1}, \dots, X_{n}$ } and $Z_{n} = n (Y_{n} - 5)$ , and let Z be standard normal. Which one of the following statements is true?

A. $lim P (1/2 < Y_{n} < 3/2) = 1$
B. $Y_{n} \to 5$ in probability as $n \to \infty$ ✓
C. $Z_{n} \to Z$ in distribution as $n \to \infty$
D. $lim P (1 < Z_{n} < 2) = Φ (2) - Φ (1)$

Solution

$Y_{n} - 5$ ~ Exp(n), so $Y_{n} \to 5$ in probability while $n (Y_{n} - 5) \to 0$ in probability (the right scaling is n, not $n)$ .

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Q31 open MLE and method of moments

The probability of a head in tossing a coin is p ∈ (0, 1). The coin is independently tossed 25 times and heads appear 10 times. The Bayes estimate of p with respect to the prior Beta(5, 5) and squared error loss is

A.3/7✓
B.3/5
C.1/2
D.2/5

Solution

Posterior is Beta(5 + 10, 5 + 15) = Beta(15, 20) with mean 15/35 = 3/7.

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Q32 open Sufficiency, completeness, UMVUE, Cramér–Rao

For n ≥ 2, let $X_{1}, \dots, X_{n}$ be a random sample with density f(x| $θ) =$ $θ x^{θ - 1}$ on $0 < x < 1, θ > 0$ unknown. Which of the following is the UMVUE for $1/ θ$ ?

A. $- (1/ n) \sum ln X_{i}$ ✓
B. $- n / \sum ln X_{i}$
C. $- (n - 1) / \sum ln X_{i}$
D. $- (2/ n) \sum ln X_{i}$

Solution

$- ln X_{i}$ ~ Exp $(θ)$ with mean $1/ θ$ , and $\sum ln X_{i}$ is complete sufficient; the sample mean of $- ln X_{i}$ is unbiased for $1/ θ$ .

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Q33 What the inference means Sufficiency, completeness, UMVUE, Cramér–Rao

For n ≥ 2, let $ε_{1}, \dots, ε_{n}$ be $i . i . d . N (0, σ^{2})$ and $Y_{i} = i α + i^{2} α^{2} + ε_{i}$ , where $σ > 0$ and $α \in R$ are unknown. Which of the following is a jointly minimal sufficient statistic for $(α, σ)$ ?

A. $(\sum Y_{i}^{2}, \sum$ iY $_{i}, \sum i^{2} Y_{i})$ ✓
B. $(\sum Y_{i}^{2}, \sum$ iY $_{i}, \sum i^{2} Y_{i}^{2})$
C. $(\sum$ iY $_{i}, \sum i^{2} Y_{i}^{2})$
D. $(\sum Y_{i}, \sum$ iY $_{i})$

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Q34 What the inference means Neyman–Pearson lemma and UMP tests

Let $X_{1}, \dots, X_{6}$ be a random sample from the gamma density f(x| $λ) = (λ^{4} /6)$ $e^{- λ x}$ $x^{3}$ for $x > 0, λ > 0$ unknown. Let $T = \sum X_{i}$ and $ψ$ be the UMP test of size 0.05 for $H_{0} : λ = 1$ against $H_{1} : λ > 1$ . With $χ_{ν, α}^{2}$ the $(1 - α) -$ th quantile of $χ_{ν}^{2}$ , the test $ψ$ rejects $H_{0}$ iff

A.T ≥ ½ $χ_{48, 0.05}^{2}$
B.T ≤ ½ $χ_{48, 0.95}^{2}$ ✓
C.T ≥ ½ $χ_{24, 0.05}^{2}$
D.T ≤ ½ $χ_{24, 0.95}^{2}$

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Q35 Execution slip Gauss–Markov, regression, ANOVA basics

Consider $Y = α + β x + ε$ with $E (ε) = 0$ . From 10 observations the OLS fit is ŷ = 1.5 + 0.8x. Suppose $\sum (y_{i} -$ ȳ $)^{2} = 5$ and $\sum (x_{i} - \overset{x}{ˉ})^{2} = 6$ . The adjusted $R^{2} ($ to two decimals) equals

A.0.74✓
B.0.83
C.0.77
D.0.84

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Q36 Execution slip Multivariate normal distribution

For n ≥ p + 1, let $X_{1}, \dots, X_{n}$ be a random sample from $N_{p} (μ, \sum)$ with $\sum$ positive definite. Define X̄ and $A = \sum (X_{i} - \overset{ˉ}{X}) (X_{i} - \overset{ˉ}{X})^{T}$ . The distribution of Trace $(A \sum$ ⁻ $^{1})$ is

A. $W_{p} (n - 1, \sum)$
B. $χ_{p}^{2}$
C. $χ_{n p}^{2}$
D. $χ_{(n - 1) p}^{2}$ ✓

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Q37 open CRD, RBD, LSD essentials

In a Latin square design, the degrees of freedom for the error sum of squares is 42. Then the degrees of freedom for the sum of squares due to treatments is

A.6
B.7✓
C.8
D.9

Solution

For a p × p Latin square the error d.f. is (p − 1)(p − 2) = 42 ⇒ p = 8, so treatments have p − 1 = 7 d.f.

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