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IIT JAM MS study materials
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IIT JAM MS practice questions

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Question 1Mathematics

For a real symmetric matrix A of order 3 with eigenvalues 1, 2, 3, which statement about A is necessarily true?

  1. A is singular
  2. A is diagonalisable by an orthogonal matrix Correct answer.
  3. A is nilpotent
  4. A has three distinct real eigenvalues but cannot be orthogonally diagonalised

Explanation

The spectral theorem states that every real symmetric matrix is orthogonally diagonalisable, that is, there exists an orthogonal matrix P with P transpose times A times P diagonal. Since the given eigenvalues 1, 2, 3 are all nonzero, A is also nonsingular, but the key structural fact is orthogonal diagonalisability.

Question 2Mathematics

Let f(x) = x^3 - 3x on the closed interval [-2, 2]. At which points of this interval does f attain its global maximum?

  1. x = 2
  2. x = -2
  3. x = -1 Correct answer.
  4. x = 1

Explanation

The derivative f'(x) = 3x^2 - 3 vanishes at x = -1 and x = 1. Evaluating f at the critical points and endpoints gives f(-2) = -2, f(-1) = 2, f(1) = -2, f(2) = 2. The largest value is 2, attained at both x = -1 and x = 2, so among the listed options only x = -1 gives the maximum value.

Question 3Mathematics

A fair six-sided die is rolled twice independently. What is the probability that the sum of the two outcomes equals 7?

  1. 1/6 Correct answer.
  2. 5/36
  3. 1/9
  4. 1/12

Explanation

There are 36 equally likely ordered pairs. The pairs summing to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), six in total. The probability is 6/36 = 1/6.

Question 4Mathematics

The group Z_12 (integers modulo 12 under addition) has how many elements of order exactly 4?

  1. 4
  2. 2 Correct answer.
  3. 3
  4. 1

Explanation

In the cyclic group Z_n, the number of elements of order d (where d divides n) equals Euler's totient phi(d). Here d = 4 and phi(4) = 2, namely the elements 3 and 9 modulo 12. Each has additive order 4.

Question 5Mathematics

What is the radius of convergence of the power series sum from n=1 to infinity of (x^n)/n^2?

  1. infinity
  2. 0
  3. 1 Correct answer.
  4. 2

Explanation

Apply the ratio test to the coefficients a_n = 1/n^2. The limit of |a_{n+1}/a_n| equals the limit of n^2/(n+1)^2, which is 1. Hence the radius of convergence is R = 1/1 = 1, and the series converges absolutely for |x| < 1.

Question 6Mathematics

If A is a 3x3 matrix with det(A) = 5, what is det(2A)?

  1. 40 Correct answer.
  2. 80
  3. 20
  4. 10

Explanation

For an n x n matrix A and scalar c, det(cA) = c^n det(A). Here n = 3 and c = 2, so det(2A) = 2^3 times 5 = 8 times 5 = 40.

Question 7Mathematics

Which of the following sequences of real numbers converges?

  1. a_n = (-1)^n
  2. a_n = log n
  3. a_n = n
  4. a_n = n sin(1/n) Correct answer.

Explanation

For a_n = n sin(1/n), write n sin(1/n) = sin(1/n)/(1/n). As n tends to infinity, 1/n tends to 0 and sin(t)/t tends to 1, so a_n converges to 1. The other sequences are unbounded or oscillate without settling.

Question 8Mathematics

The number of onto (surjective) functions from a set of 3 elements to a set of 2 elements is

  1. 3
  2. 8
  3. 2
  4. 6 Correct answer.

Explanation

The total number of functions from a 3-element set to a 2-element set is 2^3 = 8. The functions that are not onto are the two constant functions, each mapping everything to one of the two target elements. Hence the number of onto functions is 8 - 2 = 6.

Question 9Mathematics

Let X be a random variable with mean 4 and variance 9. What is the variance of Y = 2X - 3?

  1. 15
  2. 36 Correct answer.
  3. 33
  4. 18

Explanation

For constants a and b, Var(aX + b) = a^2 Var(X). Here a = 2 and b = -3, so Var(2X - 3) = 4 times 9 = 36. The additive constant does not affect the variance.

Question 10Mathematics

Consider the function f: R to R defined by f(x) = x^3 + x. Which statement is correct?

  1. f is neither injective nor surjective
  2. f is surjective but not injective
  3. f is injective but not surjective
  4. f is bijective Correct answer.

Explanation

The derivative f'(x) = 3x^2 + 1 is strictly positive for all real x, so f is strictly increasing and hence injective. As x tends to plus or minus infinity, f(x) tends to plus or minus infinity respectively, so by the intermediate value theorem f is also surjective. Therefore f is bijective.

Question 11Mathematics

In the vector space R^3, let u = (1, 0, 1) and v = (0, 1, 1). The dimension of the subspace spanned by u and v is

  1. 2 Correct answer.
  2. 3
  3. 0
  4. 1

Explanation

The vectors u and v are not scalar multiples of each other, since u has a nonzero first coordinate and zero second coordinate while v has zero first and nonzero second. Hence they are linearly independent, and the span of two independent vectors has dimension 2.

Question 12Mathematics

Evaluate the double integral over the unit square [0,1] x [0,1] of (x + y) dx dy.

  1. 1/2
  2. 2
  3. 3/2
  4. 1 Correct answer.

Explanation

By linearity, integrate x + y over the unit square. The integral of x over the square is (1/2) times 1 = 1/2, and by symmetry the integral of y is also 1/2. Summing gives 1/2 + 1/2 = 1.

Question 13Mathematics

Let G be a group of order 15. Which statement about G is necessarily true?

  1. G is isomorphic to Z_3 x Z_5 but not cyclic
  2. G is cyclic Correct answer.
  3. G is non-abelian
  4. G has no element of order 5

Explanation

By the Sylow theorems, the number of Sylow 5-subgroups divides 3 and is congruent to 1 modulo 5, so it must be 1; similarly the Sylow 3-subgroup is unique. Both are normal, and since 3 and 5 are coprime, G is their direct product, isomorphic to Z_3 x Z_5, which is cyclic because gcd(3,5) = 1.

Question 14Mathematics

Let A be a 2x2 real matrix with trace 5 and determinant 6. What are the eigenvalues of A?

  1. 2 and 3 Correct answer.
  2. 1 and 6
  3. 0 and 5
  4. 5 and 6

Explanation

For a 2x2 matrix, the characteristic polynomial is lambda^2 - (trace) lambda + (determinant). Here it is lambda^2 - 5 lambda + 6, which factors as (lambda - 2)(lambda - 3). Hence the eigenvalues are 2 and 3.

Question 15Mathematics

A random variable X has the probability density f(x) = 2x for 0 < x < 1 and f(x) = 0 otherwise. What is E[X]?

  1. 3/4
  2. 2/3 Correct answer.
  3. 1/2
  4. 1/3

Explanation

The expectation is the integral of x times f(x) over the support. Compute the integral from 0 to 1 of x times 2x dx, which equals the integral of 2x^2 dx from 0 to 1, giving 2/3.

Question 16Mathematics

The function f(x) = |x| on the interval [-1, 1] is

  1. continuous but not differentiable at x = 0 Correct answer.
  2. differentiable everywhere on (-1, 1)
  3. continuous only at x = 0
  4. discontinuous at x = 0

Explanation

The absolute value function is continuous on the whole real line, so it is continuous on [-1, 1]. At x = 0, the left derivative is -1 and the right derivative is +1, so the derivative does not exist there. Hence f is continuous but not differentiable at zero.

Question 17Mathematics

In how many ways can the letters of the word ARRANGE be arranged so that the two R's are not adjacent?

  1. 720
  2. 1260
  3. 900 Correct answer.
  4. 360

Explanation

The word ARRANGE has 7 letters with A appearing twice and R appearing twice, giving 7!/(2!2!) = 1260 total arrangements. The arrangements with the two R's adjacent can be counted by treating RR as a single block, yielding 6!/2! = 360. Subtracting gives 1260 - 360 = 900.

Question 18Mathematics

Let f: R to R be defined by f(x) = x^2. Which statement about f is correct?

  1. f is bijective
  2. f is not injective and not surjective onto R Correct answer.
  3. f is injective
  4. f is neither injective nor surjective

Explanation

The function x squared is not injective because f(1) = f(-1) = 1 with 1 not equal to -1. It is also not surjective onto R because no real x satisfies x^2 = -1. Hence f is neither injective nor surjective as a map from R to R.

Question 19Mathematics

The matrix A = [[2, 1], [1, 2]] has eigenvalues 1 and 3. Which vector is an eigenvector corresponding to the eigenvalue 3?

  1. (1, 1) Correct answer.
  2. (1, -1)
  3. (0, 1)
  4. (1, 0)

Explanation

Compute A times (1, 1) = (2 + 1, 1 + 2) = (3, 3) = 3 times (1, 1). So (1, 1) is an eigenvector with eigenvalue 3. For eigenvalue 1, the eigenvector is (1, -1).

Question 20Mathematics

Evaluate the limit as x tends to 0 of (sin 3x)/x.

  1. 3 Correct answer.
  2. 1
  3. does not exist
  4. 0

Explanation

Write (sin 3x)/x as 3 times (sin 3x)/(3x). As x tends to 0, 3x also tends to 0, so (sin 3x)/(3x) tends to 1. Therefore the limit equals 3 times 1 = 3.

Question 21Mathematics

A fair coin is tossed until the first head appears. What is the expected number of tosses?

  1. 2 Correct answer.
  2. 3
  3. 4
  4. 1

Explanation

The number of tosses follows a geometric distribution with success probability p = 1/2. The expectation of a geometric random variable counting the trial of the first success is 1/p, which is 1/(1/2) = 2.

Question 22Mathematics

Let A be a 3x3 real matrix with A^3 = 0 but A^2 not equal to 0. Which statement about A is necessarily true?

  1. A is diagonalisable over R
  2. A has three distinct eigenvalues
  3. A is invertible
  4. A is nilpotent and all its eigenvalues are 0 Correct answer.

Explanation

A matrix satisfying A^k = 0 for some positive integer k is nilpotent. If lambda is an eigenvalue with eigenvector v, then A^3 v = lambda^3 v = 0, forcing lambda = 0. Hence every eigenvalue of A is zero, and A cannot be diagonalisable since A is nonzero.

Question 23Mathematics

Let f_n(x) = x^n on [0, 1]. Consider the pointwise limit f of the sequence (f_n). Which statement is correct?

  1. f_n does not converge pointwise on [0, 1]
  2. f_n converges uniformly on [0, 1] because each f_n is continuous
  3. f_n converges uniformly to f on [0, 1]
  4. f_n converges pointwise but not uniformly to f on [0, 1] Correct answer.

Explanation

For 0 <= x < 1, x^n tends to 0, while at x = 1 each term equals 1, so the pointwise limit is f(x) = 0 for x < 1 and f(1) = 1. Since the limit function is discontinuous while each f_n is continuous, the convergence cannot be uniform on [0, 1].

Question 24Mathematics

Let X and Y be independent exponential random variables with the same rate parameter 1. What is P(X < Y)?

  1. 1/2 Correct answer.
  2. 2/3
  3. 1/4
  4. 1/3

Explanation

By symmetry, since X and Y are independent and identically distributed continuous random variables, the probability that X < Y equals the probability that Y < X, and the probability of a tie is zero. Hence each probability equals 1/2, which can also be verified by integrating the joint density.

Question 25Mathematics

Let G be a finite group and H a subgroup of index 2. Which statement is necessarily true?

  1. H has index 3 in G
  2. H is trivial
  3. H is a normal subgroup of G Correct answer.
  4. G is abelian

Explanation

A subgroup of index 2 in a finite group is always normal. The two left cosets are H and its complement, and the two right cosets are the same pair, so gH = Hg for every g in G. Normality follows without any additional hypothesis on G.

Question 26Mathematics

For the real sequence a_n = (3n^2 + 2n)/(n^2 + 5), what is the limit as n tends to infinity?

  1. 2
  2. 1
  3. 3 Correct answer.
  4. 0

Explanation

Dividing numerator and denominator by n^2 gives (3 + 2/n)/(1 + 5/n^2). As n grows, 2/n and 5/n^2 tend to 0, so the ratio tends to 3/1 = 3.

Question 27Mathematics

A relation R on a set A is called an equivalence relation when it satisfies which combination of properties?

  1. Reflexive, antisymmetric, and transitive
  2. Reflexive, symmetric, and transitive Correct answer.
  3. Irreflexive, symmetric, and transitive
  4. Symmetric and transitive only

Explanation

An equivalence relation is defined by reflexivity (aRa for all a), symmetry (aRb implies bRa), and transitivity (aRb and bRc imply aRc). These three properties guarantee that R partitions A into disjoint equivalence classes.

Question 28Mathematics

The system of linear equations x + y + z = 3, x - y + z = 1, 2x + 2z = 4 has which solution set?

  1. A two-parameter family of solutions
  2. No solution
  3. A unique solution
  4. A one-parameter family of solutions Correct answer.

Explanation

The third equation is exactly twice the first, so only two independent equations remain in three unknowns. The rank of the coefficient matrix is 2, less than the number of variables 3, so the solution set is an affine line with one free parameter.

Question 29Mathematics

If the probability generating function of a non-negative integer-valued random variable X is G(s) = 1/(2 - s), what is P(X = 1)?

  1. 1
  2. 1/2
  3. 1/3
  4. 1/4 Correct answer.

Explanation

Expanding G(s) = (1/2)(1 - s/2)^{-1} gives the geometric series with coefficients P(X = k) = 1/2^{k+1}. Setting k = 1 yields 1/4, which also equals G'(0).

Question 30Mathematics

What is the value of the limit of (1 - cos x)/x^2 as x tends to 0?

  1. 1
  2. 1/2 Correct answer.
  3. Does not exist
  4. 0

Explanation

Using 1 - cos x = 2 sin^2(x/2), the expression becomes 2 sin^2(x/2)/x^2, which tends to 2*(1/2)^2 = 1/2. L'Hopital's rule applied twice gives the same value.

Question 31Mathematics

Let f(x, y) = x^2 y + y^3. What is the gradient of f at the point (1, 2)?

  1. (2, 4)
  2. (4, 13) Correct answer.
  3. (1, 2)
  4. (4, 6)

Explanation

The partial derivatives are f_x = 2xy and f_y = x^2 + 3y^2. Evaluating at (1, 2) gives f_x = 4 and f_y = 1 + 12 = 13, so the gradient is (4, 13).

Question 32Mathematics

A fair six-sided die is rolled twice. Given that the sum of the two rolls is 8, what is the probability that the first roll is 3?

  1. 1/5 Correct answer.
  2. 2/5
  3. 1/6
  4. 1/4

Explanation

The pairs summing to 8 are (2,6), (3,5), (4,4), (5,3), (6,2), five equally likely outcomes. Exactly one has first roll 3, so the conditional probability is 1/5.

Question 33Mathematics

How many distinct necklaces can be formed using 4 beads, each either red or blue, where rotations of the necklace are considered the same?

  1. 4
  2. 16
  3. 6 Correct answer.
  4. 8

Explanation

Burnside's lemma on the cyclic group C4 acting on 2^4 = 16 colorings: the identity fixes 16, each of the two 90-degree rotations fixes 2 (monochromatic), and the 180-degree rotation fixes 4. The average is (16 + 2 + 4 + 2)/4 = 6.

Question 34Mathematics

Let X be uniformly distributed on the interval (0, 1). What is the probability density function of Y = -2 ln X?

  1. 2 e^{-2y} for y > 0
  2. (1/2) e^{-y} for y > 0
  3. e^{-y} for y > 0
  4. (1/2) e^{-y/2} for y > 0 Correct answer.

Explanation

The transformation is decreasing, so f_Y(y) = f_X(x)|dx/dy| with x = e^{-y/2} and |dx/dy| = (1/2)e^{-y/2}. Since f_X = 1 on (0,1), we get f_Y(y) = (1/2)e^{-y/2} for y > 0, an exponential distribution with mean 2.

Question 35Mathematics

Let V be the vector space of all real polynomials of degree at most 3. What is the dimension of the subspace W of polynomials p with p(1) = 0?

  1. 1
  2. 4
  3. 3 Correct answer.
  4. 2

Explanation

The evaluation map p -> p(1) is a nonzero linear functional on the 4-dimensional space V, so its kernel W has dimension 4 - 1 = 3. A basis is {(x - 1), (x - 1)^2, (x - 1)^3}.

Question 36Mathematics

The series sum over n from 1 to infinity of 1/n^p converges if and only if which condition on p holds?

  1. p >= 1
  2. p < 1
  3. p > 0
  4. p > 1 Correct answer.

Explanation

The p-series converges exactly when p > 1, by the integral test comparing with the integral of x^{-p}. At p = 1 the harmonic series diverges, and for p < 1 the terms decay too slowly.

Question 37Mathematics

In the group S_3 of permutations of three symbols, what is the order of the element (1 2 3)?

  1. 1
  2. 3 Correct answer.
  3. 6
  4. 2

Explanation

A 3-cycle has order equal to its length, so (1 2 3)^3 is the identity and no smaller positive power is. Hence the order is 3, and it generates a cyclic subgroup of order 3.

Question 38Mathematics

Let A be a 2x2 real symmetric matrix with eigenvalues 2 and 5. Which statement about A is necessarily true?

  1. A has a repeated eigenvalue
  2. A is diagonalizable by an orthogonal matrix Correct answer.
  3. A is the zero matrix
  4. A is not invertible

Explanation

The spectral theorem states that every real symmetric matrix is orthogonally diagonalizable. So A = Q D Q^T with Q orthogonal and D = diag(2, 5).

Question 39Mathematics

What is the value of the definite integral from 0 to 1 of x e^x dx?

  1. 1 Correct answer.
  2. e
  3. 1/2
  4. e - 1

Explanation

Integration by parts with u = x and dv = e^x dx gives [x e^x - e^x] from 0 to 1. Evaluating yields (e - e) - (0 - 1) = 1.

Question 40Mathematics

A function f from R to R is continuous on the closed interval [a, b]. Which conclusion follows from the extreme value theorem?

  1. f is injective on [a, b]
  2. f is differentiable on (a, b)
  3. f(a) = f(b)
  4. f attains both a maximum and a minimum on [a, b] Correct answer.

Explanation

The extreme value theorem guarantees that a continuous function on a closed bounded interval attains its maximum and minimum values there. Continuity alone does not imply differentiability, injectivity, or equal endpoint values.

Question 41Mathematics

Let f(x) = x^3 - 3x. On which interval is f strictly increasing?

  1. (-1, 1)
  2. (-infinity, -1) union (1, infinity) Correct answer.
  3. (0, infinity)
  4. (-infinity, 0)

Explanation

The derivative f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1) is positive exactly when |x| > 1. Hence f is strictly increasing on (-infinity, -1) and on (1, infinity).

Question 42Mathematics

Let X be a random variable with moment generating function M(t) = (1 - 3t)^{-1} for t < 1/3. What is Var(X)?

  1. 3
  2. 9 Correct answer.
  3. 1
  4. 6

Explanation

This is the MGF of an exponential distribution with rate 1/3, whose mean and variance are both 9. Alternatively, differentiating log M(t) = -ln(1 - 3t) twice at 0 gives variance 9.

Question 43Mathematics

Let f be a continuous function on [0, 1]. What is the limit as n tends to infinity of the integral from 0 to 1 of x^n f(x) dx?

  1. f(1)
  2. 0 Correct answer.
  3. Infinity
  4. f(0)

Explanation

Because x^n concentrates near 1 but its total mass integral x^n dx = 1/(n+1) tends to 0, and f is bounded, the integral is bounded by max|f|/(n+1), which tends to 0.

Question 44Mathematics

A binary operation * on the set of integers is defined by a * b = a + b + 2. Which element is the identity for *?

  1. -1
  2. 0
  3. 2
  4. -2 Correct answer.

Explanation

We need a * e = a, so a + e + 2 = a, giving e = -2. Checking: a * (-2) = a - 2 + 2 = a, and similarly on the other side.

Question 45Mathematics

What is the coefficient of x^5 in the expansion of (1 + x)^7?

  1. 7
  2. 5
  3. 35
  4. 21 Correct answer.

Explanation

By the binomial theorem, the coefficient of x^5 is C(7, 5) = C(7, 2) = 7*6/2 = 21.

Question 46Mathematics

A random variable X takes values 1, 2, 3 with probabilities 1/2, 1/3, 1/6 respectively. What is E[X]?

  1. 3/2
  2. 11/6
  3. 2
  4. 5/3 Correct answer.

Explanation

Compute E[X] = 1*(1/2) + 2*(1/3) + 3*(1/6) = 1/2 + 2/3 + 1/2 = 5/3. The probabilities sum to 1, confirming a valid distribution.

Question 47Mathematics

Let A be a 3x3 matrix whose characteristic polynomial is (lambda - 1)^2 (lambda - 4). Which statement about A is necessarily true?

  1. A is diagonalizable
  2. A has three distinct eigenvalues
  3. The eigenvalues 1 and 4 are the only eigenvalues of A Correct answer.
  4. A is the identity matrix

Explanation

The eigenvalues of a matrix are exactly the roots of its characteristic polynomial. The polynomial has roots 1 (multiplicity 2) and 4, so these are the only eigenvalues, though diagonalizability depends on the eigenspace dimension for lambda = 1.

Question 48Mathematics

Let f(x) = e^{-1/x^2} for x not equal to 0, and f(0) = 0. Which statement about f is correct?

  1. f is continuous but not differentiable at 0
  2. f'(0) exists and is nonzero
  3. f is infinitely differentiable and all derivatives at 0 vanish Correct answer.
  4. f is not continuous at 0

Explanation

As x tends to 0, e^{-1/x^2} tends to 0 faster than any power of x, so f is smooth at 0 with all derivatives equal to 0. Its Taylor series at 0 is identically zero, so f is not analytic there.

Question 49Mathematics

Two events A and B satisfy P(A) = 0.5, P(B) = 0.4, and P(A union B) = 0.7. What is P(A intersect B)?

  1. 0.3
  2. 0.9
  3. 0.1
  4. 0.2 Correct answer.

Explanation

By inclusion-exclusion, P(A intersect B) = P(A) + P(B) - P(A union B) = 0.5 + 0.4 - 0.7 = 0.2. This is non-negative and no larger than min(P(A), P(B)), as required.

Question 50Mathematics

A fair coin is flipped 100 times. Using the normal approximation with continuity correction, which expression best approximates the probability of getting at most 45 heads?

  1. Phi((45 - 50)/5)
  2. Phi((45.5 - 50)/5) Correct answer.
  3. Phi((45.5 - 50)/25)
  4. Phi((50 - 45.5)/5)

Explanation

The number of heads has mean 50 and standard deviation sqrt(100 * 1/4) = 5. With continuity correction, P(X <= 45) is approximated by Phi((45.5 - 50)/5) = Phi(-0.9).