ISI B.Math Bachelor of Mathematics Admission Test — Questions and Answers

Question 1: What transformation does an orthogonal matrix represent geometrically?

Scaling
Projection
Shearing
Rotation or reflection (Correct answer)

Correct answer: Rotation or reflection

Orthogonal matrices (with QᵀQ = I) preserve lengths and angles, representing rotations and reflections.

Question 2: Fermat's Little Theorem states that if p is prime and gcd(a, p) = 1, then:

a^p ≡ a (mod p)
a² ≡ 1 (mod p)
a^(p−1) ≡ 1 (mod p) (Correct answer)
p ≡ 1 (mod a)

Correct answer: a^(p−1) ≡ 1 (mod p)

Fermat's Little Theorem: a^(p−1) ≡ 1 (mod p) for prime p not dividing a, a foundation of modular exponentiation.

Question 3: In set theory, the symmetric difference A △ B is defined as:

(A ∪ B) \ (A ∩ B) (Correct answer)
A ∪ B
A ∩ B
A \ B

Correct answer: (A ∪ B) \ (A ∩ B)

The symmetric difference A △ B consists of elements in A or B but not in both, i.e., (A ∪ B) \ (A ∩ B).

Question 4: The Pigeonhole Principle states that if n+1 objects are placed into n containers, then:

At least one container holds more than one object (Correct answer)
No two containers hold the same number of objects
Every container holds exactly one object
At least one container is empty

Correct answer: At least one container holds more than one object

The Pigeonhole Principle is a basic combinatorial result: distributing more objects than containers forces repetition.

Question 5: The commutator subgroup [G, G] (also called the derived subgroup) of G is the smallest normal subgroup N such that:

N is trivial
N equals G
G/N is cyclic
G/N is abelian (Correct answer)

Correct answer: G/N is abelian

The commutator subgroup [G, G] is defined as the smallest normal subgroup N for which the quotient G/N is abelian, measuring how far G deviates from being abelian.

Question 6: The Chinese Remainder Theorem guarantees a unique solution modulo m₁m₂ when gcd(m₁, m₂) equals:

m₁
1 (Correct answer)
m₁ + m₂
m₂

Correct answer: 1

The CRT applies when the moduli are pairwise coprime (gcd = 1), ensuring the simultaneous congruences have a unique solution modulo m₁m₂.

Question 7: For the system x' = Ax with constant matrix A, the stability of the equilibrium x = 0 depends on:

The trace of A
The real parts of the eigenvalues of A (Correct answer)
The determinant of A
Whether A is symmetric

Correct answer: The real parts of the eigenvalues of A

The zero solution is asymptotically stable if all eigenvalues have negative real parts, unstable if any eigenvalue has positive real part.

Question 8: Two groups G and H are isomorphic if and only if:

There exists a bijective homomorphism from G to H (Correct answer)
They are both abelian
They have the same number of elements
They have the same generators

Correct answer: There exists a bijective homomorphism from G to H

An isomorphism is a bijective homomorphism φ: G → H; its existence guarantees that G and H are structurally identical as groups.

Question 9: Which distribution models the number of successes in n independent Bernoulli trials with probability p?

Binomial distribution (Correct answer)
Normal distribution
Exponential distribution
Poisson distribution

Correct answer: Binomial distribution

The Binomial distribution B(n, p) counts successes in n independent trials, each with success probability p.

Question 10: Euler's formula for connected planar graphs states V − E + F equals:

1
0
2 (Correct answer)
3

Correct answer: 2

Euler's formula V − E + F = 2 relates vertices V, edges E, and faces F (including the outer face) of any connected planar graph.

Question 11: For the function f(x) = x³ − 3x, what are the local extrema?

No local extrema
Local max at x = 0
Local max at x = 1, local min at x = −1
Local max at x = −1, local min at x = 1 (Correct answer)

Correct answer: Local max at x = −1, local min at x = 1

f'(x) = 3x² − 3 = 0 gives x = ±1; f''(−1) < 0 (max) and f''(1) > 0 (min).

Question 12: The number of ways to arrange n distinct objects in a row is:

n²
n! (Correct answer)
2ⁿ
n

Correct answer: n!

There are n choices for the first position, n−1 for the second, and so on, giving n! = n × (n−1) × … × 1 total permutations.

Question 13: What is the fundamental theorem of calculus (Part 1) about?

If F(x) = ∫ₐˣ f(t) dt and f is continuous, then F'(x) = f(x) (Correct answer)
The derivative of a polynomial is one degree lower
Every continuous function has an antiderivative that is also continuous
The definite integral equals the limit of Riemann sums

Correct answer: If F(x) = ∫ₐˣ f(t) dt and f is continuous, then F'(x) = f(x)

Part 1 of the FTC states that differentiation and integration are inverse operations when f is continuous.

Question 14: What is the determinant of the 2×2 matrix [[a, b], [c, d]]?

ac − bd
a + d
ad − bc (Correct answer)
ab − cd

Correct answer: ad − bc

The determinant of a 2×2 matrix is ad − bc, computed by the difference of diagonal products.

Question 15: In a simple graph, the sum of all vertex degrees equals:

The number of edges
Half the number of vertices
The number of vertices
Twice the number of edges (Correct answer)

Correct answer: Twice the number of edges

Each edge contributes exactly 2 to the total degree sum (one for each endpoint), so Σ deg(v) = 2|E|, the handshaking lemma.

Question 16: A Type I error in hypothesis testing refers to:

Accepting a true null hypothesis
Failing to reject a false null hypothesis
Rejecting a true null hypothesis (Correct answer)
Accepting a false alternative hypothesis

Correct answer: Rejecting a true null hypothesis

A Type I error (false positive) occurs when we reject H₀ even though it is actually true; its probability is the significance level α.

Question 17: Cramer's Rule provides a formula for solving Ax = b using:

Eigenvalue decomposition
Ratios of determinants (Correct answer)
Matrix multiplication
The trace of A

Correct answer: Ratios of determinants

Cramer's Rule expresses each variable xᵢ as det(Aᵢ)/det(A), where Aᵢ replaces the i-th column of A with b.

Question 18: Which of the following best describes a metric space?

A set with a function that measures distance satisfying positivity, symmetry, and the triangle inequality (Correct answer)
A vector space equipped with an inner product
A topological space where every sequence converges
A set where every subset is open

Correct answer: A set with a function that measures distance satisfying positivity, symmetry, and the triangle inequality

A metric space is a set X with a distance function d(x,y) satisfying non-negativity, d(x,y)=0 iff x=y, symmetry, and the triangle inequality.

Question 19: Cauchy's Theorem guarantees that if a prime p divides |G| for a finite group G, then G contains:

A unique subgroup of order p
A normal subgroup of order p
An element of order p²
An element of order p (Correct answer)

Correct answer: An element of order p

Cauchy's Theorem states that if prime p divides |G|, then G must contain at least one element whose order is exactly p.

Question 20: For the exponential distribution with rate λ, the mean is:

λ
1/λ²
λ²
1/λ (Correct answer)

Correct answer: 1/λ

The exponential distribution Exp(λ) has mean 1/λ and variance 1/λ², modeling waiting times between Poisson events.

Question 21: Which of the following groups is NOT cyclic?

Z_7
Z_6
Z_2 × Z_2 (Correct answer)
Z_4

Correct answer: Z_2 × Z_2

Z_2 × Z_2 (the Klein four-group) has no element of order 4, so no single element generates the entire group.

Question 22: A relation R on a set A is an equivalence relation if it is:

Symmetric and transitive only
Reflexive and antisymmetric only
Reflexive, antisymmetric, and transitive
Reflexive, symmetric, and transitive (Correct answer)

Correct answer: Reflexive, symmetric, and transitive

An equivalence relation must satisfy reflexivity (aRa), symmetry (aRb ⟹ bRa), and transitivity (aRb ∧ bRc ⟹ aRc).

Question 23: Which statement correctly defines a Cauchy sequence in a metric space?

For every ε>0 there exists N such that d(xₘ,xₙ)<ε for all m,n>N (Correct answer)
The sequence satisfies |xₙ₊₁ - xₙ| < 1/n for all n
Every subsequence converges to the same limit
The sequence is bounded and monotone

Correct answer: For every ε>0 there exists N such that d(xₘ,xₙ)<ε for all m,n>N

A Cauchy sequence requires that terms become arbitrarily close to each other (not just to a fixed limit) as the indices grow.

Question 24: What is the derivative of arctan(x)?

1/√(1 − x²)
1/(1 + x²) (Correct answer)
−1/(1 + x²)
1/(x² − 1)

Correct answer: 1/(1 + x²)

The derivative of arctan(x) is 1/(1 + x²), derived by implicit differentiation of tan(y) = x.

Question 25: For two independent random variables X and Y, Var(X + Y) equals:

Var(X) + Var(Y) (Correct answer)
(Var(X) + Var(Y))²
Var(X) · Var(Y)
Var(X) + Var(Y) + 2Cov(X,Y)

Correct answer: Var(X) + Var(Y)

Independence implies Cov(X,Y) = 0, so Var(X + Y) = Var(X) + Var(Y).

Question 26: A group G is called solvable if:

Every subgroup of G is normal
It has prime order
It is abelian
It has a subnormal series with abelian factor groups (Correct answer)

Correct answer: It has a subnormal series with abelian factor groups

G is solvable if there exists a subnormal series 1 = G_0 ⊲ G_1 ⊲ ⋯ ⊲ G_n = G where each successive quotient G_{i+1}/G_i is abelian.

ISI B.Math Bachelor of Mathematics Admission Test

The Indian Statistical Institute B.Math entrance examination tests undergraduate mathematical aptitude across algebra, calculus, number theory, combinatorics, and probability through objective (UGA) and subjective (UGB) problem-solving formats.

Questions

35 min

Time Limit

60%

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