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

Question 1: Which property distinguishes a Hilbert space from a general Banach space?

A Hilbert space is finite-dimensional
A Hilbert space has an inner product whose induced norm makes it complete (Correct answer)
A Hilbert space has a countable basis
A Hilbert space contains no non-convergent Cauchy sequences

Correct answer: A Hilbert space has an inner product whose induced norm makes it complete

A Hilbert space is a complete inner product space; all Hilbert spaces are Banach spaces but not vice versa since the Banach space norm need not come from an inner product.

Question 2: In set theory, the symmetric difference A △ B is defined as:

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

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 3: The principle of inclusion-exclusion for two sets A and B states |A ∪ B| equals:

|A| · |B|
|A| + |B|
|A| − |B| + |A ∩ B|
|A| + |B| − |A ∩ B| (Correct answer)

Correct answer: |A| + |B| − |A ∩ B|

Inclusion-exclusion corrects for double-counting: |A ∪ B| = |A| + |B| − |A ∩ B|.

Question 4: In simple linear regression y = β₀ + β₁x + ε, what does β₁ represent?

The variance of the error term
The y-intercept of the regression line
The correlation coefficient
The expected change in y for a one-unit increase in x (Correct answer)

Correct answer: The expected change in y for a one-unit increase in x

β₁ is the slope coefficient: it gives the expected change in the response variable y for each one-unit increase in the predictor x.

Question 5: A mass-spring system obeys m·x'' + kx = 0. If m = 4 kg and k = 16 N/m, what is the angular frequency ω?

2 rad/s (Correct answer)
1 rad/s
4 rad/s
8 rad/s

Correct answer: 2 rad/s

ω = √(k/m) = √(16/4) = √4 = 2 rad/s.

Question 6: For a continuous random variable X with CDF F(x), the PDF f(x) satisfies:

f(x) = F'(x) (Correct answer)
f(x) = 1 − F(x)
f(x) = ∫F(x) dx
f(x) = F(x²)

Correct answer: f(x) = F'(x)

The PDF is the derivative of the CDF: f(x) = F'(x), which gives the density of probability at each point x.

Question 7: The general solution of dy/dx = ky is:

y = Ce^(kx) (Correct answer)
y = kx + C
y = k ln x + C
y = Cx^k

Correct answer: y = Ce^(kx)

Separating variables: dy/y = k dx, integrating gives ln|y| = kx + C₀, so y = Ce^(kx).

Question 8: Lagrange's Theorem states that for a finite group G with subgroup H:

|H| divides |G| (Correct answer)
|G| = |H|²
|G| divides |H|
|H| = |G|/2 always

Correct answer: |H| divides |G|

Lagrange's Theorem states that the order of any subgroup H of a finite group G must divide the order of G.

Question 9: The variance Var(X) of a random variable X is defined as:

E[(X − E[X])²] (Correct answer)
E[X²]
E[X]²
E[X] − E[X²]

Correct answer: E[(X − E[X])²]

Variance measures the expected squared deviation from the mean: Var(X) = E[(X − μ)²] = E[X²] − μ².

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: 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)
Every container holds exactly one object
No two containers hold the same number of objects
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 12: A subgroup N of a group G is called normal if which condition holds?

N is cyclic
N has prime order
N equals the center Z(G)
gNg⁻¹ = N for all g in G (Correct answer)

Correct answer: gNg⁻¹ = N for all g in G

A normal subgroup N satisfies gNg⁻¹ = N for every g ∈ G, meaning N is closed under conjugation by any group element.

Question 13: If X ~ Poisson(λ), what is both the mean and variance of X?

Mean = λ, Variance = λ²
Mean = λ, Variance = λ (Correct answer)
Mean = λ², Variance = λ
Mean = √λ, Variance = λ

Correct answer: Mean = λ, Variance = λ

A key property of the Poisson distribution is that both its mean and variance are equal to the parameter λ.

Question 14: Which of the following correctly describes the strong form of mathematical induction?

Prove P(n) directly for all n
Prove P(0) and that P(n) holds for all n ≥ 0
Prove P(1) and that P(1) ∧ … ∧ P(k) ⟹ P(k+1) (Correct answer)
Prove P(1) and that P(k) ⟹ P(k+1)

Correct answer: Prove P(1) and that P(1) ∧ … ∧ P(k) ⟹ P(k+1)

Strong induction assumes all of P(1),…,P(k) are true (not just P(k)) to conclude P(k+1).

Question 15: For an n×n matrix A, the Cayley-Hamilton theorem states that:

A has n distinct eigenvalues
A is diagonalizable
A satisfies its own characteristic polynomial (Correct answer)
det(A) equals the product of its eigenvalues

Correct answer: A satisfies its own characteristic polynomial

The Cayley-Hamilton theorem states every square matrix satisfies its own characteristic equation p(A) = 0.

Question 16: Which pattern is characterized by numbers that can be arranged into a growing shape pattern?

Figurate Numbers (Correct answer)
Fibonacci Sequence
Arithmetic Sequence
Geometric Sequence

Correct answer: Figurate Numbers

Figurate numbers are numbers that can be represented by a regular geometric arrangement of points or pebbles. Examples include triangular numbers, square numbers, and pentagonal numbers, where the arrangement forms a growing shape. This visual and geometric property is their defining characteristic.

Question 17: How many subsets does a set with n elements have?

n²
n
n!
2ⁿ (Correct answer)

Correct answer: 2ⁿ

Each element is either included or excluded from a subset, giving 2 choices per element and 2ⁿ total subsets.

Question 18: A relation R on a set S is an equivalence relation if it is:

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

Correct answer: Reflexive, symmetric, and transitive

An equivalence relation must be reflexive (aRa), symmetric (aRb ⟹ bRa), and transitive (aRb and bRc ⟹ aRc).

Question 19: A matrix A is diagonalizable if and only if:

Its determinant is 1
It is symmetric
It has n linearly independent eigenvectors (Correct answer)
All its entries are distinct

Correct answer: It has n linearly independent eigenvectors

An n×n matrix is diagonalizable exactly when it has n linearly independent eigenvectors, allowing it to be written as A = PDP⁻¹.

Question 20: Which sequence involves each term being obtained from the previous term by multiplying by a fixed number?

Figurate Numbers
Recursive Sequence
Fibonacci Sequence
Geometric Sequence (Correct answer)

Correct answer: Geometric Sequence

A geometric sequence is characterized by a constant ratio between consecutive terms. Each term is obtained from the previous term by multiplying by a fixed, non-zero number, known as the common ratio. This consistent multiplicative relationship distinguishes it from other types of sequences.

Question 21: The number of ways to arrange n distinct objects in a row is:

n
n! (Correct answer)
2ⁿ
n²

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 22: What method does this document suggest for squaring numbers ending in 5?

Multiply the first digit by itself plus one and append "25" (Correct answer)
Multiply by 5 and add 25
Square each digit separately then add them together
None of these

Correct answer: Multiply the first digit by itself plus one and append "25"

This is a well-known mental math trick for squaring numbers ending in 5. For a number like `N5` (e.g., 35), you take the first digit `N` (3), multiply it by `N+1` (3 * 4 = 12), and then append '25' to the result. So, 35 squared is 1225.

Question 23: A group G is called solvable if:

Every subgroup of G is normal
It is abelian
It has prime order
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.

Question 24: What is the order of the group Z_12 (integers modulo 12 under addition)?

12 (Correct answer)
24
6
11

Correct answer: 12

Z_12 has elements {0, 1, 2, ..., 11}, so its order (number of elements) is exactly 12.

Question 25: Which statement about the alternating group A_n is TRUE?

|A_n| = n!
A_n = S_n for all n ≥ 2
A_n is a normal subgroup of S_n with index 2 (Correct answer)
A_n consists of all odd permutations in S_n

Correct answer: A_n is a normal subgroup of S_n with index 2

A_n consists of all even permutations, is normal in S_n, and has index 2 because |A_n| = n!/2.

Question 26: Which of the following correctly describes the Big-O notation f(n) = O(g(n))?

|f(n)| ≥ C·g(n) for some constant C
f(n) = g(n) for all sufficiently large n
f(n)/g(n) → 1 as n → ∞
There exist positive constants C and N such that |f(n)| ≤ C·g(n) for all n > N (Correct answer)

Correct answer: There exist positive constants C and N such that |f(n)| ≤ C·g(n) for all n > N

Big-O describes an asymptotic upper bound: f grows no faster than a constant multiple of g beyond some threshold.

Question 27: What does the expected value E[X] of a discrete random variable X represent?

The median of the distribution
The variance of X
The most likely outcome
The weighted average of all possible values (Correct answer)

Correct answer: The weighted average of all possible values

E[X] = Σ x·P(X = x) is the long-run average value of X over many repetitions of the experiment.

Question 28: A square matrix A is invertible if and only if:

Its trace is non-zero
It is symmetric
Its determinant is non-zero (Correct answer)
All its entries are positive

Correct answer: Its determinant is non-zero

A matrix is invertible (non-singular) precisely when its determinant is non-zero, ensuring a unique solution to Ax = b.

Question 29: The eigenvalues of the matrix [[3,1],[0,3]] are:

0 and 3
1 and 3
Both equal to 3 (repeated) (Correct answer)
3 and -3

Correct answer: Both equal to 3 (repeated)

An upper-triangular matrix has eigenvalues equal to its diagonal entries, both of which are 3.

Question 30: The Gram-Schmidt process is used to:

Compute the determinant
Convert a basis into an orthonormal basis (Correct answer)
Find eigenvalues
Diagonalize a matrix

Correct answer: Convert a basis into an orthonormal basis

Gram-Schmidt orthogonalizes a set of linearly independent vectors into a mutually orthogonal (and normalizable) set.

Question 31: The Fundamental Theorem of Arithmetic states that every integer n > 1:

Can be written as a sum of two primes
Has at most two prime factors
Has a unique prime factorization (Correct answer)
Is either prime or a perfect square

Correct answer: Has a unique prime factorization

Every integer greater than 1 can be uniquely expressed as a product of prime numbers (up to order of factors).

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

120 min

Time Limit

60%

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