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 DecemberPart Bmgf-factorisationshow ▾

Locked — Probability. Unlock the PYQ Pack

2023 DecemberPart Bextreme-value-scalingshow ▾

Let

X_{1}, X_{2}, \dots

be i.i.d. with CDF F(x) = 0 for x < 5 and 1 −

e^{5 - x}

for x ≥ 5. Define

Y_{n} = min

{

X_{1}, \dots, X_{n}

} and

Z_{n} = n (Y_{n} - 5)

, and let Z be standard normal. Which one of the following statements is true?

A. $lim P (1/2 < Y_{n} < 3/2) = 1$
B. $Y_{n} \to 5$ in probability as $n \to \infty$ ✓
C. $Z_{n} \to Z$ in distribution as $n \to \infty$
D. $lim P (1 < Z_{n} < 2) = Φ (2) - Φ (1)$

Solution

$Y_{n} - 5$ ~ Exp(n), so $Y_{n} \to 5$ in probability while $n (Y_{n} - 5) \to 0$ in probability (the right scaling is n, not $n)$ .

Topic: Limit Theorems and Markov Chains › Modes of convergence, WLLN, SLLN, CLT

2023 DecemberPart Bshow ▾

The probability of a head in tossing a coin is p ∈ (0, 1). The coin is independently tossed 25 times and heads appear 10 times. The Bayes estimate of p with respect to the prior Beta(5, 5) and squared error loss is

A.3/7✓
B.3/5
C.1/2
D.2/5

Solution

Posterior is Beta(5 + 10, 5 + 15) = Beta(15, 20) with mean 15/35 = 3/7.

Topic: Estimation › MLE and method of moments

2023 DecemberPart Bshow ▾

For n ≥ 2, let

X_{1}, \dots, X_{n}

be a random sample with density f(x|

θ) =

θ x^{θ - 1}

0 < x < 1, θ > 0

unknown. Which of the following is the UMVUE for

1/ θ

A. $- (1/ n) \sum ln X_{i}$ ✓
B. $- n / \sum ln X_{i}$
C. $- (n - 1) / \sum ln X_{i}$
D. $- (2/ n) \sum ln X_{i}$

Solution

$- ln X_{i}$ ~ Exp $(θ)$ with mean $1/ θ$ , and $\sum ln X_{i}$ is complete sufficient; the sample mean of $- ln X_{i}$ is unbiased for $1/ θ$ .

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

2023 DecemberPart Bsufficient-statistic-by-expansionshow ▾

Locked — Estimation. Unlock the PYQ Pack

2023 DecemberPart Bdirection-of-rejectionshow ▾

Locked — Hypothesis Testing. Unlock the PYQ Pack

2023 DecemberPart Badjusted-r-squaredshow ▾

Locked — Linear Models and Multivariate. Unlock the PYQ Pack

2023 DecemberPart Bwishart-traceshow ▾

Locked — Linear Models and Multivariate. Unlock the PYQ Pack

2023 DecemberPart Bshow ▾

In a Latin square design, the degrees of freedom for the error sum of squares is 42. Then the degrees of freedom for the sum of squares due to treatments is

A.6
B.7✓
C.8
D.9

Solution

For a p × p Latin square the error d.f. is (p − 1)(p − 2) = 42 ⇒ p = 8, so treatments have p − 1 = 7 d.f.

Topic: Sampling and Design of Experiments › CRD, RBD, LSD essentials

2023 JunePart Csubsequence-criteriashow ▾

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: $x_{n} = n$ .

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

2023 JunePart Crecursive-sequence-fixed-pointshow ▾

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.

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

2023 JunePart Cimplicit-function-theorem-failsshow ▾

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.

Topic: Functions of Several Variables › Inverse and implicit function theorems, extrema

2023 JunePart Csylvester-signatureshow ▾

Consider the following quadratic forms over

R : (a) 6 X^{2} - 13

+ 6 Y^{2}, (b) X^{2} -

+ 2 Y^{2}, (c) X^{2} -

- 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).

Topic: Inner Product Spaces and Forms › Quadratic forms, positive definiteness, Sylvester's law

2023 JunePart Crank-is-field-independentshow ▾

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.

Topic: Vector Spaces and Linear Maps › Bases, dimension, rank–nullity

2023 JunePart Ckernel-dimension-vs-block-sizesshow ▾

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.

Topic: Eigenvalues and Canonical Forms › Jordan canonical form

2023 JunePart Cp-squared-groups-abelian-not-cyclicshow ▾

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.

Topic: Groups › Sylow theorems and groups of small order

2023 JunePart Cconjugation-and-analyticityshow ▾

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.

Topic: Analytic Functions › Cauchy–Riemann equations, harmonic functions

2023 JunePart Cbounded-rhs-global-existenceshow ▾

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|.

Topic: Ordinary Differential Equations › Existence–uniqueness, Picard, Lipschitz

2023 JunePart Ctransport-equation-data-on-a-lineshow ▾

Let

u : R^{2} \to R

solve

\partial_{x} u + 2 \partial_{y} u = 0

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.

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

2023 JunePart Ccoupled-euler-lagrangeshow ▾

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.

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

2023 JunePart Csymmetry-of-dependenceshow ▾

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.

Topic: Probability › Axioms, conditional probability, independence, Bayes

2023 JunePart Cclt-at-the-boundaryshow ▾

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).

Topic: Limit Theorems and Markov Chains › Modes of convergence, WLLN, SLLN, CLT

2023 JunePart Cmixture-vs-linear-combinationshow ▾

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.

Topic: Probability › Standard discrete and continuous distributions

2023 JunePart Cconsistency-with-a-fixed-nuisance-termshow ▾

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.

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

2023 JunePart Csize-vs-powershow ▾

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.

Topic: Hypothesis Testing › Neyman–Pearson lemma and UMP tests