CSIR NET June 2023 Maths — Part C, all 16 questions

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 { $x_{2n}$ ₊ $_{1}$ }, { $x_{2n}$ } and { $x_{3n}$ } are convergent and have the same limit.✓ Correct
B.The subsequences { $x_{2n}$ ₊ $_{1}$ }, { $x_{2n}$ } and { $x_{3n}$ } are convergent.✓ Correct
C.The subsequences { $x_{kn}$ } $_{n}$ 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 $f : \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = 1/4 + x - x^{2}$ . Given $a \in \mathbb{R}$ , 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.✓ Correct
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.✓ Correct
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 $f : \mathbb{R}^{2} \to \mathbb{R}$ 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 $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.
B.There is exactly one continuous real-valued function g defined on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.✓ Correct
C.There is exactly one differentiable real-valued function g defined on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.
D.There are two distinct differentiable real-valued functions g on an interval of $\mathbb{R}$ 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 $\mathbb{R}: (a) 6X^{2} - 13$ XY $+ 6Y^{2}, (b) X^{2} -$ XY $+ 2Y^{2}, (c) X^{2} -$ XY $- 2Y^{2}$ . Which of the following statements are true?

A.Quadratic forms (a) and (b) are equivalent.
B.Quadratic forms (a) and (c) are equivalent.✓ Correct
C.Quadratic form (b) is positive definite.✓ Correct
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 $\mathbb{Q}$ . Assume that { $v \in \mathbb{R}^{5}$ | Bv = 0} is a three-dimensional real vector space. Which of the following statements are true?

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

Solution

Rank does not change under field extension, so rank B = 5 − 3 = 2 over $\mathbb{Q}$ as well: nullity over $\mathbb{Q}$ is 3 and the column span is 2-dimensional.

Bᵗ : $\mathbb{Q}^{3} \to \mathbb{Q}^{5}$ 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.✓ Correct
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.✓ Correct
B.G is a cyclic group.
C.G is a simple group.
D.G is not a simple group.✓ Correct

Solution

$2023 = 7 \cdot 17^{2}. n_{7} \equiv 1 (mod 7)$ and $n_{7}$ | $289 \Rightarrow n_{7} = 1; n_{17}$ | 7 and $\equiv 1 (mod 17) \Rightarrow n_{17} = 1$ .

So $G \cong \mathbb{Z}_{7} \times P$ with P of order $17^{2}$ abelian: G is abelian and not simple.

P may be $\mathbb{Z}_{17} \times \mathbb{Z}_{17}$ , 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 $\mathbb{C}$ . 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.✓ Correct
D.conj(f(z̄)) + f(z̄) is an entire function.

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Q9 Existence vs uniqueness Existence–uniqueness, Picard, Lipschitz

Let $f \in C^{1}(\mathbb{R})$ 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.✓ Correct
B.(P) has a unique solution.✓ Correct
C.(P) has infinitely many solutions.
D.The solution(s) of (P) is/are Lipschitz.✓ Correct

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

Let $u : \mathbb{R}^{2} \to \mathbb{R}$ solve $\partial_{x}u + 2\partial_yu = 0$ on $\mathbb{R}^{2}$ with u(x, y) = sin x on the line y = 3x + 1, and let $v : \mathbb{R}^{2} \to \mathbb{R}$ solve $\partial_{x}v + 2\partial_yv = 0$ 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.✓ Correct
C.v changes sign in the interior of S.✓ Correct
D.v vanishes along a line in S.✓ Correct

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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].✓ Correct
B.3z(x) + y(x) = 2 for x ∈ [0,1].
C.y(x) + z(x) = 2x for x ∈ [0,1].✓ Correct
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.✓ Correct
B.If P(A | B) = 1 then P(B | A) = 1.
C.If P(A | B) > P(A) then P(B | A) > P(B).✓ Correct
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 $Y_{n} = X_{1}^{4} + X_{2}^{4} + \cdots + X_{n}^{4}$ . Which of the following probabilities converge to 1/2 as $n \to \infty$ ?

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

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

Solution

Densities integrate to 1 ⇒ q = 0.6.

Mixture mean 0.4·10 + 0.6·3 = 5.8.

$E[Y^{2}] = 0.4(10 + 100) + 0.6(4 + 9) = 51.8$ , so $Var(Y) = 51.8 - 5.8^{2} = 18.16$ .

A mixture is not a linear combination of the variables.

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

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

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

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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 $\varphi(x) = 1$ if x ≤ 2 (else 0) and $\psi(x) = 1$ if x ≥ 19 (else 0). Which of the following statements are true?

A.Size of the test $\varphi$ is 0.1.✓ Correct
B.Size of the test $\psi$ is 0.05.
C.(Power of the test $\psi) > 0.05$ .✓ Correct
D.(Power of the test $\psi) > ($ Power of the test $\varphi)$ .✓ Correct

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CSIR NET June 2023 — Part C

Part C