CUET-PG - Statistics · India · National Testing Agency
Original practice questions written to the published syllabus · not past papersThe rigorous calculus behind limits theorems and estimation: sequences, series, continuity, differentiability and Riemann integration, at the level GATE Statistics and JAM Mathematical Statistics examine and that measure-theoretic probability later builds on.
In the exams: GATE ST 'Calculus' section (heavy); IIT JAM MS mathematics component (heavy); AIEEA-PG Group 07 Unit II-III mathematics (present); ISI admission tests (heavy); UPSC ISS Statistics I (light)
How our questions for CUET-PG spread across the units of Real Analysis for Statistics. Difficulty: 23 easy, 3 medium, 0 hard.
| Sets, countability and the real number system | 4 | |
| Sequences: convergence, monotone and Cauchy sequences | 3 | |
| Limits, continuity and uniform continuity | 3 | |
| L'Hospital's rule and indeterminate forms | 3 | |
| Series: tests of convergence, absolute and conditional convergence | 2 | |
| Power series and radius of convergence | 2 | |
| Differentiability, Rolle's and mean value theorems, Taylor's theorem | 2 | |
| Riemann integration and improper integrals | 2 | |
| Gamma and beta functions | 2 | |
| Double and triple integrals and change of variables | 2 | |
| Functions of several variables: partial derivatives, Jacobians, maxima, minima and Lagrange multipliers | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
What are the supremum and infimum of S = {1 − 1/n : n = 1, 2, 3, …}?
C. sup 1 (not attained), inf 0 (attained)
Terms are 0, 1/2, 2/3, …, increasing towards 1 without reaching it; the minimum 0 occurs at n = 1.
Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)
Which sequence diverges?
A. (−1)ⁿ
(−1)ⁿ has subsequences tending to 1 and −1. The others tend to 0, 0 and 1.
Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)
Which statement is true for real sequences?
B. Every bounded sequence has a convergent subsequence
Bolzano–Weierstrass gives the first. Monotone sequences converge only if bounded; convergent sequences need not be monotone.
Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)
The p-series Σ1/nᵖ converges if and only if:
A. p > 1
At p = 1 it is the divergent harmonic series; the integral test gives convergence exactly for p > 1.
Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)
What is the radius of convergence of Σxⁿ/n!?
C. ∞
aₙ/aₙ₊₁ = n + 1 → ∞; the series is eˣ for all x.
Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)
The radius of convergence of Σn!xⁿ is:
C. 0
aₙ₊₁/aₙ = n + 1 → ∞, so the series converges only at x = 0.
Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)
What is lim (1 − cos x)/x² as x → 0?
D. 1/2
Applying L'Hospital twice: sin x/(2x) → 1/2; or 1 − cos x ≈ x²/2.
Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)
What is lim (ln x)/x as x → ∞?
D. 0
L'Hospital gives (1/x)/1 → 0: logarithms grow more slowly than x.
Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)
What is ∫₀¹ ln x dx?
C. −1
x ln x − x from 0 to 1 = (0 − 1) − 0 = −1; the integral is improper at 0 but converges.
Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)
Ten here, 26 in the app. The Laisha app has every practice question we have written to CUET-PG in this subject, marked as you go, with timed papers, a jump-to-unanswered list and a certificate when you pass at 70 per cent.
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