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 Cliouville-obstructionshow ▾

Locked — Cauchy Theory. Unlock the PYQ Pack

2023 DecemberPart Cresidue-of-exp-over-powershow ▾

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.

Topic: Singularities and Residues › Residue theorem and standard contour integrals

2023 DecemberPart Clinear-growth-global-existenceshow ▾

Locked — Ordinary Differential Equations. Unlock the PYQ Pack

2023 DecemberPart Cequilibria-trap-solutionsshow ▾

Locked — Ordinary Differential Equations. Unlock the PYQ Pack

2023 DecemberPart Cshow ▾

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 $φ (5) = 25$ .

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

2023 DecemberPart Ccharacteristics-crossingshow ▾

Locked — Partial Differential Equations. Unlock the PYQ Pack

2023 DecemberPart Ceikonal-no-smooth-solutionshow ▾

Locked — Partial Differential Equations. Unlock the PYQ Pack

2023 DecemberPart Cnewton-form-coefficientshow ▾

Locked — Numerical Analysis. Unlock the PYQ Pack

2023 DecemberPart Crk2-order-conditionsshow ▾

Locked — Numerical Analysis. Unlock the PYQ Pack

2023 DecemberPart Cshow ▾

Locked — Calculus of Variations. Unlock the PYQ Pack

2023 DecemberPart Cweak-vs-strong-extremumshow ▾

Locked — Calculus of Variations. Unlock the PYQ Pack

2023 DecemberPart Cvolterra-to-odeshow ▾

Locked — Linear Integral Equations. Unlock the PYQ Pack

2023 DecemberPart Cfredholm-alternativeshow ▾

Locked — Linear Integral Equations. Unlock the PYQ Pack

2023 DecemberPart Cpoisson-bracket-conditionsshow ▾

Locked — Classical Mechanics. Unlock the PYQ Pack

2023 DecemberPart Ccauchy-no-momentsshow ▾

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.

Topic: Probability › Random variables, distributions, moments, MGF

2023 DecemberPart Cshow ▾

Locked — Probability. Unlock the PYQ Pack

2023 DecemberPart Cshifted-cauchy-signshow ▾

Locked — Probability. Unlock the PYQ Pack

2023 DecemberPart Cbinomial-symmetryshow ▾

Locked — Probability. Unlock the PYQ Pack

2023 DecemberPart Cmin-of-exponentialsshow ▾

Locked — Probability. Unlock the PYQ Pack

2023 DecemberPart Cmm1-formulasshow ▾

Locked — Limit Theorems and Markov Chains. Unlock the PYQ Pack

2023 DecemberPart Ccompeting-poisson-processesshow ▾

Locked — Limit Theorems and Markov Chains. Unlock the PYQ Pack

2023 DecemberPart Cshifted-exponentialshow ▾

Locked — Estimation. Unlock the PYQ Pack

2023 DecemberPart Ccomparing-test-statisticsshow ▾

Locked — Hypothesis Testing. Unlock the PYQ Pack

2023 DecemberPart Cump-mean-vs-medianshow ▾

Locked — Hypothesis Testing. Unlock the PYQ Pack

2023 DecemberPart Cone-sided-intervalsshow ▾

Locked — Hypothesis Testing. Unlock the PYQ Pack