100+ Free IIT JAM MS Practice Questions

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Key Facts: IIT JAM MS Exam

60 questions

30 MCQ + 10 MSQ + 20 NAT for 100 marks

JAM 2026 Information Brochure

180 minutes

Total exam time (3 hours, CBT)

jam2026.iitb.ac.in

INR 2100

Application fee for one paper (General male)

JAM 2026 Brochure

IIT Bombay

Organizing institute for JAM 2026

jam2026.iitb.ac.in

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IIT JAM Mathematical Statistics (MS) is a 180-minute computer-based exam with 60 questions (30 MCQ + 10 MSQ + 20 NAT) for 100 marks, blending ~40% mathematics and ~60% statistics. Negative marking applies only on MCQs. Used for M.Sc. admission at IITs, IISc, IISERs, NITs and CFTIs.

Sample IIT JAM MS Practice Questions

Try these sample questions to test your IIT JAM MS exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Which of the following sequences is convergent?

A.aₙ = (-1)ⁿ

B.aₙ = n/(n+1)

C.aₙ = n²

D.aₙ = (-1)ⁿ · n

Explanation: The sequence aₙ = n/(n+1) = 1 - 1/(n+1) approaches 1 as n → ∞, so it converges to 1. A sequence converges if and only if it has a finite limit. The other three are either oscillating or unbounded.

2By the ratio test, the series Σ n!/nⁿ from n=1 to ∞ is:

A.Convergent

B.Divergent

C.Conditionally convergent

D.Oscillatory

Explanation: Apply the ratio test: |aₙ₊₁/aₙ| = (n+1)!/(n+1)ⁿ⁺¹ · nⁿ/n! = nⁿ/(n+1)ⁿ = 1/(1 + 1/n)ⁿ → 1/e < 1. Since the limit is less than 1, the series converges absolutely.

3A sequence (aₙ) is called Cauchy if:

A.aₙ → 0 as n → ∞

B.For every ε > 0, there exists N such that |aₘ - aₙ| < ε for all m, n ≥ N

C.aₙ is monotone and bounded

D.Σ aₙ converges

Explanation: A sequence is Cauchy if its terms get arbitrarily close to each other for sufficiently large indices: for every ε > 0, there exists N such that |aₘ - aₙ| < ε whenever m, n ≥ N. In the real numbers (which are complete), Cauchy sequences are exactly the convergent sequences.

4The series Σ (-1)ⁿ⁺¹/n from n=1 to ∞ is:

A.Absolutely convergent

B.Conditionally convergent

C.Divergent

D.Equal to 0

Explanation: The alternating series Σ(-1)ⁿ⁺¹/n converges to ln 2 by the Leibniz (alternating series) test since 1/n decreases monotonically to 0. However, the absolute series Σ 1/n is the harmonic series, which diverges. So the series is conditionally convergent.

5The radius of convergence of the power series Σ xⁿ/n! from n=0 to ∞ is:

A.0

B.1

C.e

D.∞

Explanation: Using the ratio test, |aₙ₊₁/aₙ| = |x|/(n+1) → 0 for any fixed x as n → ∞. Since the limit is 0 < 1 for all x ∈ ℝ, the series converges for every real x. Therefore the radius of convergence is ∞. This series equals eˣ.

6The comparison test states that if 0 ≤ aₙ ≤ bₙ and Σbₙ converges, then:

A.Σaₙ diverges

B.Σaₙ converges

C.Σaₙ may diverge or converge

D.aₙ → bₙ

Explanation: The comparison test says: if a non-negative series Σaₙ is bounded above term-by-term by a convergent series Σbₙ, then the partial sums of Σaₙ are bounded and non-decreasing, so Σaₙ also converges. The contrapositive: if Σaₙ diverges and 0 ≤ aₙ ≤ bₙ, then Σbₙ diverges.

7The series Σ 1/n^p converges if and only if:

A.p ≥ 1

B.p > 1

C.p < 1

D.p ≤ 1

Explanation: The p-series Σ 1/n^p converges for p > 1 and diverges for p ≤ 1 (the case p = 1 is the harmonic series, which diverges). This is established by the integral test comparing to ∫1/x^p dx.

8Every monotone bounded sequence of real numbers:

A.Diverges

B.Oscillates

C.Converges

D.Has no limit point

Explanation: The Monotone Convergence Theorem (a consequence of the completeness of ℝ) guarantees that every monotone (non-decreasing or non-increasing) sequence that is bounded above (or below) converges to its supremum (or infimum). This is one of the foundational theorems of real analysis.

9The limit lim n→∞ (1 + 1/n)ⁿ equals:

A.1

B.0

C.e

D.∞

Explanation: By definition, e = lim n→∞ (1 + 1/n)ⁿ ≈ 2.71828. This is the foundational limit used to define the natural exponential base. It can also be derived from the binomial expansion or the Taylor series for eˣ at x = 1.

10Which of the following is the Cauchy root test for Σ aₙ with aₙ ≥ 0?

A.Compute lim aₙ; series converges iff this limit is 0

B.Compute L = lim sup (aₙ)^(1/n); converges if L < 1, diverges if L > 1

C.Compute L = lim aₙ₊₁/aₙ; converges if L = 1

D.Compute Σ n·aₙ and check its convergence

Explanation: The Cauchy root test computes L = lim sup (aₙ)^(1/n). If L < 1 the series converges absolutely; if L > 1 it diverges; if L = 1 the test is inconclusive. The root test is at least as powerful as the ratio test.

About the IIT JAM MS Exam

IIT JAM Mathematical Statistics (Paper Code: MS) is one of seven test papers of the Joint Admission Test for Masters (JAM), conducted annually by the IITs and IISc for admission to M.Sc., M.Sc.-Ph.D. dual, and other post-graduate programs at the IITs, IISc, IISERs, NITs and CFTIs. JAM 2026 is being organized by IIT Bombay. The MS paper combines Mathematics (~40 percent) and Statistics (~60 percent) content into a single 3-hour computer-based exam carrying 100 marks across 60 questions in three formats: 30 Multiple Choice Questions (MCQ), 10 Multiple Select Questions (MSQ) and 20 Numerical Answer Type (NAT) questions. Only the MCQ section has negative marking; MSQ requires all correct options to be selected for any marks. The exam is typically held in February with results in March.

Questions

100 scored questions

Time Limit

180 minutes (3 hours)

Passing Score

No fixed pass; merit-based ranking and category cut-offs after results

Exam Fee

INR 2100 (one paper, General/OBC-NCL/EWS male); INR 1050 (female/SC/ST/PwD); INR 3000 (two papers, General male) (Joint Admission Test for Masters (JAM) — IITs and IISc; IIT Bombay is the organizing institute for JAM 2026)

IIT JAM MS Exam Content Outline

10%

Sequences and Series of Real Numbers

Convergence; Cauchy sequences; monotone sequences; comparison, ratio, root and Leibniz tests; absolute and conditional convergence; power series and radius of convergence

10%

Differential and Integral Calculus

Functions of one and two variables; limits, continuity, differentiability; mean value theorem and Taylor's theorem; maxima/minima; Riemann integration; double integrals; gradient, divergence, curl

Matrices and Determinants

Rank; inverse; determinants; eigenvalues and eigenvectors; system of linear equations; Cayley-Hamilton theorem

10%

Descriptive Statistics and Probability

Measures of central tendency and dispersion; moments, skewness, kurtosis; counting principles; sample spaces and events; axiomatic probability; conditional probability; Bayes' theorem; independence

12%

Univariate Distributions

Discrete and continuous random variables; pmf, pdf, distribution function; expectation, variance, moments, mgf; Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric, uniform, exponential, gamma, beta, normal, lognormal, chi-square, t, F

Multivariate Distributions

Joint, marginal and conditional distributions; covariance and correlation; functions of random variables and transformations; order statistics; bivariate normal

Limit Theorems

Modes of convergence; Chebyshev's inequality; Weak and Strong Laws of Large Numbers; Central Limit Theorem

Sampling Distributions

Random sampling; sample mean and variance; chi-square, t and F distributions; distribution of order statistics; sampling from normal populations

10%

Estimation

Point estimation — unbiased, consistent, sufficient, efficient estimators; method of moments; maximum likelihood; Cramer-Rao lower bound; UMVUE; interval estimation and confidence intervals

10%

Testing of Hypotheses

Type I and II errors and power; Neyman-Pearson lemma; uniformly most powerful tests; likelihood ratio tests; tests for means, variances and proportions; chi-square, t and F tests

Nonparametric Methods

Sign test; Wilcoxon signed-rank; Mann-Whitney U; Kruskal-Wallis; Spearman rank correlation; chi-square goodness-of-fit

Stochastic Processes

Markov chains; transition probabilities; classification of states; stationary distribution; Poisson process; pure birth, pure death and birth-death processes

How to Pass the IIT JAM MS Exam

What You Need to Know

Passing score: No fixed pass; merit-based ranking and category cut-offs after results
Exam length: 100 questions
Time limit: 180 minutes (3 hours)
Exam fee: INR 2100 (one paper, General/OBC-NCL/EWS male); INR 1050 (female/SC/ST/PwD); INR 3000 (two papers, General male)

Keys to Passing

Complete 500+ practice questions
Score 80%+ consistently before scheduling
Focus on highest-weighted sections
Use our AI tutor for tough concepts

IIT JAM MS Study Tips from Top Performers

1Master the calculus and linear algebra foundations early — sequences/series, multivariable calculus, eigenvalues and Cayley-Hamilton appear in roughly 30 percent of marks

2Memorize the mean, variance, mgf and key relationships of every standard distribution (binomial, Poisson, geometric, hypergeometric, normal, exponential, gamma, beta, chi-square, t, F) — direct recall questions are common

3For NAT questions, practice computing numerical answers with 2-decimal precision; mistakes in arithmetic and unit conversion are the largest source of lost marks

4Solve at least 8-10 full-length mock papers in CBT mode under timed conditions to internalize the 3-hour pacing and the navigation interface used on the actual JAM portal

5Build derivation muscle for estimation (MLE, method of moments, CRLB, UMVUE) and testing (Neyman-Pearson, UMP, LRT) — these sections distinguish top scorers from average ones

6For MSQ, do not guess — selecting any wrong option gives 0 marks even if other correct options are chosen, so attempt only when sure of all correct alternatives

Frequently Asked Questions

What is the exam pattern for IIT JAM Mathematical Statistics 2026?

JAM MS is a 3-hour computer-based test with 60 questions for 100 marks. There are three sections: Section A has 30 Multiple Choice Questions (MCQ), Section B has 10 Multiple Select Questions (MSQ), and Section C has 20 Numerical Answer Type (NAT) questions. Negative marking applies only on MCQs (-1/3 for 1-mark, -2/3 for 2-mark). MSQs award marks only if all correct options are chosen, with no partial credit; NATs have no negative marking.

Who conducts JAM 2026 and what is the official syllabus source?

JAM 2026 is conducted by IIT Bombay on behalf of the IITs and IISc. The official portal is jam2026.iitb.ac.in. The MS syllabus combines roughly 40 percent mathematics (sequences and series, differential and integral calculus, matrices and determinants) and 60 percent statistics (probability, distributions, sampling, estimation, testing of hypotheses, nonparametric methods, stochastic processes).

Which programs accept JAM MS scores?

JAM MS scores are used for admission to M.Sc. Statistics, M.Sc. Applied Statistics and Informatics, Joint M.Sc.-Ph.D. and Master's programs in Statistics, Data Science and related areas at IITs (Bombay, Kanpur, Kharagpur, Madras, Delhi etc.), IISc Bangalore, IISERs, NITs and several CFTIs. Each institute publishes its own program-wise cut-offs.

What is the application fee for JAM 2026?

The application fee for one paper is INR 2100 for General/OBC-NCL/EWS male candidates and INR 1050 for female candidates of any category and SC/ST/PwD candidates. For two papers, the fees are INR 3000 (General male) and INR 1500 (concessional). Fees are paid online via the JOAPS portal during registration.

When is JAM 2026 held and when are results announced?

JAM 2026 is scheduled for February 2026 (typically the first or second Sunday). Admit cards are released about three weeks before. Results are usually published in March, and the JOAPS counselling and admission portal opens in April for institute applications.

What is the eligibility for JAM MS?

Any candidate with a Bachelor's degree (or pursuing the final year) from a recognized institution is eligible. Mathematics or Statistics at the +2 level is required for most M.Sc. Statistics programs; specific course-wise eligibility (e.g., minimum CGPA, prior coursework in calculus, probability and linear algebra) is set by each admitting institute and listed in the JAM Information Brochure.

How should I prepare for JAM MS in 2026?

Start by mastering the foundational mathematics (calculus, sequences and series, linear algebra) since they account for roughly 40 marks. Then build on probability and standard distributions, sampling distributions, and the central limit theorem. The high-value scoring sections are estimation, hypothesis testing, and stochastic processes — practice derivations and numerical NAT-style problems. Take at least 8-12 full-length mock JAM MS papers and solve previous-year question papers (PYQs) on the JAM portal.