CSIR NET December 2023 — Part C

All 59 Part C 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 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

Let x be a real number. Which of the following statements are true?

A.There exists an integer n ≥ 1 such that $n^{2} sin (1/ n) \geq x$ .✓
B.There exists an integer n ≥ 1 such that n cos(1/n) ≥ x.✓
C.There exists an integer n ≥ 1 such that n $e^{- n}$ ≥ x.
D.There exists an integer n ≥ 2 such that n (log $n)^{- 1}$ ≥ x.✓

Solution

$n^{2} sin (1/ n)$ ~ n, n cos(1/n) ~ n and n/log n all tend to $+ \infty$ , so each eventually exceeds any x. n $e^{- n}$ ≤ 1/e is bounded, so it fails for x > 1/e.

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Q2 Converse assumed Continuity, uniform continuity, Lipschitz

Let $f : R \to R$ be a continuous function such that |f(x) − f(y)| ≥ log(1 + |x − y|) for all $x, y \in R$ . Which of the following statements are true?

A.f is necessarily one-one.✓
B.f need not be one-one.
C.f is necessarily onto.✓
D.f need not be onto.

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Q3 Pointwise vs uniform Continuity, uniform continuity, Lipschitz

Let $f : [0, \infty) \to R$ be the periodic function of period 1 given by f(x) = 1 − |2x − 1| for x ∈ [0, 1], and define $g : [0, \infty) \to R$ by $g (x) = f (x^{2})$ . Which of the following statements are true?

A.f is continuous on $[0, \infty)$ .✓
B.f is uniformly continuous on $[0, \infty)$ .✓
C.g is continuous on $[0, \infty)$ .✓
D.g is uniformly continuous on $[0, \infty)$ .

Solution

f is a continuous tent function with f(0) = f(1) = 0, so its periodic extension is continuous; continuous periodic ⇒ uniformly continuous. g = f∘ $x^{2}$ is continuous but its oscillations speed up as $x \to \infty ($ the tents get compressed), so it is not uniformly continuous.

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Q4 Boundary and endpoint Riemann integration and criteria

For a real number $λ$ , consider the improper integrals $I_{λ} = \int_{0}^{1}$ dx $/ (1 - x)^{λ}$ and $K_{λ} = \int_{1}^{\infty}$ dx $/ x^{λ}$ . Which of the following statements are true?

A.There exists $λ$ such that $I_{λ}$ converges, but $K_{λ}$ does not converge.✓
B.There exists $λ$ such that $K_{λ}$ converges, but $I_{λ}$ does not converge.✓
C.There exists $λ$ such that $I_{λ}, K_{λ}$ both converge.
D.There exists $λ$ such that neither $I_{λ}$ nor $K_{λ}$ converges.✓

Solution

$I_{λ}$ converges iff $λ < 1; K_{λ}$ converges iff $λ > 1$ . So they never converge together, and at $λ = 1$ neither does.

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Q5 Hypothesis dropped Riemann integration and criteria

Suppose that $f : [- 1, 1] \to R$ is continuous. Which of the following imply that f is identically zero on [−1, 1]?

A. $\int$ ₋ $_{1}^{1} f (x) x^{n}$ dx = 0 for all n ≥ 0.✓
B. $\int$ ₋ $_{1}^{1} f (x) p (x)$ dx = 0 for all real polynomials p(x).✓
C. $\int$ ₋ $_{1}^{1} f (x) x^{n}$ dx = 0 for all n ≥ 0 odd.
D. $\int$ ₋ $_{1}^{1} f (x) x^{n}$ dx = 0 for all n ≥ 0 even.

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Q6 Boundary and endpoint Pointwise vs uniform convergence, M-test, Dini

Let $(f_{n})$ be the sequence of functions on [0, 1] defined by $f_{n} (x) = x^{n} lo g ((1 + x) /2)$ . Which of the following statements are true?

A. $(f_{n})$ converges pointwise on [0, 1].✓
B. $(f_{n})$ converges uniformly on compact subsets of [0, 1) but not on [0, 1).
C. $(f_{n})$ converges uniformly on [0, 1) but not on [0, 1].
D. $(f_{n})$ converges uniformly on [0, 1].✓

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Q7 open Partial derivatives, differentiability, chain rule

For real numbers a, b, c, d, e, f, consider the function $F : R^{2} \to R^{2}$ given by F(x, y) = (ax + by + c, dx + ey + f). Which of the following statements are true?

A.F is continuous.✓
B.F is uniformly continuous.✓
C.F is differentiable.✓
D.F has partial derivatives of all orders.✓

Solution

An affine map is Lipschitz (hence uniformly continuous), differentiable with constant derivative, and all higher partials are zero.

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Q8 Converse assumed Partial derivatives, differentiability, chain rule

For a differentiable surjective function f : (0,1) → (0,1), consider $F : (0, 1)^{2} \to (0, 1)^{2}, F (x, y) = (f (x), f (y))$ . If f′(x) ≠ 0 for every x ∈ (0,1), which of the following statements are true?

A.F is injective.✓
B.f is increasing.
C.For every $(x^{'}, y^{'}) \in (0, 1)^{2}$ , there exists a unique $(x, y) \in (0, 1)^{2}$ such that F(x, y) = (x′, y′).✓
D.The total derivative DF(x, y) is invertible for all $(x, y) \in (0, 1)^{2}$ .✓

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Q9 Finite-dimensional intuition Partial derivatives, differentiability, chain rule

Let $p : R^{2} \to R$ be the function p(x, y) = x. Which of the following statements are true?

A.Let $A_{1} =$ { $x^{2} + y^{2} < 1$ }. For each $γ \in p (A_{1})$ there exists $ε > 0$ with $(γ - ε, γ + ε) \subseteq p (A_{1})$ .✓
B.Let $A_{2} =$ { $x^{2} + y^{2} \leq 1$ }. For each $γ \in p (A_{2})$ there exists $ε > 0$ with $(γ - ε, γ + ε) \subseteq p (A_{2})$ .
C.Let $A_{3} =$ {xy = 0}. For each $γ \in p (A_{3})$ there exists $ε > 0$ with $(γ - ε, γ + ε) \subseteq p (A_{3})$ .✓
D.Let $A_{4} =$ {xy = 1}. For each $γ \in p (A_{4})$ there exists $ε > 0$ with $(γ - ε, γ + ε) \subseteq p (A_{4})$ .✓

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Q10 Hypothesis dropped Measurable sets and functions

Let { $A_{n}$ } $_{n \geq 1}$ be a collection of non-empty subsets of $Z$ such that $A_{n} \cap A_{m} = \emptyset$ for m ≠ n. If $Z = \cup_{n \geq 1}$ $A_{n}$ , then which of the following statements are necessarily true?

A. $A_{n}$ is finite for every integer n ≥ 1.
B. $A_{n}$ is finite for some integer n ≥ 1.
C. $A_{n}$ is infinite for some integer n ≥ 1.
D. $A_{n}$ is countable (finite or infinite) for every integer n ≥ 1.✓

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Q11 Invariants don't determine Linear transformations, matrix representation, change of basis

Let $T : R^{5} \to R^{5}$ be an $R -$ linear transformation. Suppose that (1, −1, 2, 4, 0), (4, 6, 1, 6, 0) and (5, 5, 3, 9, 0) span the null space of T. Which of the following statements are true?

A.The rank of T is equal to 2.✓
B.Suppose that for every vector $v \in R^{5}$ there exists n such that $T^{n} v = 0$ . Then $T^{2}$ must be zero.
C.Suppose that for every vector $v \in R^{5}$ there exists n such that $T^{n} v = 0$ . Then $T^{3}$ must be zero.✓
D.(−2, −8, 3, 2, 0) is contained in the null space of T.✓

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Q12 Finite-dimensional intuition Bases, dimension, rank–nullity

Let 𝔽 be a finite field and V be a finite dimensional non-zero 𝔽-vector space. Which of the following can NEVER be true?

A.V is the union of 2 proper subspaces.✓
B.V is the union of 3 proper subspaces.
C.V has a unique basis.
D.V has precisely two bases.

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Q13 Execution slip Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Consider A = [[1, 4], [2, 3]]. Suppose $A^{5} - 4 A^{4} - 7 A^{3} + 11 A^{2} - A - 10 I =$ aA + bI for some $a, b \in Z$ . Which of the following statements are true?

A.a + b > 8.
B.a + b < 7.✓
C.a + b is divisible by 2.✓
D.a > b.

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Q14 Execution slip Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let X, Y be two n × n real matrices such that XY $= X^{2} + X + I$ . Which of the following statements are necessarily true?

A.X is invertible.✓
B.X + I is invertible.
C.XY = YX.✓
D.Y is invertible.

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Q15 open Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let A be an n × n real symmetric matrix. Which of the following statements are necessarily true?

A.A is diagonalizable.✓
B.If Aᵏ = I for some positive integer k, then $A^{2} = I$ .✓
C.If Aᵏ = 0 for some positive integer k, then $A^{2} = 0$ .✓
D.All eigenvalues of A are real.✓

Solution

Spectral theorem: A = QDQᵀ with real D. Aᵏ = I forces each real eigenvalue to satisfy $λ^{k} = 1$ , so $λ = \pm 1$ and $A^{2} = I; A^{k} = 0$ forces all $λ = 0$ so A = 0.

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Q16 Invariants don't determine Jordan canonical form

Suppose a 7 × 7 block diagonal complex matrix A has blocks (0), (1), [[0, 1], [0, 0]] and the 3×3 block with $2 π i$ on the diagonal, a 1 in position (1,2) and zeros elsewhere, along the diagonal. Which of the following statements are true?

A.The characteristic polynomial of A is $x^{3} (x - 1) (x - 2 π i)^{3}$ .✓
B.The minimal polynomial of A is $x^{2} (x - 1) (x - 2 π i)^{3}$ .
C.The dimensions of the eigenspaces for $0, 1, 2 π i$ are 2, 1, 3 respectively.
D.The dimensions of the eigenspaces for $0, 1, 2 π i$ are 2, 1, 2 respectively.✓

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Q17 open Quadratic forms, positive definiteness, Sylvester's law

Let A be a real diagonal matrix with characteristic polynomial $λ^{3} - 2 λ^{2} - λ + 2$ . Define a bilinear form ⟨v, w⟩ = vᵗAw on $R^{3}$ . Which of the following statements are true?

A.A is positive definite.
B. $A^{2}$ is positive definite.✓
C.There exists a nonzero $v \in R^{3}$ such that ⟨v, v⟩ = 0.✓
D.rank A = 2.

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Q18 Standard counterexample Ideals, quotient rings, prime & maximal ideals, CRT

Let $R = Z [X] / (X^{2} + 1)$ and $ψ : Z [X] \to R$ be the natural quotient map. Which of the following statements are true?

A.R is isomorphic to a subring of $C$ .✓
B.For any prime number $p \in Z$ , the ideal generated by $ψ (p)$ is a proper ideal of R.✓
C.R has infinitely many prime ideals.✓
D.The ideal generated by $ψ (X)$ is a prime ideal in R.

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Q19 open Ideals, quotient rings, prime & maximal ideals, CRT

Let $n \in Z$ be such that n ≡ 1 (mod 7) and n ≡ 4 (mod 15). Which of the following statements are true?

A.n ≡ 1 (mod 3).✓
B.n ≡ 1 (mod 35).
C.n ≡ 1 (mod 21).✓
D.n ≡ 1 (mod 5).

Solution

n ≡ 4 (mod 15) gives n ≡ 1 (mod 3) and n ≡ 4 (mod 5). With n ≡ 1 (mod 7), CRT gives n ≡ 1 (mod 21); n ≢ 1 (mod 5) kills the mod-35 claim.

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Q20 Base field or ring Polynomial rings and irreducibility tests

Let $f (X) = X^{2} + X + 1$ and $g (X) = X^{2} + X - 2$ be polynomials in $Z [X]$ . Which of the following statements are true?

A.For all prime numbers p, f(X) mod p is irreducible in $(Z / p Z) [X]$ .
B.There exists a prime number p such that g(X) mod p is irreducible in $(Z / p Z) [X]$ .
C.g(X) is irreducible in $Q [X]$ .
D.f(X) is irreducible in $Q [X]$ .✓

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Q21 Execution slip Group actions, class equation, p-groups

Let G be the group (under matrix multiplication) of 2 × 2 invertible matrices with entries from $Z /9 Z$ . Let a be the order of G. Which of the following statements are true?

A.a is divisible by $3^{4}$ .✓
B.a is divisible by $2^{4}$ .✓
C.a is not divisible by 48.
D.a is divisible by $3^{6}$ .

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Q22 open Galois theory essentials

Let $f (X) = X^{3} - 2 \in Q [X]$ and let $K \subset C$ be the splitting field of f(X) over $Q$ . Let $ω =$ $e^{2 π i /3}$ . Which of the following statements are true?

A.The Galois group of K over $Q$ is the symmetric group $S_{3}$ .✓
B.The Galois group of K over $Q (ω)$ is the symmetric group $S_{3}$ .
C.The Galois group of K over $Q$ is $Z /3 Z$ .
D.The Galois group of K over $Q (ω)$ is $Z /3 Z$ .✓

Solution

$K = Q (32, ω)$ has degree 6 over $Q$ with non-abelian Galois group $S_{3}$ ; over $Q (ω)$ the degree is 3, so the group is $Z /3$ .

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Q23 Property not inherited Continuity, homeomorphism, separation axioms

Consider $R^{2}$ with the Euclidean topology and $Q^{2} \subset R^{2}$ with the subspace topology. Which of the following statements are true?

A. $Q^{2}$ is connected.
B.If A is a non-empty connected subset of $Q^{2}$ , then A has exactly one element.✓
C. $Q^{2}$ is Hausdorff.✓
D.{ $(x, y) \in Q^{2}$ | $x^{2} + y^{2} = 1$ } is compact in the subspace topology.

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Q24 Standard counterexample Power series and analyticity

For every n ≥ 1, consider the entire function $p_{n} (z) = \sum_{k = 0}^{n}$ zᵏ/k!. Which of the following statements are true?

A.The sequence $(p_{n})$ converges to an entire function uniformly on compact subsets of $C$ .✓
B.For all $n \geq 1, p_{n}$ has a zero in the set { $z \in C$ : |z| ≤ 2023}.
C.There exists a sequence $(z_{n})$ of complex numbers such that lim | $z_{n}$ | $= \infty$ and $p_{n} (z_{n}) = 0$ for all n ≥ 1.✓
D.Let $S_{n}$ be the set of all zeros of $p_{n}$ . If $a_{n} =$ $mi n_{z \in S_{n}}$ |z|, then $a_{n} \to \infty$ as $n \to \infty$ .✓

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Q25 Standard counterexample Liouville, Morera, maximum modulus principle

Let X be an uncountable subset of $C$ and let $f : C \to C$ be an entire function. Assume that for every z ∈ X there exists an integer n ≥ 1 such that $f^{(n)} (z)$ = 0. Which of the following statements are necessarily true?

A.f = 0.
B.f is a constant function.
C.There exists a compact subset K of $C$ such that $f^{- 1} (K)$ is not compact.
D.f is a polynomial.✓

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Q26 Standard counterexample Liouville, Morera, maximum modulus principle

Let $Ω_{1} =$ { $z \in C$ : |z| < 1} and $Ω_{2} = C$ . Which of the following statements are true?

A.There exists a holomorphic surjective map $f : Ω_{1} \to Ω_{2}$ .✓
B.There exists a holomorphic surjective map $f : Ω_{2} \to Ω_{1}$ .
C.There exists a holomorphic injective map $f : Ω_{1} \to Ω_{2}$ .✓
D.There exists a holomorphic injective map $f : Ω_{2} \to Ω_{1}$ .

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Q27 Execution slip Residue theorem and standard contour integrals

For an integer k, consider the contour integral $I_{k} = \int_{∣ z ∣ = 1}$ eᶻ/zᵏ dz. Which of the following statements are true?

A. $I_{k} = 0$ for every integer k.
B. $I_{k} \neq = 0$ if k ≥ 1.✓
C.| $I_{k}$ | ≤ | $I_{k}$ ₊ $_{1}$ | for every integer k.
D. $l i m_{k \to \infty}$ | $I_{k}$ | $= \infty$ .

Solution

$I_{k} = 2 π i \cdot ($ coefficient of $z^{k - 1}$ in eᶻ $) = 2 π i / (k - 1)$ ! for k ≥ 1 and 0 for k ≤ 0. These values decrease to 0, so (3) and (4) fail.

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Q28 Existence vs uniqueness Existence–uniqueness, Picard, Lipschitz

Consider the initial value problem y′ = y + ½| $sin (y^{2})$ |, x > 0, y(0) = −1. Which of the following statements are true?

A.There exists an $α \in (0, \infty)$ such that $l i m_{x \to α^{-}}$ |y(x)| $= \infty$ .
B.y(x) exists on $(0, \infty)$ and it is monotone.✓
C.y(x) exists on $(0, \infty)$ , but not bounded below.✓
D.y(x) exists on $(0, \infty)$ , but not bounded above.

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Q29 Existence vs uniqueness Existence–uniqueness, Picard, Lipschitz

Consider the problem $y^{'} = (1 - y^{2})^{10} cos y, y (0) = 0$ . Let J be the maximal interval of existence and K be the range of the solution. Which of the following statements are true?

A. $J = R$ ✓
B.K = (−1, 1)✓
C.J = (−1, 1)
D.K = [−1, 1]

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Q30 open Existence–uniqueness, Picard, Lipschitz

Consider the initial value problem $x^{2} y^{''} - 2 x^{2} y^{'} + (4 x - 2) y = 0, y (0) = 0$ . Suppose $y = φ (x)$ is a polynomial solution satisfying $φ (1) = 1$ . Which of the following statements are true?

A. $φ (4) = 16$ ✓
B. $φ (2) = 2$
C. $φ (5) = 25$ ✓
D. $φ (3) = 3$

Solution

Try $y = x^{2} : 2 x^{2} - 4 x^{3} + 4 x^{3} - 2 x^{2} = 0$ . So $φ (x) = x^{2}$ , giving $φ (4) = 16$ and $φ (5) = 25$ .

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Q31 Existence vs uniqueness First-order PDE: Lagrange, Charpit, characteristics

Consider the Cauchy problem $u \cdot u_{x} + u_{y} = 1$ for $(x, y) \in R \times (0, \infty)$ , with u(x, 0) = kx, $x \in R$ , for a real parameter k. For which of the following values of k does the problem have a solution defined on $R \times (0, \infty)$ ?

A.k = 0✓
B.k = −2
C.k = 4✓
D.k = 1✓

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Q32 Existence vs uniqueness Laplace, heat and wave equations: separation of variables

Let B be the open unit disc in $R^{2}, \partial B$ its boundary and $\overset{ˉ}{B} = B \cup \partial B$ . For $λ \in (0, \infty)$ , let $S_{λ}$ be the set of twice continuously differentiable functions in B, continuous on B̄, satisfying $(u_{x})^{2} + λ (u_{y})^{2} = 1$ in B and u = 0 on $\partial B$ . Which of the following statements are true?

A. $S_{1} = \emptyset$ ✓
B. $S_{2} = \emptyset$ ✓
C. $S_{1}$ has exactly one element and $S_{2}$ has exactly two elements.
D. $S_{1}$ and $S_{2}$ are both infinite.

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Q33 Execution slip Interpolation and numerical integration with error terms

The coefficient of $x^{3}$ in the interpolating polynomial for the data x = 0, 1, 2, 3, 4 with y = 1, 2, 1, 3, 5 is

A.−1/3
B.−1/2
C.5/6
D.17/6✓

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Q34 Numerical convergence Numerical ODE: Euler, Runge–Kutta

For the IVP dy/dx $= f (x, y), y (x_{0}) = y_{0}$ , with step size h, let the first iterate of a second-order scheme be $y_{1} = y_{0} +$ Pk $_{1} +$ Qk $_{2}$ , where $k_{1} = h f (x_{0}, y_{0}), k_{2} = h f (x_{0} + α_{0} h, y_{0} + β_{0} k_{1})$ and $P, Q, α_{0}, β_{0} \in R$ . Which of the following statements are correct?

A.If $α_{0} = 2$ , then $β_{0} = 2, P = 3/4, Q = 1/4$ ✓
B.If $β_{0} = 3$ , then $α_{0} = 3, P = 5/6, Q = 1/6$ ✓
C.If $α_{0} = 2$ , then $β_{0} = 2, P = 1/4, Q = 3/4$
D.If $β_{0} = 3$ , then $α_{0} = 3, P = 1/6, Q = 5/6$

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Q35 open Euler–Lagrange equation and standard functionals

Among the curves connecting the points (1, 2) and (2, 8), let $γ$ be the curve on which an extremal of $J [y] = \int_{1}^{2} (1 + x^{3} y^{'}) y^{'}$ dx can be attained. Which of the following points lie on the curve $γ$ ?

A. $(2, 3)$
B. $(2, 6)$ ✓
C. $(3, 22/3)$ ✓
D. $(3, 23/3)$

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Q36 Existence vs uniqueness Euler–Lagrange equation and standard functionals

Define S = { $y \in C^{1} [0, π] : y (0) = y (π) = 0$ }, ‖f‖ $_{\infty} = max$ |f|, $B_{0} (0, ε) =$ {f ∈ S : ‖f‖ $_{\infty} < ε$ } and $B_{1} (0, ε) =$ {f ∈ S : ‖f‖ $_{\infty} +$ ‖f′‖ $_{\infty} < ε$ }. For the functional $J [y] = \int_{0}^{π} (1 - (y^{'})^{2}) y^{2}$ dx, there exists $ε > 0$ such that

A.J[y] ≤ J[0] for all $y \in B_{0} (0, ε)$
B.J[y] ≤ J[0] for all $y \in B_{1} (0, ε)$
C.J[y] ≥ J[0] for all $y \in B_{0} (0, ε)$
D.J[y] ≥ J[0] for all $y \in B_{1} (0, ε)$ ✓

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Q37 Execution slip Fredholm and Volterra equations

Let y be the solution to the Volterra integral equation y(x) = eˣ $+ \int_{0}$ ˣ $((1 + x^{2}) / (1 + t^{2})) y (t)$ dt. Which of the following statements are true?

A. $y (1) = (1 + π /4) e$
B. $y (1) = (1 + π /2) e$ ✓
C. $y (3) = (1 +$ $3 π /4) e^{3}$
D. $y (3) = (1 +$ $4 π /3) e^{3}$ ✓

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Q38 Existence vs uniqueness Fredholm and Volterra equations

Consider the Fredholm integral equation $y (x) - 3 \int_{0}^{1} t x y (t)$ dt = f(x), with f continuous on [0, 1]. Which of the following choices for f(x) have the property that the equation admits at least one solution?

A. $f (x) = x^{2} - 1/2$ ✓
B.f(x) = eˣ
C.f(x) = 2 − 3x✓
D.f(x) = x − 1

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Q39 Hypothesis dropped Hamiltonian formalism and conservation laws

Let $q_{1}, q_{2}$ be generalized coordinates and $p_{1}, p_{2}$ their conjugate momenta. Let a and b be such that $Q_{1} = q_{1}, P_{1} = a p_{1} + 16 p_{2}, Q_{2} = p_{2}, P_{2} = 2 q_{1} + b q_{2}$ is a canonical transformation. Which of the following statements are true?

A. $a^{2} + b^{2} = 2$ ✓
B.a − b = 2✓
C.a + b = 2
D.a = 1, b = 1

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Q40 Moments and tails Random variables, distributions, moments, MGF

Suppose U ~ Uniform(0,1) and $X = tan (π (U -$ ½)). Then which of the following statements are true?

A. $E (X^{4}) = 3$
B.P(X ∈ {1, 2, 5}) = 1/2
C.E(eˣ) does not exist✓
D.P(X ≤ 0) = 1/2✓

Solution

X is standard Cauchy: symmetric about 0, continuous (so any finite set has probability 0), and with no finite moments — E(eˣ) diverges too.

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Q41 open Random variables, distributions, moments, MGF

Let X be a discrete random variable with support S = {−1, 0, 1} and P(X = 0) = 1/3. Then which of the following statements are true?

A.E(X) ≤ 2/3✓
B. $E (X^{2}) = 2/3$ ✓
C.E(|X|) = 2/3✓
D.Var(X) > 2/3

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Q42 Boundary and endpoint Random variables, distributions, moments, MGF

Suppose X is a continuous random variable with density $f (x) = (1/ π) \cdot 1/ (1 + (x + 1)^{2})$ , and Y = X/|X| for X ≠ 0, Y = 0 for X = 0. Then which of the following statements are true?

A.E(Y) = 0
B.P(Y > 0) < P(Y < 0)✓
C.P(Y < −1) < P(Y > 1)
D. $E (Y^{2}) = 1$ ✓

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Q43 Execution slip Random variables, distributions, moments, MGF

Let X be a discrete random variable with support {0, 1, …, 25} and $P (X = x) = C (25, x) / 2^{25}$ . Then which of the following statements are true?

A.The distributions of X − 12.5 and 12.5 − X are identical✓
B.P(X ≤ 4) = P(X ≥ 22)
C.The coefficient of variation (in percentage) of X is 20✓
D.P(X ≤ 4.9) = P(X ≥ 20.1)✓

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Q44 Dependence misread Joint distributions, transformations, order statistics

Let $X_{i} (i = 1, 2)$ be independent with densities $f_{i} (x) = i$ $e^{- i x}$ for x ≥ 0. Consider a series system of two independent components with lifetimes $X_{1}$ and $X_{2}$ , and let X denote the lifetime of the system. Which of the following statements are true?

A.P(X > 4) = P(X > 1)P(X > 2)
B.P(X > 4 | X > 2) = P(X > 2)✓
C.E(X) = 1/3✓
D.6X ~ $χ_{3}^{2}$

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Q45 Execution slip Markov chains: classification of states, stationary distributions

Consider an M/M/1 queue with arrival rate $λ = 15$ per hour and service rate $μ = 45$ per hour. Let N(t) be the number of customers in the system at time t, and $T_{1}, T_{2}$ the time a customer spends in the queue and in the system. Which of the following statements are true?

A. $l i m_{t \to \infty}$ P(N(t) = 1) = 2/9✓
B. $P (T_{1} > 0) = 1/3$ ✓
C. $E (T_{1}) = 1/90$ ✓
D. $E (T_{2}) = 1/35$

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Q46 Dependence misread Markov chains: classification of states, stationary distributions

Suppose {X(t)} and {Y(t)} are two independent homogeneous Poisson processes with the same rate $λ = 2$ . Let $W_{n}$ ˣ and $W_{n}$ ʸ be the waiting times for the n-th arrival in each process. Which of the following statements are true?

A. $P (W_{2}$ ˣ $< W_{3}$ ʸ) = 11/16✓
B. $P (W_{1}$ ˣ $< W_{1}$ ʸ) = 1/2✓
C. $P (W_{2}$ ˣ $< W_{3}$ ʸ) = 13/16
D. $P (W_{1}$ ˣ $< W_{1}$ ʸ) = 1/4

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Q47 Boundary and endpoint MLE and method of moments

Let $X_{1}, \dots, X_{n}$ be a random sample with density f(x| $θ) =$ $e^{θ - x}$ for $x \geq θ$ , where $θ \in R$ is unknown. Define X̄ and $X_{(1)} = min X_{i}$ . Which of the following statements are true?

A.X̄ is the method of moments estimator of $θ$
B. $X_{(1)}$ is the maximum likelihood estimator of $θ$ ✓
C. $X_{(1)} - 1/ n$ is the uniformly minimum variance unbiased estimator of $θ$ ✓
D. $X_{(1)}$ is a sufficient statistic for $θ$ ✓

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Q48 What the inference means Neyman–Pearson lemma and UMP tests

Let $X_{1}, \dots, X_{25}$ be i.i.d. Bernoulli(p), X̄ their mean, $T_{1} = 5 (\overset{ˉ}{X} - 0.5) / \overset{ˉ}{X} (1 - \overset{ˉ}{X})$ for 0 < X̄ < 1 (±5 at the endpoints) and $T_{2} = 10 (\overset{ˉ}{X} - 0.5)$ . For $H_{0} : p = 0.5$ vs $H_{1} : p > 0.5$ , test $ψ_{i}$ rejects iff $T_{i} > 2$ . If the observed X̄ ∈ (0.5, 0.75), which of the following statements are true?

A.If $ψ_{1}$ rejects $H_{0}$ , then $ψ_{2}$ also rejects $H_{0}$
B.If $ψ_{1}$ does not reject $H_{0}$ , then $ψ_{2}$ also does not reject $H_{0}$ ✓
C.If $ψ_{2}$ rejects $H_{0}$ , then $ψ_{1}$ also rejects $H_{0}$ ✓
D.If $ψ_{2}$ does not reject $H_{0}$ , then $ψ_{1}$ also does not reject $H_{0}$

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Q49 What the inference means Neyman–Pearson lemma and UMP tests

Let $X_{1}, \dots, X_{n}$ be a random sample from $N (μ, 1)$ . To test $H_{0} : μ = μ_{0}$ against $H_{1} : μ > μ_{0}$ , consider (A) reject iff $\overset{ˉ}{X}_{n} > c_{1}$ with $P_{μ_{0}} (X_{n} > c_{1}) = α$ , and (B) reject iff Median{ $X_{1}, \dots, X_{n}$ } $> c_{2}$ with $P_{μ_{0}} (M e d ian$ $> c_{2}) = α$ . Which of the following statements are true?

A.The test in (A) is the uniformly most powerful test of size $α$ ✓
B.The test in (B) is the uniformly most powerful test of size $α$
C. $P_{μ} (\overset{ˉ}{X}_{n} > c_{1}) \to 1$ as $n \to \infty$ for all $μ > μ_{0}$ ✓
D. $P_{μ_{0}} (M e d ian$ { $X_{1}, \dots, X_{n}$ } $> μ_{0}) = 1/2$ ✓

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Q50 Boundary and endpoint Likelihood ratio and standard tests

For n ≥ 2, let $X_{1}, \dots, X_{n}$ be from $N (μ, σ^{2})$ with both unknown, X̄ the mean and $S^{2}$ the sample variance. With $z_{α}$ and $t_{m, α}$ the $(1 - α) -$ th quantiles, which of the following represent 90% confidence intervals for $μ$ ?

A. $(\overset{ˉ}{X} - (S / n)$ $t_{n - 1, 0.05}$ , $\overset{ˉ}{X} + (S / n)$ $t_{n - 1, 0.05})$ ✓
B. $(\overset{ˉ}{X} - (σ / n)$ $z_{0.05}$ , $\overset{ˉ}{X} + (σ / n)$ $z_{0.05})$
C. $[\overset{ˉ}{X} - (S / n)$ $t_{n - 1, 0.9}$ , $\infty)$
D. $(- \infty, \overset{ˉ}{X} - (S / n)$ $t_{n - 1, 0.9})$ ✓

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Q51 Moments and tails Likelihood ratio and standard tests

Let $X_{1}, \dots, X_{n}$ be a random sample from an unknown absolutely continuous CDF F, and $F_{0} a$ specified absolutely continuous CDF. For $H_{0} : F = F_{0}$ vs $H_{1} : F \neq = F_{0}$ , consider $T_{1, n}$ = sup_x | $F_{n} (x) - F_{0} (x)$ | and $T_{2, n}$ = n·sup_x | $F_{n} (x) - F_{0} (x)$ |, where $F_{n}$ is the empirical CDF. Which of the following statements are true?

A. $T_{1, n}$ → 0 in probability as $n \to \infty$ under $H_{0}$ ✓
B. $T_{2, n}$ → 0 in probability as $n \to \infty$ under $H_{0}$
C.lim P_F( $T_{2, n}$ > 1) = 1 for all F✓
D. $T_{2, n}$ converges in distribution to a degenerate random variable under $H_{0}$

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Q52 Hypothesis dropped Gauss–Markov, regression, ANOVA basics

Consider the one-way fixed effects ANOVA model $Y_{ij} = μ + α_{i} + ε_{ij}, j = 1, \dots, n_{i}; i = 1, \dots, k$ , with uncorrelated errors of mean 0 and variance $σ^{2}$ . Let Ȳ $_{i}$ be the i-th group mean. Which of the following statements are true?

A. $(1/ \sum n_{i}) \sum_{i} \sum_{j} Y_{ij}$ is an unbiased estimator of $μ$
B. $2 μ + α_{1} + α_{2}$ is an estimable linear parametric function✓
C. $μ + α_{1} + α_{2}$ is an estimable linear parametric function
D. $(1/ n_{2}) \sum_{j} (Y_{2 j} -$ Ȳ $_{2})$ is an unbiased estimator of $α_{2}$

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Q53 Invariants don't determine Gauss–Markov, regression, ANOVA basics

Consider $Y = X β + ε$ with X a fixed n × (p + 1) matrix of rank p + 1 and $ε$ ~ $N_{n} (0, σ^{2} I)$ . If $β$ ̂ is the OLS estimator, which of the following statements are true?

A. $(1/ σ^{2}) Y^{T} X β$ ̂ has a central $χ_{p + 1}^{2}$ distribution
B. $(1/ σ^{2}) (Y - X β$ ̂ $)^{T} (Y - X β$ ̂) has a central $χ_{n - p - 1}^{2}$ distribution✓
C. $X β$ ̂ and $(Y - X β$ ̂ $)^{T} (Y - X β$ ̂) are independently distributed✓
D. $(1/ σ^{2}) \sum (Y_{i} -$ Ȳ $)^{2}$ has a central $χ_{n - 1}^{2}$ distribution

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Q54 Execution slip Multivariate normal distribution

Suppose $A = (a_{ij})$ ~ $W_{3} (5, \sum)$ , where $\sum = [[2, 1, 1], [1, 2, 0], [1, 0, 2]]$ . Then which of the following statements are true?

A. $a_{22}$ ~ $χ_{3}^{2}$
B.½ $a_{22}$ ~ $χ_{5}^{2}$ ✓
C. $(1/33) (a_{11} - 4 a_{13} + 4 a_{33})$ ~ $χ_{3}^{2}$
D. $(1/9) (a_{11} - 4 a_{13} + 4 a_{33})$ ~ $χ_{5}^{2}$

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Q55 Hypothesis dropped Multivariate normal distribution

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), (X_{3}, Y_{3})$ be i.i.d. bivariate normal with mean (0, 0) and correlation matrix $[[1, ρ], [ρ, 1]]$ , | $ρ$ | < 1. Let $S_{ρ} = 3 E ($ sgn $(X_{1} - X_{2}) (Y_{1} - Y_{3}))$ . Which of the following statements are true?

A.If $X_{1}$ and $Y_{1}$ are independent, then $S_{ρ} = 0$ ✓
B. $S_{ρ} = (6/ π) sin$ ⁻ $^{1} (ρ /2)$ ✓
C.If $S_{ρ} = 0$ , then $X_{1}$ and $Y_{1}$ are independent✓
D.If $X_{1}$ and $Y_{1}$ are independent, then $S_{ρ} = 1/2$

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Q56 open SRS, stratified and systematic sampling

Consider a population of 3 units having values 2, 4 and 6. A simple random sample (without replacement) of 2 units is drawn. Let M be the sample mean. Which of the following statements are true?

A.E(M) = 4✓
B. $E (M^{2}) = 17$
C. $E (M^{3}) = 72$ ✓
D.Var(M) = 1

Solution

M takes the values 3, 4, 5 with probability 1/3 each: $E (M) = 4, E (M^{2}) = 50/3, E (M^{3}) = 72, V a r (M) = 2/3$ .

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Q57 Execution slip SRS, stratified and systematic sampling

Two groups $G_{1}$ and $G_{2}$ have 10 and 30 patients with mean diastolic blood pressures 80 and 100 mmHg and variances 4 and 2 mmHg $^{2}$ . Let $\overset{ˉ}{X}, S^{2}, C$ and R be the mean, variance (with divisor n), coefficient of variation (%) and range of the combined group. Which of the following statements are true?

A.X̄ = 95✓
B. $S^{2} = 77$
C.C > 180/19
D.R > 8✓

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Q58 Standard counterexample Continuity, uniform continuity, Lipschitz

Which of the following statements are true?

A.f(x) = [x] sin(1/x) for x ≠ 0, f(0) = 0 (where [x] is the integer part) has a discontinuity at 0 which is removable.
B. $f : [0, \infty) \to R, f (x) = sin (lo g x)$ for x ≠ 0, f(0) = 0, has a discontinuity at 0 which is NOT removable.✓
C.f(x) = $e^{1/ x}$ for x < 0, f(x) = $e^{1/ (x + 1)}$ for x ≥ 0, has a jump discontinuity at 0.✓
D.If $f, g : [0, 1] \to R$ are of bounded variation, then fg has at most countably many discontinuities.✓

Solution

(1) As x → 0⁻, [x] = −1 and −sin(1/x) oscillates: no left limit, not removable. (2) sin(log x) oscillates as x → 0⁺. (3) Left limit 0, right limit e. (4) A product of BV functions is BV, and BV functions have only countably many (jump) discontinuities.

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Q59 open Subgroups, cosets, Lagrange, cyclic groups

Which of the following statements are true?

A.Let $G_{1}$ and $G_{2}$ be finite groups such that | $G_{1}$ | and | $G_{2}$ | are coprime. Then any homomorphism from $G_{1}$ to $G_{2}$ is trivial.✓
B.Let G be a finite group and f : G → G a homomorphism that fixes more than half of the elements of G. Then f(x) = x for all x ∈ G.✓
C.Let G be a finite group having exactly 3 subgroups. Then G is of order $p^{2}$ for some prime p.✓
D.Any finite abelian group G has at least d(|G|) subgroups, where d(m) is the number of positive divisors of m.✓

Solution

(1) the image has order dividing both. (2) the fixed set is a subgroup of index < 2, hence all of G. (3) exactly three subgroups forces cyclic of order $p^{2} . (4)$ an abelian group has a subgroup of every order dividing |G|.

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