Previous year questions

The complete June 2023 paper is solved and free to read — every question, with the reasoning behind each option.

Every PYQ solved, tagged by topic and trap type. Free samples are open; the full set needs the PYQ Pack.

2023 JunePart Bconditional-vs-absoluteshow ▾

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).

Topic: The Real Line › Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements

2023 JunePart Blimits-of-perturbed-argumentsshow ▾

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.

Topic: The Real Line › limsup, liminf and subsequential limits

2023 JunePart Bextension-to-closed-intervalshow ▾

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.

Topic: Continuity and Differentiation › Continuity, uniform continuity, Lipschitz

2023 JunePart Balgebraic-vs-geometric-multiplicityshow ▾

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).

Topic: Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

2023 JunePart Bchar-vs-min-polynomialshow ▾

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.

Topic: Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

2023 JunePart Bpositive-definitenessshow ▾

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).

Topic: Inner Product Spaces and Forms › Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

2023 JunePart Bnormal-subgroups-of-p-groupsshow ▾

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.

Topic: Groups › Group actions, class equation, p-groups

2023 JunePart Bcrt-countingshow ▾

The number of solutions of the equation

x^{2} = 1

in the ring

Z /105 Z

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 $2^{3} = 8$ .

Topic: Rings and Fields › Ideals, quotient rings, prime & maximal ideals, CRT

2023 JunePart Bshow ▾

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.

Topic: Rings and Fields › Polynomial rings and irreducibility tests

2023 JunePart Bcardinal-arithmeticshow ▾

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.

Topic: Lebesgue Measure and Integration › Measurable sets and functions

2023 JunePart Bresidue-by-series-productshow ▾

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 $k = m + 1 : \sum_{m} 1/ (m! (m + 1)!)$ .

Topic: Singularities and Residues › Laurent series, classification of singularities, Casorati–Weierstrass

2023 JunePart Bgenerating-functionshow ▾

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.

Topic: Analytic Functions › Power series and analyticity

2023 JunePart Beigenvalue-growth-ratesshow ▾

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.

Topic: Ordinary Differential Equations › Linear ODE, Wronskian, variation of parameters, systems

2023 JunePart Bburgers-characteristicsshow ▾

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.

Topic: Partial Differential Equations › First-order PDE: Lagrange, Charpit, characteristics

2023 JunePart Bnull-lagrangian-termshow ▾

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.

Topic: Calculus of Variations › Euler–Lagrange equation and standard functionals

2023 JunePart Bgreens-function-dirichletshow ▾

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 $y = 2 \int G y$ .

Topic: Linear Integral Equations › Fredholm and Volterra equations

2023 JunePart Border-statistics-of-bernoullishow ▾

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.

Topic: Probability › Standard discrete and continuous distributions

2023 JunePart Bshow ▾

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.

Topic: Estimation › Sufficiency, completeness, UMVUE, Cramér–Rao

2023 JunePart Bruns-distributionshow ▾

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

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.

Topic: Hypothesis Testing › Likelihood ratio and standard tests

2023 JunePart Bweighted-least-squaresshow ▾

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

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 $1/ x_{i}^{2})$ .

Topic: Linear Models and Multivariate › Gauss–Markov, regression, ANOVA basics

2023 JunePart Bshow ▾

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

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

Topic: Linear Models and Multivariate › Multivariate normal distribution

2023 JunePart Bwishart-linear-formshow ▾

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}

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 $W_{1} (2, a^{T} \sum a) = W_{1} (2, 2)$ .

Topic: Linear Models and Multivariate › Multivariate normal distribution

2023 DecemberPart Bshow ▾

Consider the following subset of

R : U =

{

x \in R : x^{2} - 9 x + 18 \leq 0, x^{2} - 7 x + 12 \leq 0

}. Which one of the following statements is true?

A.inf U = 5.
B.inf U = 4.
C.inf U = 3.✓
D.inf U = 2.

Solution

$x^{2} - 9 x + 18 \leq 0 \Leftrightarrow x \in [3, 6]; x^{2} - 7 x + 12 \leq 0 \Leftrightarrow x \in [3, 4]$ . So U = [3, 4] and inf U = 3.

Topic: The Real Line › Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

2023 DecemberPart Bperiodic-subsequenceshow ▾

Locked — The Real Line. Unlock the PYQ Pack

2023 DecemberPart Bshow ▾

Consider the following infinite series:

(a) \sum_{n \geq 1}

sin (nπ /2) / n, (b) \sum_{n \geq 1}

lo g (1 + 1/ n^{2})

. Which one of the following statements is true?

A.(a) is convergent, but (b) is not convergent.
B.(a) is not convergent, but (b) is convergent.
C.Both (a) and (b) are convergent.✓
D.Neither (a) nor (b) is convergent.

Solution

(a): the non-zero terms are $(- 1)^{k} / 2 k + 1$ , an alternating series with terms decreasing to 0 (Leibniz $) . (b) : lo g (1 + 1/ n^{2})$ ~ $1/ n^{2}$ , comparison with a convergent p-series.

Topic: The Real Line › Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements