CSIR NET June 2023 — Part C

All 16 Part C 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 C

One or more correct options. 4.75 marks, no negative marking, and credit only for exactly the right set.

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

Under which of the following conditions is the sequence { $x_{n}$ } of real numbers convergent?

A.The subsequences { $x_{2 n}$ ₊ $_{1}$ }, { $x_{2 n}$ } and { $x_{3 n}$ } are convergent and have the same limit.✓
B.The subsequences { $x_{2 n}$ ₊ $_{1}$ }, { $x_{2 n}$ } and { $x_{3 n}$ } are convergent.✓
C.The subsequences { $x_{k n}$ } $_{n}$ are convergent for every k ≥ 2.
D.lim | $x_{n}$ ₊ $_{1} - x_{n}$ | = 0.

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Q2 Limit assumed to exist Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

Let $f : R \to R$ be defined as $f (x) = 1/4 + x - x^{2}$ . Given $a \in R$ , define the sequence { $x_{n}$ } by $x_{0} = a$ and $x_{n} = f (x_{n}$ ₋ $_{1})$ for n ≥ 1. Which of the following statements are true?

A.If a = 0, then the sequence { $x_{n}$ } converges to 1/2.✓
B.If a = 0, then the sequence { $x_{n}$ } converges to −1/2.
C.The sequence { $x_{n}$ } converges for every a ∈ (−1/2, 3/2), and it converges to 1/2.✓
D.If a = 0, then the sequence { $x_{n}$ } does not converge.

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Q3 Hypothesis dropped Inverse and implicit function theorems, extrema

Consider the function $f : R^{2} \to R$ defined by $f (x, y) = x^{2} - y^{3}$ . Which of the following statements are true?

A.There is no continuous real-valued function g defined on any interval of $R$ containing 0 such that f(x, g(x)) = 0.
B.There is exactly one continuous real-valued function g defined on an interval of $R$ containing 0 such that f(x, g(x)) = 0.✓
C.There is exactly one differentiable real-valued function g defined on an interval of $R$ containing 0 such that f(x, g(x)) = 0.
D.There are two distinct differentiable real-valued functions g on an interval of $R$ containing 0 such that f(x, g(x)) = 0.

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Q4 Invariants don't determine Quadratic forms, positive definiteness, Sylvester's law

Consider the following quadratic forms over $R : (a) 6 X^{2} - 13$ XY $+ 6 Y^{2}, (b) X^{2} -$ XY $+ 2 Y^{2}, (c) X^{2} -$ XY $- 2 Y^{2}$ . Which of the following statements are true?

A.Quadratic forms (a) and (b) are equivalent.
B.Quadratic forms (a) and (c) are equivalent.✓
C.Quadratic form (b) is positive definite.✓
D.Quadratic form (c) is positive definite.

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Q5 Base field or ring Bases, dimension, rank–nullity

Let B be a 3×5 matrix with entries from $Q$ . Assume that { $v \in R^{5}$ | Bv = 0} is a three-dimensional real vector space. Which of the following statements are true?

A.{ $v \in Q^{5}$ | Bv = 0} is a three-dimensional vector space over $Q$ .✓
B.The linear transformation $T : Q^{3} \to Q^{5}$ given by T(v) = Bᵗv is injective.
C.The column span of B is two-dimensional.✓
D.The linear transformation $T : Q^{3} \to Q^{3}$ given by T(v) = BBᵗv is injective.

Solution

Rank does not change under field extension, so rank B = 5 − 3 = 2 over $Q$ as well: nullity over $Q$ is 3 and the column span is 2-dimensional. Bᵗ : $Q^{3} \to Q^{5}$ has rank 2 < 3, not injective; BBᵗ has rank 2 < 3, not injective.

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Q6 Invariants don't determine Jordan canonical form

Let V be a finite dimensional real vector space and $T_{1}, T_{2}$ be two nilpotent operators on V. Let $W_{1} =$ { $v \in V : T_{1} (v) = 0$ } and $W_{2} =$ { $v \in V : T_{2} (v) = 0$ }. Which of the following statements are FALSE?

A.If $T_{1}$ and $T_{2}$ are similar, then $W_{1}$ and $W_{2}$ are isomorphic vector spaces.
B.If $W_{1}$ and $W_{2}$ are isomorphic vector spaces, then $T_{1}$ and $T_{2}$ have the same minimal polynomial.✓
C.If $W_{1} = W_{2} = V$ , then $T_{1}$ and $T_{2}$ are similar.
D.If $W_{1}$ and $W_{2}$ are isomorphic, then $T_{1}$ and $T_{2}$ have the same characteristic polynomial.

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Q7 Invariants don't determine Sylow theorems and groups of small order

Let G be a group of order 2023. Which of the following statements are true?

A.G is an Abelian group.✓
B.G is a cyclic group.
C.G is a simple group.
D.G is not a simple group.✓

Solution

$2023 = 7 \cdot 1 7^{2} . n_{7} \equiv 1 (m o d 7)$ and $n_{7}$ | $289 \Rightarrow n_{7} = 1; n_{17}$ | 7 and $\equiv 1 (m o d 17) \Rightarrow n_{17} = 1$ . So $G ≅ Z_{7} \times P$ with P of order $1 7^{2}$ abelian: G is abelian and not simple. P may be $Z_{17} \times Z_{17}$ , so G need not be cyclic.

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Q8 Base field or ring Cauchy–Riemann equations, harmonic functions

Let f(z) be an entire function on $C$ . Which of the following statements are true?

A.f(z̄) is an entire function.
B.conj(f(z)) is an entire function.
C.conj(f(z̄)) is an entire function.✓
D.conj(f(z̄)) + f(z̄) is an entire function.

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Q9 Existence vs uniqueness Existence–uniqueness, Picard, Lipschitz

Let $f \in C^{1} (R)$ be bounded. Consider the initial-value problem (P): x′(t) = f(x(t)), t > 0, x(0) = 0. Which of the following statements are true?

A.(P) has solution(s) defined for all t > 0.✓
B.(P) has a unique solution.✓
C.(P) has infinitely many solutions.
D.The solution(s) of (P) is/are Lipschitz.✓

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Q10 Execution slip First-order PDE: Lagrange, Charpit, characteristics

Let $u : R^{2} \to R$ solve $\partial_{x} u + 2 \partial_{y} u = 0$ on $R^{2}$ with u(x, y) = sin x on the line y = 3x + 1, and let $v : R^{2} \to R$ solve $\partial_{x} v + 2 \partial_{y} v = 0$ with v(x, 0) = sin x. Let S = [0,1] × [0,1]. Which of the following statements are true?

A.u changes sign in the interior of S.
B.u(x, y) = v(x, y) along a line in S.✓
C.v changes sign in the interior of S.✓
D.v vanishes along a line in S.✓

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Q11 Execution slip Euler–Lagrange equation and standard functionals

Let y(x) and z(x) be the stationary functions (extremals) of $J (y, z) = \int_{0}^{1} [(y^{'})^{2} + (z^{'})^{2} + y^{'} z^{'}]$ dx subject to y(0) = 1, y(1) = 0, z(0) = −1, z(1) = 2. Which of the following statements are correct?

A.z(x) + 3y(x) = 2 for x ∈ [0,1].✓
B.3z(x) + y(x) = 2 for x ∈ [0,1].
C.y(x) + z(x) = 2x for x ∈ [0,1].✓
D.y(x) + z(x) = x for x ∈ [0,1].

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Q12 Dependence misread Axioms, conditional probability, independence, Bayes

Let A, B be two events in a discrete probability space with P(A) > 0 and P(B) > 0. Which of the following are necessarily true?

A.If P(A | B) = 0 then P(B | A) = 0.✓
B.If P(A | B) = 1 then P(B | A) = 1.
C.If P(A | B) > P(A) then P(B | A) > P(B).✓
D.If P(A | B) > P(B) then P(B | A) > P(A).

Solution

(1) P(A∩B) = 0 is symmetric. (3) P(A|B) > P(A) ⇔ P(A∩B) > P(A)P(B), symmetric in A, B. (2) A ⊇ B (a.s.) does not give B ⊇ A. (4) Take B ⊂ A with P(B) small.

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Q13 Boundary and endpoint Modes of convergence, WLLN, SLLN, CLT

Suppose $X_{1}, X_{2}, \dots$ are independent and identically distributed N(0,1) random variables and $Y_{n} = X_{1}^{4} + X_{2}^{4} + \dots + X_{n}^{4}$ . Which of the following probabilities converge to 1/2 as $n \to \infty$ ?

A.P{ $Y_{n} \in [0, 2 n]$ }
B.P{ $Y_{n} \in [n, 3 n]$ }✓
C.P{ $Y_{n} \in [2 n, 4 n]$ }
D.P{ $Y_{n} \in [3 n, 5 n]$ }✓

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Q14 Invariants don't determine Standard discrete and continuous distributions

Let $X_{1}$ and $X_{2}$ be independent, $X_{1}$ gamma with mean 10 and variance 10, and $X_{2}$ ~ N(3, 4). Let $f_{1}, f_{2}$ be their densities. Define Y with density $f (y) = 0.4 f_{1} (y) + q f_{2} (y)$ . Which of the following are true?

A.q = 0.6✓
B.E[Y] = 5.8✓
C.Var(Y) = 3.04
D. $Y = 0.4 X_{1} +$ qX $_{2}$

Solution

Densities integrate to 1 ⇒ q = 0.6. Mixture mean $0.4 \cdot 10 + 0.6 \cdot 3 = 5.8. E [Y^{2}] = 0.4 (10 + 100) + 0.6 (4 + 9) = 51.8$ , so $V a r (Y) = 51.8 - 5. 8^{2} = 18.16. A$ mixture is not a linear combination of the variables.

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Q15 Hypothesis dropped Sufficiency, completeness, UMVUE, Cramér–Rao

Let { $X_{i} : 1 \leq i \leq 2 n$ } be i.i.d. normal with mean $μ$ and variance 1, independent of a standard Cauchy random variable W. Which of the following statistics are consistent for $μ$ ?

A.n⁻ $^{1} \sum_{i = 1}^{n}$ $X_{i}$ ✓
B.n⁻ $^{1} \sum_{i = 1}^{2 n}$ $X_{i}$
C.n⁻ $^{1} \sum_{i = 1}^{n}$ $X_{2 i}$ ₋ $_{1}$ ✓
D.n⁻ $^{1} (\sum_{i = 1}^{n}$ $X_{i} + W)$ ✓

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Q16 What the inference means Neyman–Pearson lemma and UMP tests

Under H: X ~ p with p(x) = 1/20, and under K: X ~ q with q(x) = x/210, x ∈ {1, …, 20}. Define test functions $φ (x) = 1$ if x ≤ 2 (else 0) and $ψ (x) = 1$ if x ≥ 19 (else 0). Which of the following statements are true?

A.Size of the test $φ$ is 0.1.✓
B.Size of the test $ψ$ is 0.05.
C.(Power of the test $ψ) > 0.05$ .✓
D.(Power of the test $ψ) > ($ Power of the test $φ)$ .✓

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