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 Cunbounded-vs-bounded-sequencesshow ▾

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.

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

2023 DecemberPart Cinjective-continuous-implies-monotoneshow ▾

Locked — Continuity and Differentiation. Unlock the PYQ Pack

2023 DecemberPart Cperiodic-uniform-continuityshow ▾

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

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.

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

2023 DecemberPart Cp-integral-thresholdsshow ▾

For a real number

λ

, consider the improper integrals

I_{λ} = \int_{0}^{1}

/ (1 - x)^{λ}

and

K_{λ} = \int_{1}^{\infty}

/ 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.

Topic: Integration › Riemann integration and criteria

2023 DecemberPart Cmoments-determine-functionshow ▾

Locked — Integration. Unlock the PYQ Pack

2023 DecemberPart Cvanishing-factor-at-endpointshow ▾

Locked — Sequences and Series of Functions. Unlock the PYQ Pack

2023 DecemberPart Cshow ▾

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.

Topic: Functions of Several Variables › Partial derivatives, differentiability, chain rule

2023 DecemberPart Cmonotone-not-necessarily-increasingshow ▾

Locked — Functions of Several Variables. Unlock the PYQ Pack

2023 DecemberPart Cprojection-is-open-mapshow ▾

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2023 DecemberPart Cpartition-counterexamplesshow ▾

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2023 DecemberPart Cnilpotent-index-from-nullityshow ▾

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2023 DecemberPart Cunion-of-subspacesshow ▾

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2023 DecemberPart Ccayley-hamilton-remaindershow ▾

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2023 DecemberPart Cinvertibility-from-matrix-identityshow ▾

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2023 DecemberPart Cshow ▾

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.

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

2023 DecemberPart Creading-jordan-blocksshow ▾

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2023 DecemberPart Cshow ▾

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2023 DecemberPart Cgaussian-integersshow ▾

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2023 DecemberPart Cshow ▾

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.

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

2023 DecemberPart Cirreducible-over-Q-not-mod-pshow ▾

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2023 DecemberPart Corder-of-gl2-over-z-9show ▾

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2023 DecemberPart Cshow ▾

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 $Z /3$ .

Topic: Rings and Fields › Galois theory essentials

2023 DecemberPart Ctotally-disconnectedshow ▾

Locked — Topology. Unlock the PYQ Pack

2023 DecemberPart Churwitz-zeros-escapeshow ▾

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2023 DecemberPart Cidentity-theorem-pigeonholeshow ▾

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