JAM Practice Set 1

JAM · 2026 · with answers and explanations

questions
30
Total marks
60
Duration
180 minutes
Negative marking
−0.67

Subjects: Mathematics

Question 1

Mathematics · 2 marks · −0.67

The sequence aₙ = 1/n converges to:

  1. A.1
  2. B.0
  3. C.∞
  4. D.The sequence diverges

Correct answer: B

Explanation: As n → ∞, 1/n → 0. For any ε > 0, choosing N = ⌈1/ε⌉ ensures |1/n − 0| < ε for all n > N.

Question 2

Mathematics · 2 marks · −0.67

The series ∑(n=1 to ∞) (−1)ⁿ/n is:

  1. A.Absolutely convergent
  2. B.Conditionally convergent
  3. C.Divergent
  4. D.Oscillating

Correct answer: B

Explanation: The series ∑(−1)ⁿ/n converges by the alternating series test (Leibniz test), but ∑|(-1)ⁿ/n| = ∑1/n (harmonic series) diverges. Hence it is conditionally convergent.

Question 3

Mathematics · 2 marks · −0.67

The radius of convergence of the power series ∑(n=0 to ∞) xⁿ/n! is:

  1. A.0
  2. B.1
  3. C.e
  4. D.∞

Correct answer: D

Explanation: Using the ratio test: lim |aₙ₊₁/aₙ| = lim |x|/(n+1) = 0 for all x. Since the limit is 0 < 1 for every x, the series converges for all real x, giving an infinite radius of convergence. This is the Taylor series for eˣ.

Question 4

Mathematics · 2 marks · −0.67

Rolle's theorem guarantees that if f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c ∈ (a,b) such that:

  1. A.f(c) = 0
  2. B.f'(c) = 0
  3. C.f''(c) = 0
  4. D.f(c) = (f(a) + f(b))/2

Correct answer: B

Explanation: Rolle's theorem guarantees at least one point c in (a,b) where f'(c) = 0, meaning the tangent line is horizontal. This is a special case of the mean value theorem when f(a) = f(b).

Question 5

Mathematics · 2 marks · −0.67

The function f(x, y) = x² + y² − 2x − 4y + 5 has a minimum at the point:

  1. A.(0, 0)
  2. B.(1, 2)
  3. C.(2, 4)
  4. D.(−1, −2)

Correct answer: B

Explanation: Setting ∂f/∂x = 2x − 2 = 0 and ∂f/∂y = 2y − 4 = 0 gives the critical point (1, 2). The Hessian determinant fₓₓfᵧᵧ − fₓᵧ² = 4 > 0 and fₓₓ = 2 > 0, confirming a local minimum.

Question 6

Mathematics · 2 marks · −0.67

If f(x, y) = x³y + xy³, then the mixed partial derivative ∂²f/∂x∂y at (1, 1) is:

  1. A.4
  2. B.6
  3. C.8
  4. D.2

Correct answer: B

Explanation: ∂f/∂y = x³ + 3xy². Then ∂²f/∂x∂y = ∂/∂x(x³ + 3xy²) = 3x² + 3y². At (1, 1): 3(1)² + 3(1)² = 6.

Question 7

Mathematics · 2 marks · −0.67

By the Fundamental Theorem of Calculus, the value of d/dx[∫₀ˣ sin(t²) dt] equals:

  1. A.cos(x²)
  2. B.sin(x²)
  3. C.2x sin(x²)
  4. D.−cos(x²)

Correct answer: B

Explanation: By the Fundamental Theorem of Calculus (Part 1), if F(x) = ∫₀ˣ f(t) dt, then F'(x) = f(x). Here f(t) = sin(t²), so the derivative is sin(x²).

Question 8

Mathematics · 2 marks · −0.67

The value of the Gamma function Γ(1/2) is:

  1. A.1
  2. B.√π
  3. C.π
  4. D.1/2

Correct answer: B

Explanation: Γ(1/2) = ∫₀^∞ t^(−1/2)e^(−t) dt = √π. This is derived using the substitution t = u² and evaluating the resulting Gaussian integral ∫₀^∞ e^(−u²) du = √π/2.

Question 9

Mathematics · 2 marks · −0.67

The rank of the zero matrix of size m × n is:

  1. A.0
  2. B.1
  3. C.min(m, n)
  4. D.max(m, n)

Correct answer: A

Explanation: The zero matrix has no non-zero rows (or columns), so its row echelon form has no pivots. Therefore its rank is 0.

Question 10

Mathematics · 2 marks · −0.67

If A is a 2×2 matrix with eigenvalues 3 and 7, what is det(A)?

  1. A.10
  2. B.4
  3. C.21
  4. D.−4

Correct answer: C

Explanation: The determinant of a matrix equals the product of its eigenvalues. det(A) = 3 × 7 = 21.

Question 11

Mathematics · 2 marks · −0.67

For the data set {2, 4, 6, 8, 10}, the arithmetic mean is:

  1. A.5
  2. B.6
  3. C.7
  4. D.8

Correct answer: B

Explanation: Mean = (2 + 4 + 6 + 8 + 10)/5 = 30/5 = 6.

Question 12

Mathematics · 2 marks · −0.67

A box contains 3 red and 2 blue balls. Two balls are drawn without replacement. Given that the first ball is red, the probability that the second ball is also red is:

  1. A.3/5
  2. B.2/5
  3. C.1/2
  4. D.1/4

Correct answer: C

Explanation: After drawing a red ball first, 4 balls remain: 2 red and 2 blue. P(second red | first red) = 2/4 = 1/2.

Question 13

Mathematics · 2 marks · −0.67

If the moment generating function of X is M_X(t) = e^(3t + 2t²), then E[X] is:

  1. A.2
  2. B.3
  3. C.5
  4. D.6

Correct answer: B

Explanation: E[X] = M'_X(0). Differentiating: M'_X(t) = (3 + 4t)e^(3t + 2t²). At t = 0: M'_X(0) = 3 × 1 = 3.

Question 14

Mathematics · 2 marks · −0.67

If X ~ Binomial(10, 0.3), what is E[X]?

  1. A.3
  2. B.0.3
  3. C.2.1
  4. D.7

Correct answer: A

Explanation: For a Binomial(n, p) distribution, E[X] = np = 10 × 0.3 = 3.

Question 15

Mathematics · 2 marks · −0.67

If X follows an Exponential distribution with rate λ = 2, what is P(X > 1)?

  1. A.e⁻²
  2. B.1 − e⁻²
  3. C.2e⁻²
  4. D.e⁻¹

Correct answer: A

Explanation: For an Exponential(λ) distribution, P(X > x) = e^(−λx). With λ = 2: P(X > 1) = e^(−2) ≈ 0.1353.

Question 16

Mathematics · 2 marks · −0.67

If (X, Y) follows a bivariate normal distribution with correlation coefficient ρ = 0, then X and Y are:

  1. A.Uncorrelated but possibly dependent
  2. B.Independent
  3. C.Positively correlated
  4. D.Identically distributed

Correct answer: B

Explanation: For the bivariate normal distribution, zero correlation (ρ = 0) implies independence. This is a special property unique to the bivariate normal — for arbitrary distributions, zero correlation does not generally imply independence.

Question 17

Mathematics · 2 marks · −0.67

If Z₁, Z₂, ..., Zₙ are independent standard normal random variables, then ∑ᵢ₌₁ⁿ Zᵢ² follows:

  1. A.Normal distribution
  2. B.t distribution with n degrees of freedom
  3. C.Chi-square distribution with n degrees of freedom
  4. D.F distribution

Correct answer: C

Explanation: By definition, the sum of squares of n independent standard normal random variables follows a Chi-square distribution with n degrees of freedom: ∑Zᵢ² ~ χ²(n).

Question 18

Mathematics · 2 marks · −0.67

For a sample of size n from a continuous distribution, the CDF of the maximum order statistic X₍ₙ₎ is:

  1. A.[F(x)]ⁿ
  2. B.1 − [1 − F(x)]ⁿ
  3. C.nF(x)[1 − F(x)]ⁿ⁻¹
  4. D.F(x)/n

Correct answer: A

Explanation: The CDF of the maximum X₍ₙ₎ is P(X₍ₙ₎ ≤ x) = P(all Xᵢ ≤ x) = [F(x)]ⁿ, since the observations are independent with common CDF F(x).

Question 19

Mathematics · 2 marks · −0.67

A sufficient statistic T(X) for parameter θ captures:

  1. A.All information about θ contained in the sample
  2. B.Only the first two moments
  3. C.The median of the distribution
  4. D.The maximum likelihood estimate only

Correct answer: A

Explanation: By the sufficiency principle, a sufficient statistic T(X) contains all the information about θ present in the entire sample. The conditional distribution of the sample given T does not depend on θ (Fisher-Neyman factorization).

Question 20

Mathematics · 2 marks · −0.67

The maximum likelihood estimator of the mean μ for a normal population N(μ, σ²) based on a random sample is:

  1. A.The sample median
  2. B.The sample mean X̄
  3. C.The sample mode
  4. D.(X₍₁₎ + X₍ₙ₎)/2

Correct answer: B

Explanation: Maximizing the likelihood function with respect to μ yields the MLE μ̂ = X̄ = (1/n)∑xᵢ, the sample mean.

Question 21

Mathematics · 2 marks · −0.67

A 95% confidence interval for a population mean becomes narrower when:

  1. A.The confidence level is increased
  2. B.The sample size is increased
  3. C.The population variance increases
  4. D.The sample size is decreased

Correct answer: B

Explanation: The width of a confidence interval is proportional to 1/√n. Increasing the sample size n reduces the standard error and narrows the interval, giving a more precise estimate.

Question 22

Mathematics · 2 marks · −0.67

The Neyman-Pearson lemma states that the most powerful test of size α for H₀: θ = θ₀ versus H₁: θ = θ₁ rejects H₀ when:

  1. A.The p-value is less than α
  2. B.The likelihood ratio L(θ₁)/L(θ₀) exceeds a constant
  3. C.The sample mean exceeds a threshold
  4. D.The test statistic follows a chi-square distribution

Correct answer: B

Explanation: The Neyman-Pearson lemma states that rejecting H₀ when the likelihood ratio L(θ₁|x)/L(θ₀|x) exceeds a constant k (chosen to achieve size α) yields the most powerful test among all tests of size α.

Question 23

Mathematics · 2 marks · −0.67

A uniformly most powerful (UMP) test of size α typically exists when:

  1. A.Both H₀ and H₁ are simple
  2. B.The family has a monotone likelihood ratio
  3. C.The test statistic is symmetric
  4. D.The alternative is two-sided

Correct answer: B

Explanation: UMP tests exist for one-sided alternatives when the family of distributions has the monotone likelihood ratio (MLR) property. For two-sided alternatives, UMP tests generally do not exist.

Question 24

Mathematics · 2 marks · −0.67

In simple linear regression, the least squares estimator of the slope β₁ is:

  1. A.S_XY / S_XX
  2. B.S_XX / S_XY
  3. C.S_YY / S_XY
  4. D.Ȳ − β₀X̄

Correct answer: A

Explanation: β̂₁ = S_XY / S_XX = ∑(xᵢ − x̄)(yᵢ − ȳ) / ∑(xᵢ − x̄)², obtained by minimizing the sum of squared residuals.

Question 25

Mathematics · 2 marks · −0.67

In one-way ANOVA with k groups and total sample size N, the degrees of freedom for the 'between groups' sum of squares is:

  1. A.N − 1
  2. B.k − 1
  3. C.N − k
  4. D.k(N − 1)

Correct answer: B

Explanation: In one-way ANOVA, the between-groups degrees of freedom = k − 1 (number of groups minus 1). The within-groups df = N − k, and the total df = N − 1.

Question 26

Mathematics · 2 marks · −0.67

In a finite irreducible aperiodic Markov chain, the stationary distribution:

  1. A.May not exist
  2. B.Exists but is not unique
  3. C.Exists and is unique
  4. D.Depends on the initial state

Correct answer: C

Explanation: A finite Markov chain that is irreducible and aperiodic has a unique stationary distribution π. Moreover, the chain converges to this distribution regardless of the initial state.

Question 27

Mathematics · 2 marks · −0.67

Which of the following improper integrals converges?

  1. A.∫₁^∞ 1/x dx
  2. B.∫₁^∞ 1/x² dx
  3. C.∫₀¹ 1/x dx
  4. D.∫₁^∞ 1/√x dx

Correct answer: B

Explanation: ∫₁^∞ 1/x² dx = [−1/x]₁^∞ = 0 − (−1) = 1, which is finite. The others diverge: ∫1/x (p = 1) and ∫1/√x (p = 1/2) diverge at infinity, and ∫₀¹ 1/x diverges at 0.

Question 28

Mathematics · 2 marks · −0.67

A square matrix A is invertible if and only if:

  1. A.A is symmetric
  2. B.det(A) ≠ 0
  3. C.A has at least one zero eigenvalue
  4. D.trace(A) = 0

Correct answer: B

Explanation: A square matrix is invertible (non-singular) if and only if its determinant is non-zero. This is equivalent to all eigenvalues being non-zero and the columns being linearly independent.

Question 29

Mathematics · 2 marks · −0.67

If X₁, ..., Xₙ are i.i.d. from N(μ, σ²), then the sample mean X̄ follows:

  1. A.N(μ, σ²)
  2. B.N(μ, σ²/n)
  3. C.N(0, 1)
  4. D.χ²(n)

Correct answer: B

Explanation: The sample mean of i.i.d. normal random variables is normally distributed: X̄ ~ N(μ, σ²/n).

Question 30

Mathematics · 2 marks · −0.67

The Kolmogorov-Smirnov one-sample test statistic is based on the:

  1. A.Sample mean
  2. B.Sample variance
  3. C.Maximum absolute difference between the empirical and hypothesized CDFs
  4. D.Chi-square statistic

Correct answer: C

Explanation: The KS test statistic is Dₙ = sup|Fₙ(x) − F₀(x)|, the supremum of the absolute difference between the empirical CDF and the hypothesized CDF. It is distribution-free and tests overall goodness of fit.

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