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PYQ Paper

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.

Q1Hypothesis droppedSeries: comparison, ratio, root, Raabe, condensation, alternating, rearrangements

Consider the series ∑n=1∞\sum_{n=1}^\infty ana_{n}, where an=a_{n} = (−1)n+1(n+1−n)(-1)^{n+1}(\sqrt{n+1} - \sqrt{n}). Which of the following statements is true?

  1. A.The series is divergent.
  2. B.The series is convergent.
  3. C.The series is conditionally convergent.✓ Correct
  4. D.The series is absolutely convergent.

Solution

n+1−n=1/(n+1+n)\sqrt{n+1} - \sqrt{n} = 1/(\sqrt{n+1} + \sqrt{n}) decreases to 0, so the alternating series converges (Leibniz).

The absolute series behaves like ∑1/(2n)\sum 1/(2\sqrt{n}), which diverges.

Hence conditionally convergent — option 3 is the most precise true statement (official key: 3).

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Q2Limit assumed to existlimsup, liminf and subsequential limits

Which of the following assertions is correct?

  1. A.limsupnexp⁡(cos⁡((nπ+(−1)n⋅2e)/(2n)))>1limsup_{n} \exp(\cos((n\pi + (-1)^{n}\cdot2e)/(2n))) > 1.
  2. B.limnexp⁡(loge((nπ2+(−1)ne2)/(7n)))lim_{n} \exp(log_{e}((n\pi^{2} + (-1)^{n}e^{2})/(7n))) does not exist.
  3. C.liminfnexp⁡(sin⁡((nπ+(−1)n⋅2e)/(2n)))<πliminf_{n} \exp(\sin((n\pi + (-1)^{n}\cdot2e)/(2n))) < \pi.✓ Correct
  4. D.limnexp⁡(tan⁡((nπ2+(−1)ne2)/(7n)))lim_{n} \exp(\tan((n\pi^{2} + (-1)^{n}e^{2})/(7n))) does not exist.

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

Q3Pointwise vs uniformContinuity, uniform continuity, Lipschitz

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

  1. A.f(x) = sin(1/x)
  2. B.f(x) = e−1/x2e^{-1/x^{2}}✓ Correct
  3. C.f(x) = eˣ cos(1/x)
  4. D.f(x)=cos⁡x⋅cos⁡(π/x)f(x) = \cos x \cdot \cos(\pi/x)

Solution

A continuous function on (0,1) is uniformly continuous iff it extends continuously to [0,1]. e−1/x2e^{-1/x^{2}} → 0 as x → 0⁺, so it extends; the other three oscillate without a limit at 0.

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Q4Invariants don't determineEigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let A be a 3×3 real matrix whose characteristic polynomial p(T) is divisible by T2T^{2}. Which of the following statements is true?

  1. A.The eigenspace of A for the eigenvalue 0 is two-dimensional.
  2. B.All the eigenvalues of A are real.✓ Correct
  3. C.A3=0A^{3} = 0.
  4. D.A is diagonalizable.

Solution

p(T)=T2(T−c)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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Q5Invariants don't determineEigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let T be a linear operator on R3\mathbb{R}^{3}. Let f(X)∈R[X]f(X) \in \mathbb{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]\mathbb{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]\mathbb{R}[X], then the linear operator g(T) is zero. Which of the following is true?

  1. A.Both (a) and (b) are true.
  2. B.Both (a) and (b) are false.
  3. C.(a) is true and (b) is false.
  4. D.(a) is false and (b) is true.✓ Correct

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

Q6Execution slipGram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

Let x=(x1,…,xn)x = (x_{1}, \dots, x_{n}) and y=(y1,…,yn)y = (y_{1}, \dots, y_{n}) denote vectors in Rn\mathbb{R}^{n} for a fixed n ≥ 2. Which of the following defines an inner product on Rn\mathbb{R}^{n}?

  1. A.⟨x, y⟩ =∑i,j=1n= \sum_{i,j=1}^n xiyjx_{i}y_{j}
  2. B.⟨x, y⟩ =∑i,j=1n= \sum_{i,j=1}^n (xi2+yj2)(x_{i}^{2} + y_{j}^{2})
  3. C.⟨x, y⟩ =∑j=1n= \sum_{j=1}^n j3xjyjj^{3} x_{j}y_{j}✓ Correct
  4. D.⟨x, y⟩ =∑j=1n= \sum_{j=1}^n xjynx_{j} y_{n}₋j_{j}₊1_{1}

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

Q7Property not inheritedGroup 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∈Nn \in \mathbb{N} such that gpng^{p^{n}} = 1. Which of the following statements is FALSE?

  1. A.If |G| =p6= p^{6}, then G has a subgroup of index p2p^{2}.
  2. B.If |G| =p6= p^{6}, then G has at least five normal subgroups.
  3. C.Center of G can be infinite.
  4. D.There exists G with |G| =p6= p^{6} such that G has exactly six normal subgroups.✓ Correct

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

Q8Execution slipIdeals, quotient rings, prime & maximal ideals, CRT

The number of solutions of the equation x2=1x^{2} = 1 in the ring Z/105Z\mathbb{Z}/105\mathbb{Z} is

  1. A.0
  2. B.2
  3. C.4
  4. D.8✓ Correct

Solution

By CRT, Z/105Z≅Z/3×Z/5×Z/7;x2=1\mathbb{Z}/105\mathbb{Z} \cong \mathbb{Z}/3 \times \mathbb{Z}/5 \times \mathbb{Z}/7; x^{2} = 1 has exactly 2 solutions in each odd prime field, giving 23=82^{3} = 8.

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Q9openPolynomial rings and irreducibility tests

How many real roots does the polynomial x3+3x−2023x^{3} + 3x - 2023 have?

  1. A.0
  2. B.1✓ Correct
  3. C.2
  4. D.3

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

Q10Finite-dimensional intuitionMeasurable sets and functions

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

  1. A.S is in bijection with the set of rational numbers.
  2. B.S is in bijection with the set of real numbers.
  3. C.S is in bijection with S × S.✓ Correct
  4. 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.

Q11Execution slipLaurent series, classification of singularities, Casorati–Weierstrass

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

  1. A.∑l=0∞\sum_{l=0}^\infty 1/(l+1)!
  2. B.∑l=0∞\sum_{l=0}^\infty 1/(l!(l+1))
  3. C.∑l=0∞\sum_{l=0}^\infty 1/(l!(l+1)!)✓ Correct
  4. D.∑l=0∞\sum_{l=0}^\infty 1/((l2+l)!)1/((l^{2}+l)!)

Solution

Multiply the series eᶻ =∑zm/m= \sum z^{m}/m! and e1/ze^{1/z} =∑= \sum z−k/kz^{-k}/k!; the coefficient of z−1z^{-1} comes from k=m+1:∑m1/(m!(m+1)!)k = m + 1: \sum_m 1/(m!(m+1)!).

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Q12Execution slipPower series and analyticity

Consider the function f defined by f(z)=1/(1−z−z2)f(z) = 1/(1 - z - z^{2}) for z∈Cz \in \mathbb{C} such that 1−z−z2≠01 - z - z^{2} \ne 0. Which of the following statements is true?

  1. A.f is an entire function.
  2. B.f has a simple pole at z = 0.
  3. C.f has a Taylor series expansion f(z)=∑anznf(z) = \sum a_{n}z^{n}, where a0=1,a1=0a_{0} = 1, a_{1} = 0 and ana_{n}₊2=an+an_{2} = a_{n} + a_{n}₊1_{1} for n ≥ 0.
  4. D.f has a Taylor series expansion f(z)=∑anznf(z) = \sum a_{n}z^{n}, where a0=1,a1=1a_{0} = 1, a_{1} = 1 and ana_{n}₊2=an+an_{2} = a_{n} + a_{n}₊1_{1} for n ≥ 0.✓ Correct

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

Q13Numerical convergenceLinear ODE, Wronskian, variation of parameters, systems

Suppose x(t) is the solution of the initial value problem in R2\mathbb{R}^{2}: ẋ = Ax, x(0)=x0x(0) = x_{0}, where A = [[5, 4], [1, 2]]. Which of the following statements is true?

  1. A.x(t) is a bounded solution for some x0≠0x_{0} \ne 0.
  2. B.e−6te^{-6t}|x(t)| → 0 as t→∞t \to \infty, for all x0≠0x_{0} \ne 0.
  3. C.e−te^{-t}|x(t)| →∞\to \infty as t→∞t \to \infty, for all x0≠0x_{0} \ne 0.
  4. D.e−10te^{-10t}|x(t)| → 0 as t→∞t \to \infty, for all x0≠0x_{0} \ne 0.✓ Correct

Solution

Eigenvalues of A are 6 and 1 (trace 7, det 6), both positive: every non-zero solution grows like e6te^{6t} or ete^{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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Q14Execution slipFirst-order PDE: Lagrange, Charpit, characteristics

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

  1. A.2
  2. B.3
  3. C.1/2✓ Correct
  4. D.1/3

Solution

Characteristics: dx/dy = u with u constant along them, so x=x0+u⋅yx = x_{0} + u\cdot{}y and u=x0u = x_{0}.

Hence u = x/(1 + y), and u(2, 3) = 1/2.

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Q15Hypothesis droppedEuler–Lagrange equation and standard functionals

Consider the variational problem (P):J(y)=∫01[(y′)2−y(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?

  1. A.(P) has no stationary function (extremal).
  2. B.y ≡ 0 is the only stationary function (extremal) for (P).
  3. C.(P) has a unique stationary function (extremal) y not identically equal to 0.✓ Correct
  4. 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.

Q16Execution slipFredholm and Volterra equations

For the unknown y:[0,1]→Ry : [0,1] \to \mathbb{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∫01K(x,t)y(t)y(x) = 2\int_{0}^{1} K(x, t) y(t) dt. Which of the following is the kernel K(x, t)?

  1. A.K(x,t) = t(1 − x) for t < x; x(1 − t) for t > x✓ Correct
  2. B.K(x,t)=t2(1−x)K(x,t) = t^{2}(1 - x) for t<x;x2(1−t)t < x; x^{2}(1 - t) for t > x
  3. C.K(x,t)=t(1−x)K(x,t) = \sqrt{t}(1 - x) for t<x;x(1−t)t < x; \sqrt{x}(1 - t) for t > x
  4. D.K(x,t)=t3(1−x)K(x,t) = \sqrt{t^{3}}(1 - x) for t<x;x3(1−t)t < x; \sqrt{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∫Gyy = 2\int{}G y.

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Q17Execution slipStandard discrete and continuous distributions

Let X1,X2,X3,X4X_{1}, X_{2}, X_{3}, X_{4} be i.i.d. Bernoulli(1/3) and X(1)≤⋯≤X(4)X_{(1)} \le \cdots \le X_{(4)} the order statistics. Which of the following is true?

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

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

Q18openSufficiency, completeness, UMVUE, Cramér–Rao

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

  1. A.λ\lambda⁻1^{1}✓ Correct
  2. B.λ\lambda
  3. C.λ2\lambda^{2}
  4. D.e−λe^{-\lambda}

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

Q19Execution slipLikelihood ratio and standard tests

Let X1,…,X7X_{1}, \dots, X_{7} and Y1,…,Y9Y_{1}, \dots, Y_{9} be independent random samples from continuous CDFs F and G. For the Wald–Wolfowitz run test of H0:F=GH_{0}: F = G vs H1:F≠GH_{1}: F \ne G, let R be the total number of runs in the combined ordered arrangement. Which of the following is true?

  1. A.PH0(RP_{H_{0}}(R = 6) = 28/286, PH0(RP_{H_{0}}(R = 9) = 28/143.
  2. B.PH0(RP_{H_{0}}(R = 6) = 21/286, PH0(RP_{H_{0}}(R = 9) = 15/286.
  3. C.PH0(RP_{H_{0}}(R = 6) = 21/286, PH0(RP_{H_{0}}(R = 9) = 28/143.✓ Correct
  4. D.PH0(RP_{H_{0}}(R = 6) = 21/286, PH0(RP_{H_{0}}(R = 9) = 15/286.

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

Q20Execution slipGauss–Markov, regression, ANOVA basics

Consider the simple linear regression model Yi=βxi+εi,i=1,…,nY_{i} = \beta{}x_{i} + \varepsilon_{i}, i = 1, \dots, n, where E(εi)=0,Cov(εi,εk)=0E(\varepsilon_{i}) = 0, Cov(\varepsilon_{i}, \varepsilon_{k}) = 0 for i ≠ k and Var(εi)=xi2σ2Var(\varepsilon_{i}) = x_{i}^{2}\sigma^{2}. The best linear unbiased estimator of β\beta is

  1. A.∑Yixi/∑xi2\sum Y_{i}x_{i} / \sum x_{i}^{2}
  2. B.∑Yi/∑xi\sum Y_{i} / \sum x_{i}
  3. C.(1/n)∑Yi/xi(1/n) \sum Y_{i}/x_{i}✓ Correct
  4. D.(1/n)∑Yixi/xi2(1/n) \sum Y_{i}x_{i} / x_{i}^{2}

Solution

Heteroscedastic errors: divide by xix_{i} to get Yi/xi=β+εi/xiY_{i}/x_{i} = \beta + \varepsilon_{i}/x_{i} with constant variance σ2\sigma^{2}; the BLUE is the plain mean of Yi/xi(Y_{i}/x_{i} (weighted least squares with weights 1/xi2)1/x_{i}^{2}).

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Q21openMultivariate normal distribution

Let X=(X1,X2)TX = (X_{1}, X_{2})^{T} be bivariate normal with mean (0, 0)ᵀ and covariance ∑=[[5,−3],[−3,10]]\sum = [[5, -3], [-3, 10]]. The mean vector and covariance matrix of Y=(X1,5−2X2)TY = (X_{1}, 5 - 2X_{2})^{T} are

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

Solution

E[5−2X2]=5;Var(5−2X2)=4⋅10=40;Cov(X1,5−2X2)=−2⋅(−3)=6E[5 - 2X_{2}] = 5; Var(5 - 2X_{2}) = 4\cdot10 = 40; Cov(X_{1}, 5 - 2X_{2}) = -2\cdot(-3) = 6.

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Q22Execution slipMultivariate normal distribution

Suppose X=(X1,X2,X3,X4)TX = (X_{1}, X_{2}, X_{3}, X_{4})^{T} ~ N4(0,I2N_{4}(0, I_{2} ⊗ ∑)\sum) with ∑=[[2,−1],[−1,2]]\sum = [[2, -1], [-1, 2]]. Define Z=[[X1,X2],[X3,X4]]Z = [[X_{1}, X_{2}], [X_{3}, X_{4}]] and Q = ZᵀZ. If Wm(n,∑)W_{m}(n, \sum) denotes a Wishart distribution of order m with n degrees of freedom, the distribution of Q11+Q12+Q21+Q22Q_{11} + Q_{12} + Q_{21} + Q_{22} is

  1. A.W1(2,2)W_{1}(2, 2)✓ Correct
  2. B.W1(1,2)W_{1}(1, 2)
  3. C.W1(2,1)W_{1}(2, 1)
  4. D.2χ422\chi^{2}_{4}

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