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CSIR NET June 2023 Mathematical Sciences — all questions solved

Every Part B and Part C question from this paper, worked out in full — not just the answer key, but why each option holds or fails and which trap it tests. Free to read, no account needed.

38 questionsPart B: 22Part C: 16fully free

Part B

One correct option. 3 marks, −0.75 for a wrong answer.

Q1conditional-vs-absoluteSeries: comparison, ratio, root, Raabe, condensation, alternating, rearrangements

Consider the series $\sum_{n = 1}^{\infty}$ $a_{n}$ , where $a_{n} =$ $(- 1)^{n + 1} (n + 1 - n)$ . Which of the following statements is true?

A.The series is divergent.
B.The series is convergent.
C.The series is conditionally convergent.✓
D.The series is absolutely convergent.

Solution

$n + 1 - n = 1/ (n + 1 + n)$ decreases to 0, so the alternating series converges (Leibniz). The absolute series behaves like $\sum 1/ (2 n)$ , which diverges. Hence conditionally convergent — option 3 is the most precise true statement (official key: 3).

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Q2limits-of-perturbed-argumentslimsup, liminf and subsequential limits

Which of the following assertions is correct?

A. $l im s u p_{n} exp (cos ((nπ + (- 1)^{n} \cdot 2 e) / (2 n))) > 1$ .
B. $l i m_{n} exp (l o g_{e} ((n π^{2} + (- 1)^{n} e^{2}) / (7 n)))$ does not exist.
C. $l imin f_{n} exp (sin ((nπ + (- 1)^{n} \cdot 2 e) / (2 n))) < π$ .✓
D. $l i m_{n} exp (tan ((n π^{2} + (- 1)^{n} e^{2}) / (7 n)))$ does not exist.

Solution

All the inner arguments converge: $(nπ + (- 1)^{n} 2 e) / (2 n) \to π /2$ and $(n π^{2} + (- 1)^{n} e^{2}) / (7 n) \to π^{2} /7$ . So the limits in (2) and (4) exist; in (1) the limit is $exp (cos π /2) = 1$ , not > 1; in (3) the limit is $exp (sin π /2) = e < π$ . Official key: 3.

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Q3extension-to-closed-intervalContinuity, uniform continuity, Lipschitz

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

A.f(x) = sin(1/x)
B.f(x) = $e^{- 1/ x^{2}}$ ✓
C.f(x) = eˣ cos(1/x)
D. $f (x) = cos x \cdot cos (π / x)$

Solution

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

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Q4algebraic-vs-geometric-multiplicityEigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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

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

Solution

$p (T) = T^{2} (T - c)$ 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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Q5char-vs-min-polynomialEigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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

A.Both (a) and (b) are true.
B.Both (a) and (b) are false.
C.(a) is true and (b) is false.
D.(a) is false and (b) is true.✓

Solution

(a) fails: for the nilpotent Jordan block on $R^{3}, f = X^{3}, g = X^{2}$ , and $g (T) = T^{2} \neq = 0. (b)$ If 0 has geometric multiplicity ≥ 2 in dimension 3, the 0-Jordan blocks all have size 1, so the minimal polynomial is X(X − c) (or X if T = 0), which divides g(X) = X(X − c); hence g(T) = 0.

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Q6positive-definitenessGram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

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

A.⟨x, y⟩ $= \sum_{i, j = 1}^{n}$ $x_{i} y_{j}$
B.⟨x, y⟩ $= \sum_{i, j = 1}^{n}$ $(x_{i}^{2} + y_{j}^{2})$
C.⟨x, y⟩ $= \sum_{j = 1}^{n}$ $j^{3} x_{j} y_{j}$ ✓
D.⟨x, y⟩ $= \sum_{j = 1}^{n}$ $x_{j} y_{n}$ ₋ $_{j}$ ₊ $_{1}$

Solution

A weighted sum $\sum w_{j} x_{j} y_{j}$ with positive weights is an inner product. (1) is $(\sum x_{i}) (\sum y_{j})$ , not positive definite; (2) is not bilinear; (4) is symmetric but ⟨x, x⟩ can be ≤ 0 (e.g. x = (1, −1) for n = 2).

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Q7normal-subgroups-of-p-groupsGroup 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 $n \in N$ such that $g^{p^{n}}$ = 1. Which of the following statements is FALSE?

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

Solution

A finite p-group has a normal subgroup of every order pᵏ dividing |G|, so |G| $= p^{6}$ gives at least seven normal subgroups (orders $1, p, \dots, p^{6})$ — 'exactly six' is impossible. (1), (2) follow from the same fact; (3) an infinite abelian p-group (e.g. the Prüfer group) has infinite centre.

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Q8crt-countingIdeals, quotient rings, prime & maximal ideals, CRT

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

A.0
B.2
C.4
D.8✓

Solution

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

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Q9Polynomial rings and irreducibility tests

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

A.0
B.1✓
C.2
D.3

Solution

The derivative $3 x^{2} + 3 > 0$ , so the cubic is strictly increasing: exactly one real root.

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Q10cardinal-arithmeticMeasurable sets and functions

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

A.S is in bijection with the set of rational numbers.
B.S is in bijection with the set of real numbers.
C.S is in bijection with S × S.✓
D.S is in bijection with the power set of S.

Solution

With AC, |S × S| = |S| for every infinite S. (1), (2) fix a specific cardinality; (4) contradicts Cantor's theorem.

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Q11residue-by-series-productLaurent series, classification of singularities, Casorati–Weierstrass

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

A. $\sum_{l = 0}^{\infty}$ 1/(l+1)!
B. $\sum_{l = 0}^{\infty}$ 1/(l!(l+1))
C. $\sum_{l = 0}^{\infty}$ 1/(l!(l+1)!)✓
D. $\sum_{l = 0}^{\infty}$ $1/ ((l^{2} + l)!)$

Solution

Multiply the series eᶻ $= \sum z^{m} / m$ ! and $e^{1/ z}$ $= \sum$ $z^{- k} / k$ !; the coefficient of $z^{- 1}$ comes from $k = m + 1 : \sum_{m} 1/ (m! (m + 1)!)$ .

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Q12generating-functionPower series and analyticity

Consider the function f defined by $f (z) = 1/ (1 - z - z^{2})$ for $z \in C$ such that $1 - z - z^{2} \neq = 0$ . Which of the following statements is true?

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

Solution

$(1 - z - z^{2}) \sum a_{n} z^{n} = 1$ gives $a_{0} = 1, a_{1} - a_{0} = 0 \Rightarrow a_{1} = 1$ , and $a_{n}$ ₊ $_{2} = a_{n}$ ₊ $_{1} + a_{n}$ : the Fibonacci generating function. Poles are at the roots of $1 - z - z^{2}$ , not at 0.

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Q13eigenvalue-growth-ratesLinear ODE, Wronskian, variation of parameters, systems

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

A.x(t) is a bounded solution for some $x_{0} \neq = 0$ .
B. $e^{- 6 t}$ |x(t)| → 0 as $t \to \infty$ , for all $x_{0} \neq = 0$ .
C. $e^{- t}$ |x(t)| $\to \infty$ as $t \to \infty$ , for all $x_{0} \neq = 0$ .
D. $e^{- 10 t}$ |x(t)| → 0 as $t \to \infty$ , for all $x_{0} \neq = 0$ .✓

Solution

Eigenvalues of A are 6 and 1 (trace 7, det 6), both positive: every non-zero solution grows like $e^{6 t}$ or $e^{t}$ . 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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Q14burgers-characteristicsFirst-order PDE: Lagrange, Charpit, characteristics

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

A.2
B.3
C.1/2✓
D.1/3

Solution

Characteristics: dx/dy = u with u constant along them, so $x = x_{0} + u \cdot y$ and $u = x_{0}$ . Hence u = x/(1 + y), and u(2, 3) = 1/2.

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Q15null-lagrangian-termEuler–Lagrange equation and standard functionals

Consider the variational problem $(P) : J (y) = \int_{0}^{1} [(y^{'})^{2} - y$ |y|·y′ + xy] dx, y(0) = 0, y(1) = 0. Which of the following statements is correct?

A.(P) has no stationary function (extremal).
B.y ≡ 0 is the only stationary function (extremal) for (P).
C.(P) has a unique stationary function (extremal) y not identically equal to 0.✓
D.(P) has infinitely many stationary functions (extremals).

Solution

The term y|y|y′ is an exact derivative ((|y| $^{3} /3)^{'}$ up to sign), so it does not affect the Euler–Lagrange equation, which reduces to 2y″ = x. With y(0) = y(1) = 0 this gives the unique extremal $y = (x^{3} - x) /12$ , not identically zero.

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Q16greens-function-dirichletFredholm and Volterra equations

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

A.K(x,t) = t(1 − x) for t < x; x(1 − t) for t > x✓
B. $K (x, t) = t^{2} (1 - x)$ for $t < x; x^{2} (1 - t)$ for t > x
C. $K (x, t) = t (1 - x)$ for $t < x; x (1 - t)$ for t > x
D. $K (x, t) = t^{3} (1 - x)$ for $t < x; x^{3} (1 - t)$ 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 $y = 2 \int G y$ .

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Q17order-statistics-of-bernoulliStandard discrete and continuous distributions

Let $X_{1}, X_{2}, X_{3}, X_{4}$ be i.i.d. Bernoulli(1/3) and $X_{(1)} \leq \dots \leq X_{(4)}$ the order statistics. Which of the following is true?

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

Solution

$X_{(2)} = 1$ iff at least three of the four are $1 : P = 4 (1/3)^{3} (2/3) + (1/3)^{4} = 9/81 = 1/9. A$ Bernoulli(1/9) has variance (1/9)(8/9) = 8/81.

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Q18Sufficiency, completeness, UMVUE, Cramér–Rao

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

A. $λ$ ⁻ $^{1}$ ✓
B. $λ$
C. $λ^{2}$
D. $e^{- λ}$

Solution

An unbiased estimator g(X) must satisfy $\sum g (k) λ^{k} e^{- λ} / k$ ! $= h (λ)$ ; the left side is $e^{- λ}$ times a power series in $λ$ , which can never equal $λ$ ⁻ $^{1} ($ unbounded at $0) . λ, λ^{2}$ and $e^{- λ}$ = P(X = 0) are all estimable.

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Q19runs-distributionLikelihood ratio and standard tests

Let $X_{1}, \dots, X_{7}$ and $Y_{1}, \dots, Y_{9}$ be independent random samples from continuous CDFs F and G. For the Wald–Wolfowitz run test of $H_{0} : F = G$ vs $H_{1} : F \neq = G$ , let R be the total number of runs in the combined ordered arrangement. Which of the following is true?

A. $P_{H_{0}} (R$ = 6) = 28/286, $P_{H_{0}} (R$ = 9) = 28/143.
B. $P_{H_{0}} (R$ = 6) = 21/286, $P_{H_{0}} (R$ = 9) = 15/286.
C. $P_{H_{0}} (R$ = 6) = 21/286, $P_{H_{0}} (R$ = 9) = 28/143.✓
D. $P_{H_{0}} (R$ = 6) = 21/286, $P_{H_{0}} (R$ = 9) = 15/286.

Solution

With m = 7, n = 9 and C(16,7) = 11440 arrangements: P(R = 2k) = 2·C(m−1,k−1)·C(n−1,k−1)/C(16,7) gives P(R=6) = 2·15·28/11440 = 21/286; P(R = 2k+1) = [C(m−1,k)C(n−1,k−1) + C(m−1,k−1)C(n−1,k)]/C(16,7) gives P(R=9) = (15·56 + 20·70)/11440 = 28/143.

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Q20weighted-least-squaresGauss–Markov, regression, ANOVA basics

Consider the simple linear regression model $Y_{i} = β x_{i} + ε_{i}, i = 1, \dots, n$ , where $E (ε_{i}) = 0, C o v (ε_{i}, ε_{k}) = 0$ for i ≠ k and $V a r (ε_{i}) = x_{i}^{2} σ^{2}$ . The best linear unbiased estimator of $β$ is

A. $\sum Y_{i} x_{i} / \sum x_{i}^{2}$
B. $\sum Y_{i} / \sum x_{i}$
C. $(1/ n) \sum Y_{i} / x_{i}$ ✓
D. $(1/ n) \sum Y_{i} x_{i} / x_{i}^{2}$

Solution

Heteroscedastic errors: divide by $x_{i}$ to get $Y_{i} / x_{i} = β + ε_{i} / x_{i}$ with constant variance $σ^{2}$ ; the BLUE is the plain mean of $Y_{i} / x_{i} ($ weighted least squares with weights $1/ x_{i}^{2})$ .

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Q21Multivariate normal distribution

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

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

Solution

$E [5 - 2 X_{2}] = 5; V a r (5 - 2 X_{2}) = 4 \cdot 10 = 40; C o v (X_{1}, 5 - 2 X_{2}) = - 2 \cdot (- 3) = 6$ .

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Q22wishart-linear-formMultivariate normal distribution

Suppose $X = (X_{1}, X_{2}, X_{3}, X_{4})^{T}$ ~ $N_{4} (0, I_{2}$ ⊗ $\sum)$ with $\sum = [[2, - 1], [- 1, 2]]$ . Define $Z = [[X_{1}, X_{2}], [X_{3}, X_{4}]]$ and Q = ZᵀZ. If $W_{m} (n, \sum)$ denotes a Wishart distribution of order m with n degrees of freedom, the distribution of $Q_{11} + Q_{12} + Q_{21} + Q_{22}$ is

A. $W_{1} (2, 2)$ ✓
B. $W_{1} (1, 2)$
C. $W_{1} (2, 1)$
D. $2 χ_{4}^{2}$

Solution

The rows of Z are $i . i . d . N_{2} (0, \sum)$ , so Q ~ $W_{2} (2, \sum)$ . The sum of all entries of Q is aᵀQa with a = (1, 1)ᵀ, which is $W_{1} (2, a^{T} \sum a) = W_{1} (2, 2)$ .

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Part C

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

Q1subsequence-criteriaSequences: 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.

Solution

(1) Even and odd subsequences with a common limit ⇒ convergent. (2) { $x_{6 n}$ } lies in both { $x_{2 n}$ } and { $x_{3 n}$ }, so those limits agree; { $x_{3 (2 n}$ ₊ $_{1)}$ } lies in both { $x_{2 n}$ ₊ $_{1}$ } and { $x_{3 n}$ }, so all three limits agree ⇒ convergent. (3) Fails: $x_{n} = 1$ if n is prime, 0 otherwise — every { $x_{k n}$ } is eventually 0 but $x_{n}$ diverges. (4) Fails: $x_{n} = n$ .

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Q2recursive-sequence-fixed-pointSequences: 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.

Solution

Write $f (x) = 1/2 - (x - 1/2)^{2}$ . Fixed points are ±1/2. For |a − 1/2| < 1 the distance $d_{n} = 1/2 - x_{n}$ satisfies $d_{n}$ ₊ $_{1} = d_{n}^{2}$ , which tends to 0, so $x_{n} \to 1/2$ for all a ∈ (−1/2, 3/2); a = 0 is included.

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Q3implicit-function-theorem-failsInverse 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.

Solution

$g (x)^{3} = x^{2}$ forces g(x) = $x^{2/3}$ (the real cube root is a bijection), so there is exactly one continuous solution. It is not differentiable at 0, so no differentiable g exists. The implicit function theorem does not apply since f_y(0,0) = 0.

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Q4sylvester-signatureQuadratic 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.

Solution

Discriminants $b^{2} - 4$ ac: (a) 169 − 144 > 0 indefinite; (b) 1 − 8 < 0 with a > 0 ⇒ positive definite; (c) 1 + 8 > 0 indefinite. Over $R$ , non-degenerate binary forms are equivalent iff they have the same signature, so (a) ~ (c).

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Q5rank-is-field-independentBases, 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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Q6kernel-dimension-vs-block-sizesJordan 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.

Solution

dim W = number of Jordan blocks. Equal numbers of blocks do not fix the largest block (minimal polynomial): blocks {2,2} vs {3,1} in dimension 4. (1) similar ⇒ equal kernel dimension. (3) W = V ⇒ T = 0. (4) nilpotent ⇒ characteristic polynomial $x^{n}$ always.

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Q7p-squared-groups-abelian-not-cyclicSylow 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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Q8conjugation-and-analyticityCauchy–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.

Solution

If $f = \sum a_{n} z^{n}$ then conj $(f (\overset{z}{ˉ})) = \sum$ ā $_{n} z^{n}$ is entire. z ↦ f(z̄) and z ↦ conj(f(z)) are anti-holomorphic (entire only if f is constant). (4) is entire plus non-entire ⇒ not entire in general.

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Q9bounded-rhs-global-existenceExistence–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.✓

Solution

$f \in C^{1} \Rightarrow$ locally Lipschitz ⇒ unique local solution (Picard). |x′| ≤ sup|f| $< \infty \Rightarrow$ the solution cannot blow up, so it is global, and it is Lipschitz with constant sup|f|.

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Q10transport-equation-data-on-a-lineFirst-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.✓

Solution

Both are constant on lines y − 2x = c. v = sin(x − y/2), which vanishes on x = y/2 and changes sign across it inside S. For u, the data line y = 3x + 1 meets the characteristic through (x, y) at $x_{0} = y - 2 x - 1$ , so u = sin(y − 2x − 1); on S the argument lies in $[- 3, 0] \subset (- π, 0]$ , so u ≤ 0 and does not change sign. u = v where y − 2x − 1 = x − y/2, i.e. on the line y = 2x + 2/3, which crosses S.

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Q11coupled-euler-lagrangeEuler–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].

Solution

Euler–Lagrange: 2y″ + z″ = 0 and 2z″ + y″ = 0 ⇒ y″ = z″ = 0. So y = 1 − x and z = −1 + 3x. Then y + z = 2x and z + 3y = 2.

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Q12symmetry-of-dependenceAxioms, 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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Q13clt-at-the-boundaryModes 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{ $Y_{n} \in [3 n, 5 n]$ }✓

Solution

$E [X^{4}] = 3$ , so $Y_{n} / n \to 3 ($ LLN) and $(Y_{n} - 3 n) / n$ is asymptotically normal (CLT). Intervals with 3n as an endpoint have probability → 1/2; [2n, 4n] contains 3n in its interior (→ 1); [0, 2n] excludes it (→ 0).

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Q14mixture-vs-linear-combinationStandard 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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Q15consistency-with-a-fixed-nuisance-termSufficiency, 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⁻ $^{1} (\sum_{i = 1}^{n}$ $X_{i} + W)$ ✓

Solution

(1), (3) are sample means of n i.i.d. terms. (2) converges to $2 μ . (4) W / n \to 0$ in probability since W is a fixed random variable, so consistency is preserved.

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Q16size-vs-powerNeyman–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 $φ)$ .✓

Solution

Sizes: P_H(X ≤ 2) = 2/20 = 0.1 and P_H(X ≥ 19) = 0.1. Powers: P_K(X ≥ 19) = 39/210 ≈ 0.186 and $P_{K} (X \leq 2) = 3/210 \approx 0.014. ψ$ rejects where the likelihood ratio is largest — the Neyman–Pearson direction.

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