MA/MSc Mathematics Entrance Exam 2026: Most Important Questions (Part 1)

Correct Answer

Detailed Explanation

For a finite set containing \(n\) elements, every element has exactly two choices while forming a subset:

Include the element.
Exclude the element.

Hence, the total number of subsets is given by

\[ \boxed{ \text{Number of Subsets}=2^n } \]

Since \(n=5\), \[ 2^5=32. \] Therefore, the given set contains exactly \(32\) subsets, including the empty set and the set itself.

\[ \boxed{ 2^5=32 } \]

Hence, the correct answer is Option (C).

ALGEBRA • EASY

Question 12

According to Lagrange's Theorem, the order of every subgroup of a finite group \(G\) must

Detailed Explanation

A sequence \(\{a_n\}\) converges to a limit \(L\) if

\[ \boxed{ \lim_{n\to\infty}a_n=L } \]

Now examine each option.

\(n\to\infty\), so it does not converge.
\((-1)^n\) oscillates between \(-1\) and \(1\), so the limit does not exist.
\(\dfrac1n\to0\) as \(n\to\infty\).
\(\sqrt n\to\infty\), so it diverges.

Hence,

\[ \boxed{ \lim_{n\to\infty}\frac1n=0 } \]

Therefore, the only sequence that converges to zero is Option (C).

LINEAR ALGEBRA • MODERATE

Question 16

If \[ A= \begin{bmatrix} 1 & 2\\ 2 & 4 \end{bmatrix}, \] then the rank of the matrix \(A\) is

(A) \(0\)
(B) \(1\)
(C) \(2\)
(D) \(3\)

Correct Answer

(B) \(1\)

Detailed Explanation

The rank of a matrix is the maximum number of linearly independent rows or columns.

\[ \boxed{\operatorname{Rank}(A)=\text{Maximum number of linearly independent rows (or columns)}} \]

Observe that \[ R_2=2R_1. \] Thus, the second row is a scalar multiple of the first row, so the two rows are linearly dependent. Also, \[ |A| = \begin{vmatrix} 1&2\\ 2&4 \end{vmatrix} = 4-4 = 0, \] which confirms that the matrix is singular. Since there is only one linearly independent row (or column),

\[ \boxed{\operatorname{Rank}(A)=1} \]

Hence, the correct answer is Option (B).

DIFFERENTIAL CALCULUS • MODERATE

Question 17

If \[ f(x)=x^3-6x^2+9x+1, \] then the value of \(x\) at which the function attains its local minimum is

(A) \(1\)
(B) \(2\)
(C) \(3\)
(D) \(4\)

The gradient of a scalar function gives the direction of the maximum rate of increase of the function.

\[ \boxed{ \nabla\phi= \frac{\partial\phi}{\partial x}\hat{i} + \frac{\partial\phi}{\partial y}\hat{j} + \frac{\partial\phi}{\partial z}\hat{k} } \]

For \[ \phi=x^2+y^2+z^2, \] we have \[ \frac{\partial\phi}{\partial x}=2x,\qquad \frac{\partial\phi}{\partial y}=2y,\qquad \frac{\partial\phi}{\partial z}=2z. \] Hence, \[ \nabla\phi = 2x\hat{i}+2y\hat{j}+2z\hat{k}. \]

\[ \boxed{ \nabla(x^2+y^2+z^2) = 2x\hat{i}+2y\hat{j}+2z\hat{k} } \]

Hence, the correct answer is Option (A).

NUMERICAL ANALYSIS • MODERATE

Question 33

The Newton-Raphson iteration formula for solving \[ f(x)=0 \] is

(A) \[ x_{n+1}=x_n+\frac{f(x_n)}{f'(x_n)} \]
(B) \[ x_{n+1}=x_n-\frac{f'(x_n)}{f(x_n)} \]
(C) \[ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} \]
(D) \[ x_{n+1}=\frac{x_n+f(x_n)}2 \]

Correct Answer

Detailed Explanation

The Newton-Raphson Method is based on the tangent line approximation of a nonlinear function.

\[ \boxed{ x_{n+1} = x_n-\frac{f(x_n)}{f'(x_n)} } \]

Starting from an initial approximation \(x_0\), successive approximations are computed until convergence. For the method to work efficiently,

\(f(x)\) should be differentiable.
\(f'(x)\neq0\) near the root.
The initial approximation should be chosen close to the actual root.

Correct Answer

(B) \(2z\)

Detailed Explanation

The function \[ f(z)=z^2 \] is a polynomial in the complex variable \(z\). Every polynomial is analytic over the entire complex plane. Differentiation follows exactly the same rules as ordinary calculus.

\[ \boxed{ \frac{d}{dz}(z^n)=nz^{\,n-1} } \]

Applying the power rule, \[ \begin{aligned} \frac{d}{dz}(z^2) &=2z^{2-1}\\[2mm] &=2z. \end{aligned} \] Since \(f(z)\) is analytic everywhere, the derivative exists for every complex number \(z\).

\[ \boxed{ \frac{d}{dz}(z^2)=2z } \]

Hence, the correct answer is Option (B).

DIFFERENTIAL CALCULUS • MODERATE+

Question 40

The function \[ f(x)=x^2-4x+3 \] satisfies the hypotheses of Rolle's Theorem on the interval \([1,3]\). The value of \(c\in(1,3)\) satisfying Rolle's Theorem is

(A) \(1\)
(B) \(2\)
(C) \(3\)
(D) \(4\)

Correct Answer

(B) \(2\)

Detailed Explanation

Rolle's Theorem states that if a function is continuous on a closed interval, differentiable on the corresponding open interval, and satisfies \[ f(a)=f(b), \] then there exists at least one point \(c\in(a,b)\) such that

\[ \boxed{ f'(c)=0 } \]

First verify the endpoint values: \[ \begin{aligned} f(1)&=1-4+3=0,\\ f(3)&=9-12+3=0. \end{aligned} \] Thus, \[ f(1)=f(3), \] so Rolle's Theorem is applicable. Now differentiate: \[ f'(x)=2x-4. \] Setting the derivative equal to zero, \[ 2x-4=0, \] gives \[ x=2. \] Since \(2\in(1,3)\), the required value is

\[ \boxed{ c=2 } \]

Hence, the correct answer is Option (B).

LINEAR ALGEBRA • MODERATE+

Question 41

Let \[ A= \begin{bmatrix} 2 & 1\\ 1 & 2 \end{bmatrix}. \] The sum of the eigenvalues of the matrix \(A\) is

(A) \(2\)
(B) \(3\)
(C) \(4\)
(D) \(5\)

Detailed Explanation

The given series is a \(p\)-series of the form \[ \sum_{n=1}^{\infty}\frac{1}{n^p}. \] A \(p\)-series converges if and only if

\[ \boxed{ \sum_{n=1}^{\infty}\frac{1}{n^p} \text{ converges if }p>1 } \]

Here, \[ p=2>1. \] Hence, \[ \sum_{n=1}^{\infty}\frac1{n^2} \] is convergent. Since every term is positive, \[ \sum_{n=1}^{\infty}\left|\frac1{n^2}\right| = \sum_{n=1}^{\infty}\frac1{n^2} \] also converges. Therefore, the series is absolutely convergent.

\[ \boxed{ p=2>1 \Longrightarrow \sum_{n=1}^{\infty}\frac1{n^2} \text{ converges} } \]

Hence, the correct answer is Option (C).

COMPLEX ANALYSIS • MODERATE+

Question 45

If \[ f(z)=u(x,y)+iv(x,y) \] is analytic, then the Cauchy-Riemann equations are

(A) \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} \]
(B) \[ \frac{\partial u}{\partial x} = - \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = \frac{\partial v}{\partial x} \]
(C) \[ \frac{\partial u}{\partial x} = \frac{\partial u}{\partial y} \]
(D) \[ \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} \]

Correct Answer

(A)

Detailed Explanation

A complex function \[ f(z)=u(x,y)+iv(x,y) \] is analytic if the first-order partial derivatives exist, are continuous, and satisfy the Cauchy-Riemann equations.

\[ \boxed{ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} } \]

These equations are necessary conditions for differentiability of a complex function. For many entrance examinations, checking the Cauchy-Riemann equations is the first step in determining whether a function is analytic. Hence, whenever these equations hold (along with continuity of the partial derivatives), the function is analytic at that point.

\[ \boxed{ u_x=v_y, \qquad u_y=-v_x } \]

Therefore, the correct answer is Option (A).

VECTOR CALCULUS • MODERATE+

Question 46

If \[ \vec{F}(x,y,z)=x^2\hat{i}+y^2\hat{j}+z^2\hat{k}, \] then the divergence of \(\vec{F}\) is

(A) \(2x+2y+2z\)
(B) \(x+y+z\)
(C) \(2(x+y+z)\)
(D) \(0\)

Detailed Explanation

For any complex number \[ z=a+bi, \] its modulus is defined by

\[ \boxed{ |z|=\sqrt{a^2+b^2} } \]

Here, \[ a=3, \qquad b=-4. \] Therefore, \[ \begin{aligned} |z| &=\sqrt{3^2+(-4)^2}\\[2mm] &=\sqrt{9+16}\\[2mm] &=\sqrt{25}\\[2mm] &=5. \end{aligned} \] Also, observe that \[ z\overline z=|z|^2=25. \] This is a standard property of complex numbers.

\[ \boxed{ |3-4i|=5 } \]

Hence, the correct answer is Option (C).

LINEAR ALGEBRA • MODERATE+

Question 56

Let \[ A= \begin{bmatrix} 1 & 2\\ 2 & 1 \end{bmatrix}. \] The characteristic polynomial of the matrix \(A\) is

(A) \(\lambda^2-2\lambda-3\)
(B) \(\lambda^2+2\lambda-3\)
(C) \(\lambda^2-3\lambda+2\)
(D) \(\lambda^2-4\lambda+3\)

Correct Answer

(A) \(\lambda^2-2\lambda-3\)

Detailed Explanation

The characteristic polynomial of a square matrix is obtained from

\[ \boxed{ \det(A-\lambda I)=0 } \]

For \[ A= \begin{bmatrix} 1 & 2\\ 2 & 1 \end{bmatrix}, \] we have \[ A-\lambda I= \begin{bmatrix} 1-\lambda & 2\\ 2 & 1-\lambda \end{bmatrix}. \] Therefore, \[ \begin{aligned} \det(A-\lambda I) &=(1-\lambda)^2-4\\[2mm] &=\lambda^2-2\lambda-3. \end{aligned} \] Hence, the characteristic equation is \[ \lambda^2-2\lambda-3=0. \] Factoring, \[ (\lambda-3)(\lambda+1)=0. \] Thus, the eigenvalues are \[ 3,\;-1. \]

\[ \boxed{ \lambda^2-2\lambda-3 } \]

Hence, the correct answer is Option (A).

DIFFERENTIAL CALCULUS • MODERATE+

Question 57

If \[ f(x)=\tan x, \] then the second derivative \[ \frac{d^2y}{dx^2} \] is

(A) \[ 2\sec^2x\tan x \]
(B) \[ \sec^2x \]
(C) \[ 2\tan^2x \]
(D) \[ \sec x\tan x \]

Correct Answer

Correct Answer

Detailed Explanation

A Fourier Series represents a periodic function as an infinite sum of sine and cosine functions having different frequencies.

\[ \boxed{ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n\cos nx + b_n\sin nx \right) } \]

Here, \[ a_n \] and \[ b_n \] are called the Fourier coefficients. The Fourier Series is widely used in heat conduction, signal processing, vibration analysis, electrical engineering, and many branches of applied mathematics. Thus, a periodic function is expanded using both sine and cosine functions.

\[ \boxed{ \text{Fourier Series} = \text{Sine Series} + \text{Cosine Series} } \]

Hence, the correct answer is Option (C).

BETA & GAMMA FUNCTIONS • ENTRANCE LEVEL

Question 64

The Gamma Function satisfies the identity

(A) \[ \Gamma(n)=n! \]
(B) \[ \Gamma(n+1)=n\,\Gamma(n) \]
(C) \[ \Gamma(n)=n\,\Gamma(n+1) \]
(D) \[ \Gamma(n)=\frac1{n!} \]

Correct Answer

(B) \[ \Gamma(n+1)=n\,\Gamma(n) \]

Detailed Explanation

The Gamma Function is defined by

\[ \boxed{ \Gamma(x) = \int_0^\infty t^{x-1}e^{-t}\,dt, \qquad x>0 } \]

One of its most important recurrence relations is \[ \Gamma(n+1)=n\,\Gamma(n). \] Using this relation repeatedly, \[ \Gamma(n+1)=n!. \] For example, \[ \Gamma(5)=4!=24. \] This recurrence formula is extensively used in probability theory, special functions, and definite integrals.

\[ \boxed{ \Gamma(n+1)=n\,\Gamma(n) } \]

Hence, the correct answer is Option (B).

ABSTRACT ALGEBRA • ENTRANCE LEVEL

Question 65

Which one of the following groups is always an abelian group?

(A) Symmetric Group \(S_3\)
(B) Dihedral Group \(D_4\)
(C) \((\mathbb{Z},+)\)
(D) Symmetric Group \(S_4\)

(D) \[ d(x,y)<0 \]

Detailed Explanation

A metric is a function that measures the distance between two points. For every metric space \((X,d)\), the following axioms must hold.

\[ \boxed{ \begin{aligned} &d(x,y)\ge0,\\ &d(x,y)=0\iff x=y,\\ &d(x,y)=d(y,x),\\ &d(x,z)\le d(x,y)+d(y,z). \end{aligned} } \]

These are known as:

Non-negativity
Identity of indiscernibles
Symmetry
Triangle inequality

This follows directly from the complement laws: \[ A+A'=1, \] and \[ A\cdot A'=0. \] Substituting \[ A=1, \] gives \[ 1'=0. \] Boolean complements play a fundamental role in digital logic design, switching circuits, and computer science.

\[ \boxed{ 1'=0 } \]

Hence, the correct answer is Option (B).

VECTOR CALCULUS • ENTRANCE LEVEL

Question 72

If \[ \vec{F}(x,y,z)=yz\,\hat{i}+xz\,\hat{j}+xy\,\hat{k}, \] then the divergence of \(\vec{F}\) is

(A) \(x+y+z\)
(B) \(xy+yz+xz\)
(C) \(0\)
(D) \(3xyz\)

Detailed Explanation

For \[ f(t)=1, \] we have \[ \begin{aligned} \mathcal{L}\{1\} &= \int_0^\infty e^{-st}\,dt\\[2mm] &= \left[ -\frac{e^{-st}}{s} \right]_0^\infty\\[2mm] &= \frac1s, \qquad s>0. \end{aligned} \] This is one of the most frequently used standard Laplace transforms.

\[ \boxed{ \mathcal{L}\{1\} = \frac1s } \]

Hence, the correct answer is Option (A).

ABSTRACT ALGEBRA • ENTRANCE LEVEL

Question 81

The identity element of the group \[ (\mathbb{Z},+) \] is

(A) \(0\)
(B) \(1\)
(C) \(-1\)
(D) Does not exist

Correct Answer

(A) \(0\)

Detailed Explanation

In any group, the identity element is an element that leaves every group element unchanged under the group operation.

\[ \boxed{ a+e=e+a=a, \qquad \forall a\in\mathbb{Z} } \]

For the additive group \[ (\mathbb{Z},+), \] the integer \[ 0 \] satisfies \[ a+0=0+a=a \] for every integer \(a\). Hence, \[ 0 \] is the identity element. The inverse of any integer \(a\) in this group is \[ -a. \]

\[ \boxed{ e=0 } \]

Hence, the correct answer is Option (A).

ABSTRACT ALGEBRA • ENTRANCE LEVEL

Question 82

Which one of the following sets forms a group under ordinary multiplication?

(A) \(\mathbb{N}\)
(B) \(\mathbb{Z}\)
(C) \(\mathbb{Q}\setminus\{0\}\)
(D) \(\mathbb{R}\)

Correct Answer

Detailed Explanation

A set forms a group under multiplication if it satisfies the group axioms:

Closure
Associativity
Identity element
Inverse for every element

\[ \boxed{ (\mathbb{Q}\setminus\{0\},\times) \text{ is a group} } \]

The multiplicative identity is \[ 1, \] and every non-zero rational number \[ a \] has the inverse \[ \frac1a. \] The remaining options fail because:

\(\mathbb N\): inverses are not always natural numbers.
\(\mathbb Z\): inverses such as \(1/2\) are not integers.
\(\mathbb R\): contains \(0\), which has no multiplicative inverse.

Indeed, if \[ a\in H, \] then \[ a^{-1}\in H. \] Hence, \[ aa^{-1}=e\in H. \] Therefore, every subgroup necessarily contains the identity element. This is one of the most fundamental results in Group Theory and appears frequently in postgraduate entrance examinations.

\[ \boxed{ \text{Every subgroup contains the identity element} } \]

Hence, the correct answer is Option (C).

REAL ANALYSIS • ENTRANCE LEVEL

Question 86

Evaluate \[ \lim_{x\to0}\frac{\sin x}{x}. \]

(A) \(0\)
(B) \(1\)
(C) \(\infty\)
(D) Does not exist

Correct Answer

(B) \(1\)

Detailed Explanation

This is one of the most fundamental limits in Calculus and Real Analysis.

\[ \boxed{ \lim_{x\to0}\frac{\sin x}{x}=1 } \]

The result can be established using the Squeeze Theorem: \[ \cos x \le \frac{\sin x}{x} \le 1, \qquad 0

\[ \boxed{ \lim_{x\to0}\frac{\sin x}{x}=1 } \]

Hence, the correct answer is Option (B).

PROBABILITY THEORY • ENTRANCE LEVEL

Question 87

If two events \(A\) and \(B\) are independent, then \[ P(A\cap B) \] is equal to

(A) \[ P(A)+P(B) \]
(B) \[ P(A)-P(B) \]
(C) \[ P(A)\,P(B) \]
(D) \[ \frac{P(A)}{P(B)} \]

Correct Answer

Detailed Explanation

Two events are called independent if the occurrence of one event does not affect the probability of the other.

\[ \boxed{ P(A\cap B) = P(A)\,P(B) } \]

For dependent events, \[ P(A\cap B) = P(A)\,P(B|A). \] However, for independent events, \[ P(B|A)=P(B). \] Therefore, \[ P(A\cap B) = P(A)\,P(B). \] This is one of the most frequently tested formulas in probability theory.

\[ \boxed{ P(A\cap B) = P(A)\,P(B) } \]

Hence, the correct answer is Option (C).

LINEAR ALGEBRA • ENTRANCE LEVEL

Question 88

If \[ A= \begin{bmatrix} 2 & 0\\ 0 & 5 \end{bmatrix}, \] then the eigenvalues of \(A\) are

(A) \(2,\;5\)
(B) \(7,\;10\)
(C) \(2,\;-5\)
(D) \(10,\;25\)

Correct Answer

(A) \(2,\;5\)

Detailed Explanation

The eigenvalues of a diagonal matrix are simply its diagonal entries.

\[ \boxed{ \text{If } A=\operatorname{diag}(a_1,a_2,\ldots,a_n), \text{ then } \lambda_i=a_i. } \]

Since \[ A= \begin{bmatrix} 2 & 0\\ 0 & 5 \end{bmatrix}, \] its diagonal entries are \[ 2 \quad\text{and}\quad 5. \] Hence, the eigenvalues are \[ 2,\;5. \] Also, \[ \operatorname{tr}(A)=2+5=7, \] which equals the sum of the eigenvalues, and \[ |A|=2\times5=10, \] which equals their product.

\[ \boxed{ \lambda_1=2,\qquad \lambda_2=5 } \]

Hence, the correct answer is Option (A).

COMPLEX ANALYSIS • ENTRANCE LEVEL

Question 89

If \[ z=1+i, \] then the complex conjugate of \(z\) is

(A) \(1+i\)
(B) \(-1+i\)
(C) \(1-i\)
(D) \(-1-i\)

Detailed Explanation

A fair coin has two equally likely outcomes.

\[ \boxed{ S=\{H,T\} } \]

The total number of possible outcomes is \[ 2. \] The favorable outcome for obtaining a head is \[ \{H\}, \] which contains one outcome. Therefore, \[ \begin{aligned} P(\text{Head}) &= \frac{\text{Number of favorable outcomes}} {\text{Total number of outcomes}}\\[2mm] &= \frac12. \end{aligned} \] This is one of the basic applications of classical probability.

\[ \boxed{ P(\text{Head})=\frac12 } \]

Hence, the correct answer is Option (C).

COMPLEX ANALYSIS • ENTRANCE LEVEL

Question 94

If \[ z=i, \] then the value of \[ i^{\,4} \] is

(A) \(i\)
(B) \(-1\)
(C) \(1\)
(D) \(-i\)

(A) \[ (\lambda-2)^2=0 \]

Detailed Explanation

The characteristic equation of a square matrix is obtained by solving

\[ \boxed{ \det(A-\lambda I)=0 } \]

For \[ A= \begin{bmatrix} 2 & 0\\ 0 & 2 \end{bmatrix}, \] we have \[ A-\lambda I= \begin{bmatrix} 2-\lambda & 0\\ 0 & 2-\lambda \end{bmatrix}. \] Therefore, \[ \begin{aligned} \det(A-\lambda I) &=(2-\lambda)(2-\lambda)\\[2mm] &=(2-\lambda)^2. \end{aligned} \] Equating the determinant to zero, \[ (2-\lambda)^2=0, \] which is equivalent to \[ (\lambda-2)^2=0. \] Thus, the matrix has a repeated eigenvalue \[ \lambda=2. \]

\[ \boxed{ (\lambda-2)^2=0 } \]

Hence, the correct answer is Option (A).

🎉 Set–1 Completed Successfully

This completes Set–1 (Questions 1–100) for the M.Sc./M.A. Mathematics Entrance Examination. The paper covers a broad range of B.Sc. Mathematics topics commonly asked in entrance examinations of universities across India, including:

Linear Algebra
Real Analysis
Differential Calculus
Integral Calculus
Differential Equations
Complex Analysis
Vector Calculus
Laplace Transform
Fourier Series
Abstract Algebra
Metric Spaces
Topology
Probability Theory
Graph Theory
Boolean Algebra
Beta & Gamma Functions
Green's Theorem
Stokes' Theorem

The next paper (Set–2: Questions 101–200) should ideally contain a significantly higher proportion of advanced, university-level entrance questions, including more conceptual and numerical problems from: Cayley–Hamilton Theorem, Rank–Nullity Theorem, Jordan Canonical Form, Eigenvalue Applications, Riemann Integration, Sequences & Series, Uniform Convergence, Group Homomorphism, Ring Theory, Field Theory, Metric Spaces, Topological Spaces, Multiple Integrals, Gamma/Beta Applications, PDEs, Fourier Transform, Probability Distributions, and advanced PYQ-style problems. This will ensure that the complete **200-question bank** closely matches the difficulty level of actual **M.Sc./M.A. Mathematics Entrance Examinations** conducted by universities such as DDU, BHU, CUET PG, AMU, JNU, DU, HPU, Allahabad University, Lucknow University, and other state universities.

नोट: यह हमारी MA/MSc Mathematics Entrance Exam सीरीज का पहला भाग (Part 1) है। हम जल्द ही Part 2 लेकर आएंगे जिसमें Abstract Algebra (अमूर्त बीजगणित) और Differential Equations (अवकल समीकरण) के महत्वपूर्ण प्रश्न शामिल होंगे। तब तक इन प्रश्नों का अच्छे से रिवीजन करते रहें!

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