HPSC Assistant Professor Mathematics Exam 2025 – Complete Question Paper With Answer Key”

HPSC Assistant Professor Mathematics Exam 2025 – 28 September Question Paper

This page provides the complete HPSC Assistant Professor Mathematics Question Paper 2025 held on 28 September 2025. The exam consisted of 100 Multiple Choice Questions (MCQs) covering Algebra, Real Analysis, Complex Analysis, Functional Analysis, Linear Algebra, Differential Equations, Topology, Probability, Statistics, Numerical Methods, and Applied Mathematics. These previous year questions are very useful for HPSC exam preparation, practice, and revision.

By going through this complete set of HPSC Assistant Professor Maths previous year question paper with answers, aspirants can understand the exam pattern, the level of difficulty, and the important topics that frequently appear. Solving these questions will boost confidence and improve speed and accuracy for upcoming exams.

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Disclaimer:

This set of questions belongs to HPSC. We are not the owners of these questions and do not claim ownership.
We accept no responsibility for any errors or inaccuracies in the questions.
This material is provided for educational purposes only.

Q.1

Let \(V=\mathbb{R}^2(\mathbb{R})\) and \(u=(1,-1)\). Let \(W=\{(x,y): y=2x\}\) be a subspace of \(V\). Then the 90° rotation of the orthogonal projection of \(u\) on \(W\) of the inner product space \(V\) is:

Option I: \(\left(\tfrac{2}{\sqrt{5}}, -\tfrac{1}{\sqrt{5}}\right)\)
Option II: \(\left(-\tfrac{1}{\sqrt{5}}, \tfrac{2}{\sqrt{5}}\right)\)

Choose the correct answer:

1. Both option I and option II are true.
2. Both option I and option II are false.
3. Option I is correct but option II is false.
4. Option I is incorrect but option II is true.
5. Question not attempted.

✅ Correct Answer: C.

Q.2

Let \(A\in M_{2\times2}(\mathbb{R})\) such that \(A^2=0\). Then:

1. \(A\) is similar to \(\begin{bmatrix}0 & 0\\ 1 & 0\end{bmatrix}\).
2. \(A\) is similar to \(\begin{bmatrix}0 & 0\\ 0 & 1\end{bmatrix}\) where \(A\neq 0\).
3. \(A\) is similar to \(\begin{bmatrix}0 & 0\\ 1 & 0\end{bmatrix}\) only if \(A\neq 0\).

Choose the correct answer:

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted.

✅ Correct Answer: A.

Q.3

Consider the following statements:

1. Let \(f:\mathbb{R}^2 \to \mathbb{R}\) by \(f(x,y)= x^{\lfloor y \rfloor}\), then for every \(x \in \mathbb{R}\), the map \(y \mapsto f(x,y)\) is continuous on \(\mathbb{R}\), where \(\lfloor y \rfloor\) is the greatest integer function.
2. There exists a continuous function \(f:\mathbb{R}\to \mathbb{R}\) that takes every value thrice.
3. There exists a continuous function \(f:\mathbb{R}\to \mathbb{R}\) that takes every value twice.

How many of the statements given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: A.

Q.4

Consider the following statements:

1. If \(A\) and \(B\) are closed subsets of \(\mathbb{R}\), then the Minkowski sum \(A+B\) is also closed in \(\mathbb{R}\).
2. If \(A\) and \(B\) are connected subsets of \(\mathbb{R}\), then the Minkowski sum \(A+B\) is also connected.
3. Let \(a,b \in \mathbb{R}\) with \( a < b \), let \(f : (a, b) \to \mathbb{R}\) be a continous function. Then \(f\) is uniformaly continuous if and only if it is bounded.

How many of the statements given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: A.

Q.5

Given below are two statements:

Statement I: Let \(f \in C^{1}[0,1]\) satisfying \(f(0)=f(1)=0\). Then \[ \max_{0 \leq x \leq 1} |f'(x)| \geq 4 \int_0^1 |f(x)|\,dx \]

Statement II: Let \(g \in C'[0,1]\) satisfying \(g(0)=g(1)=0\). Then \( \max_{0 \leq x \leq 1} |g'(x)| \geq T \int_0^1 |g(x)|\,dx\)
\(\text{where } T > 4.\)

In the light of the above statements, choose the correct answer:

1. Both Statement I and Statement II are true.
2. Both Statement I and Statement II are false.
3. Statement I is correct but Statement II is false.
4. Statement I is incorrect but Statement II is true.
5. Question not attempted.

✅ Correct Answer: C.

Q.6

Consider the following statement and the options given below:

Let \(W(F_1)\) and \(V(F_2)\) be two vector spaces. Then a map \(T: W(F_1) \to V(F_2)\) becomes a linear transformation if:

1. For any choices of \(F_1\) and \(F_2\).
2. If one of them contains the other.
3. If \(F_1 \subseteq F_2\).
4. If \(F_1 = F_2\).

How many of the options given above are correct?

1. Only one
2. Only two
3. Only three
4. All four
5. Question not attempted

✅ Correct Answer: B.

Q.7

Given below are two statements:

Statement I: Let \(S=\{(1,1,0),(1,0,2),(0,1,1)\}\) be a subset of the vector space \(\mathbb{F}^3(\mathbb{F})\). Then \(S\) is linearly independent set for any choices of the field \(\mathbb{F}\).

Statement II: The set \(S\) defined in I is a basis of \(V\) if \(\mathbb{F}=\mathbb{R}\).

In the light of the above statements, choose the correct answer from the options given below:

1. Both statement I and statement II are true.
2. Both statement I and statement II are false.
3. Statement I is correct but statement II is false.
4. Statement I is incorrect but statement II is true.
5. Question not attempted

✅ Correct Answer: D.

Q.8

Given below are two statements:

Statement I: Let \(X\) be a countable set. There exists a metric \(d\) on \(X\) such that \((X,d)\) is complete.

Statement II: Let \(X\) be a countable set. Then there exists a metric \(d\) on \(X\) such that \((X,d)\) is compact.

In the light of the above statements, choose the correct answer from the options given below:

1. Both Statement I and Statement II are true.
2. Both Statement I and Statement II are false.
3. Statement I is correct but Statement II is false.
4. Statement I is incorrect but Statement II is true.
5. Question not attempted

✅ Correct Answer: A.

Q.9

Consider the following pairs:

1. \(C[0,1]\) with \(\|f\|_\infty\): It is complete
2. \(C[0,1]\) with the norm \(\|f\|_\infty + |f(1/2)|\): It is complete
3. \(C[0,1]\) with \(\|f\|_2\): It is complete

How many of the above pairs are correctly matched?

1. Only one pair
2. Only two pairs
3. All three pair
4. None of the pairs
5. Question not attempted

✅ Correct Answer: Question Removed.

Q.10

Consider the following statement and the options given below:

If \(\lambda\) is an eigenvalue of \[ A=\begin{bmatrix} 1 & 2+4i & 5i & 1\\ i & 3+4i & 6i & 1\\ 0 & 1 & -1 & 0\\ 1 & 0 & 0 & 1 \end{bmatrix} \] then

1. \(|\lambda|\leq 7+\sqrt{34}\)
2. \(|\lambda|\leq 6+2\sqrt{5}\)
3. \(|\lambda|\leq 1+3\sqrt{5}+\sqrt{34}\)
4. \(|\lambda|\leq 13\)

How many of the options given above are correct?

1. Only one
2. Only two
3. Only three
4. All four
5. Question not attempted

✅ Correct Answer: C.

Q.11

Consider the following statement and the options given below:

Let \(A \in GL_n(\mathbb{R})\) such that \(\text{adj}(\text{adj}(\text{adj} A)) = \begin{bmatrix} 2 & 4 & 0\\ 0 & 128 & 0\\ 100 & 3 & 1 \end{bmatrix}\).

Let \(f(\lambda)=\lambda^3+a\lambda^2+b\lambda+c=0 \in \mathbb{Z}[A]\) be the characteristic equation of \(A\). Then \(c=\)?

1. 2
2. -2
3. \(\dfrac{24}{\pi^2}\sum_{n=1}^{\infty}\dfrac{1}{2n^2}\)
4. \(-\dfrac{24}{\pi^2}\sum_{n=1}^{\infty}\dfrac{1}{2n^2}\)

How many of the options given above are correct?

1. Only one
2. Only two
3. Only three
4. All four
5. Question not attempted

✅ Correct Answer: Question Removed.

Q.12

Consider the following statement and the options given below:

Let \(\lambda_1=1\) and \(\lambda_2=2\) are two eigenvalues of a matrix \(A\in M_{2\times 2}(\mathbb{R})\). Then \(\det(\text{adj} A)=|\text{adj} A|\) and \(\text{trace}(\text{adj} A)\) respectively.

1. \(2\sum_{n=1}^{\infty}\dfrac{(-1)^{n-1}(2\pi)^{2n}}{(2n)!}, 3\)
2. -2, 3
3. \(2, 3\sum_{n=1}^{\infty}\dfrac{({\dfrac{\pi}{2})}^{2n-1}(-1)^{n-1}}{(2n-1)!}\)
4. -1, 3

How many of the options given above are correct?

1. Only one
2. Only two
3. Only three
4. All four
5. Question not attempted

✅ Correct Answer: A.

Q.13

Consider the following statements:

1. \(A\) and \(B\) are positive definite matrices iff \(A+B\) is positive definite always.
2. If \(A\) is a negative definite matrix, then \(A^k\) is negative definite iff \(k\) is odd.
3. Let \(A\) be a negative definite matrix, then \(A^k\) is positive definite if \(k\) is even.

How many of the statements given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: Question Removed.

Q.14

Given below are two statements:

Let \(A\) be a semi-positive definite symmetric matrix and \(AB+BA=0\) for some \(B \in \mathbb{C}_{n \times n}(\mathbb{C})\). Then

Statement I: \(AB\) and \(BA\) are nilpotent matrices of index one.

Statement II: \(AB\) and \(BA\) are nilpotent matrices of index two.

In the light of the above statements, choose the correct answer:

1. Both statement I and statement II are true.
2. Both statement I and statement II are false.
3. Statement I is correct but statement II is false.
4. Statement I is incorrect but statement II is true.
5. Question not attempted

✅ Correct Answer: Question Removed.

Q.15

Consider the following pairs:

1. \(P_2(\mathbb{R})/\mathbb{R}\) : dimension 2
2. \(M_2(\mathbb{C})/\mathbb{R}\) : dimension 4
3. \(M_2(\mathbb{R})/\mathbb{Q}\) : dimension infinite

How many of the above pairs are correctly matched?

1. Only one pair
2. Only two pairs
3. All three pairs
4. None
5. Question not attempted

✅ Correct Answer: A.

Q.16

Given below are two statements:

Statement I: Two matrices have same characteristic polynomials are similar.

Statement II: Two matrices have same minimal polynomial are similar.

In the light of the above statements, choose the correct answer:

1. Both Statement I and Statement II are true.
2. Both Statement I and Statement II are false.
3. Statement I is correct but Statement II is false.
4. Statement I is incorrect but Statement II is true.
5. Question not attempted

✅ Correct Answer: B.

Q.17

Consider the following statement and the options given below:

Let \(A = (e_2 \ e_3 \ e_4 \ e_1 \ e_6 \ e_7 \ e_5)\), where \(e_i = \begin{bmatrix}0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0\end{bmatrix}_{1\times7}\) with 1 in the i-th position.

Then the possible \(k \in \mathbb{N}\) such that \(A^k=I\) is:

1. 7
2. 10
3. 12
4. 24

How many of the options given above are correct?

1. Only one
2. Only two
3. Only three
4. All four
5. Question not attempted

✅ Correct Answer: B.

Q.18

Given below are two statements:

Let \(\lambda\) be an eigenvalue of \[ A = \begin{bmatrix} 1 & 2+4i & 5 & i \\ 1 & 3+4i & 6 & i \\ 0 & i & -1 & 0 \\ 1 & 0 & 0 & i \end{bmatrix}. \]

Statement I: There exists an eigenvalue \(\lambda\) of \(A\) such that \(12 \leq |\lambda| \leq 13\).

Statement II: If \(\lambda\) is an eigenvalue of \(A\) then \(|\lambda| \leq 12\).

Choose the correct answer:

1. Both Statement I and Statement II are true.
2. Both Statement I and Statement II are false.
3. Statement I is correct but Statement II is false.
4. Statement I is incorrect but Statement II is true.
5. Question not attempted

✅ Correct Answer: D.

Q.19

Consider the following statement and the options given below:

The absolute value of the integral \[ \int_0^{\pi/2}\frac{x(1+x^2)}{(2i+2x)^2}\sin x \, dx, \ x \neq i \] is bounded by:

1. \(\dfrac{\pi^{\frac{3}{2}}}{\cdot16\sqrt{6}}\)
2. \(\dfrac{\pi^2}{16\sqrt{6}}\)
3. \(\dfrac{\pi^2}{8\sqrt{6}}\)
4. \(\dfrac{\pi}{8\sqrt{6}}\)

How many of the options given above are correct?

1. Only one
2. Only two
3. Only three
4. All four
5. Question not attempted

✅ Correct Answer: B.

Q.20

Consider the following statement and the options given below:

Let \(A_{n\times n}\) and \(B_{n\times n}\) be two idempotent matrices. Then

1. If \(\text{Nullity}(A)=\text{Nullity}(B)\), then \(A\) is similar to \(B\).
2. If \(\text{Rank}(A)=\text{Rank}(B)\), then \(A\) is similar to \(B\).
3. 1 and 2 both are correct.

How many of the options given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: C.

Q.21

Consider the following statement and the options given below:

Let \(A=\begin{bmatrix}1 & 5 & 6 \\ 11 & a & 0 \\ 2 & -1 & 5\end{bmatrix}\), then

1. \(A\) is positive definite for some \(a \in \mathbb{R}\).
2. \(A\) is negative definite for some \(a \in \mathbb{R}\).
3. \(A\) is indefinite if \(7a < -341\).

How many of the options given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: D.

Q.22

Consider the following statement and the options given below:

Let \(A \in M_{n\times n}(\mathbb{R})\) with characteristic polynomial \(f(x)=(x-1)^k g(x)\), where \[ k=2025 \lim_{n\to\infty}\exp\left(\sum_{t=1}^n \frac{1}{n+t}\right). \]

1. If \(A\) is diagonalizable, then \(\dim(E_{\lambda=1})=2025\), for \(g(1)\neq 0\).
2. If \(A\) is diagonalizable, then \(\dim(E_{\lambda=1})=4050\), for \(g(1)\neq 0\).
3. If \(A\) is diagonalizable, then \(\dim(E_{\lambda=1})=\dfrac{48600}{\pi^2}\sum_{n=1}^{\infty}\dfrac{(-1)^{n-1}}{n^2}\), for \(g(1)\neq 0\).
4. If \(A\) is diagonalizable, then \(\dim(E_{\lambda=1})=6075\), for \(g(1)\neq 0\).

How many of the options given above are correct?

1. Only one
2. Only two
3. Only three
4. All four
5. Question not attempted

✅ Correct Answer: B.

Q.23

Consider the following statement and the options given below:

If \(q(x,y)=x^2+2xy+2y^2-3z^2\) be the given quadratic form, then its rank, index and signature are, respectively

1. 1, 2, 3
2. 3, 2, 1
3. 3, 1, -1

How many of the options given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: Question Removed.

Q.24

Consider the following statement and the options given below:

Let \(A=\begin{bmatrix}0 & 0 & 0 & -2\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 5\\ 0 & 0 & 1 & -2\end{bmatrix}\), then its eigenvalues are:

1. 1, 1, \(-4\pm \sqrt{2}\)
2. 1, 1, 2, \(\pm \sqrt{2}\)
3. 1, -1, -2, \(\pm \sqrt{2}\)

How many of the options given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: D.

Q.25

Consider the following statement and the options given below:

Let \(A\in \mathbb{C}^{m\times n}, B\in \mathbb{C}^{n\times p}, C\in \mathbb{C}^{p\times q}\) and \(D\in \mathbb{C}^{q\times r}\), then which of the following inequalities holds:

1. \(\rho(ABCD)+\rho(C)\geq \rho(ABC)+\rho(CD)\) , \( \rho(A)=\text{rank}(A)\)
2. \(\rho(ABCD)+\rho(BC)\geq \rho(ABC)\)
   \(+\rho(BCD)\)
3. \(\rho(ABCD)+q \leq \rho(ABC)+\rho(D)\)

How many of the options given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: B.

Q.26

Consider the following statement and the options given below:

Let \(T:\mathbb{R}^2 \to \mathbb{R}^2\) be a linear transformation defined as \(T(x,y)=(y,-x)\). Then

1. Eigenvalues of \(T\) are \(\pm i\).
2. Eigenvalues of \(T\) are \(\pm 1\).
3. \(T\) has no eigenvalues.

How many of the options given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: A.

Q.27

Given below are two statements:

Let \(H\) be a 11-Sylow subgroup of the group \(GL_{33}(\mathbb{F}_{11})\) of invertible matrices \(33 \times 33\) with entries from the field \(\mathbb{F}_{11}\).

Statement I: Let \(O(H)=k\) and \(k \equiv t \pmod{13}\), then \(t=11\).

Statement II: Let \(O(H)=k\) and \(k \equiv t \pmod{13}\), then \(t=1\).

Choose the correct answer:

1. Both Statement I and Statement II are true.
2. Both Statement I and Statement II are false.
3. Statement I is correct but Statement II is false.
4. Statement I is incorrect but Statement II is true.
5. Question not attempted

✅ Correct Answer: D.

Q.28

Consider the following statements:

Let \(U(n)=\{ m : 1 \leq m < n, \ (m,n)=1, \ n>1 \}.\)

1. Number of elements of order 4 in \(U(32)\) are 2.
2. \(U(8) \cong V_4\) (Klein 4-group).
3. \(\varphi(U(128))=64.\)

How many of the statements given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: B.

Q.29

Consider the following statements:

Locate the zeroes for \(f(z)=z^9-8z^2+5\).

1. No zeroes lying inside circle \(|z|=\tfrac{1}{2}\).
2. Two zeroes lie within \(\dfrac{1}{2}\leq |z| < 1 \)
3. Seven zeroes lie within \(1 \leq |z| < \tfrac{3}{2}\).

How many of the statements given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: C.

Q.30

Consider the following statements:

Let \(f(z)=u(x,y)+iv(x,y)\) be entire function.

1. If \(u(x,y)\leq x \ \forall z=x+iy\) then \(f(z)\) is a polynomial of degree at most one.
2. If \(v(x,y)\geq x \ \forall z=x+iy\) then \(f(z)\) is a polynomial of degree 1.
3. If \(f(z)\) is non-constant then \(f(\mathbb{C})\) is dense in \(\mathbb{C}\).

How many of the statements given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: C.

Q.31

Consider the following Statements:

1. If all the zeroes of a polynomial lie in a half plane, then zeroes of derivative also lie in the same half plane.
2. If all zeroes of a polynomial lie in the unit circle then zeroes of derivative lie in closed unit disk \(\{ z\in\mathbb{C} : |z|\leq 1\}\).
3. If all zeroes of a polynomial are reals then zeroes of derivative are also reals.
4. If all zeroes of a polynomial are purely imaginary then zeroes of derivative are also purely imaginary.

How many of the statements given above are correct?

1. Only one statement
2. Only two statements
3. All three statements
4. All four statements
5. Question not attempted

✅ Correct Answer: C.

Q.32

Consider the following pairs:

1. \(f_n(x)=xe^{-nx^2}, x\in\mathbb{R}\) : Uniformly convergent on \(\mathbb{R}\).
2. \(f_n(x)=xe^{-nx^2}, x\in\mathbb{R}\) : Uniformly convergent only on compact subset of \(\mathbb{R}\).
3. \(f_n(x)=xe^{-nx^2}, x\in\mathbb{R}\) : Bounded and not uniformly convergent on \(\mathbb{R}\).

How many of the above pairs are correctly matched?

1. Only one pair
2. Only two pairs
3. All three pairs
4. None of the pairs
5. Question not attempted

✅ Correct Answer: A.

Q.33

Consider the following statements:

1. Let \(f_n:[0,1]\to\mathbb{R}\) be a continuous function for each positive integer \(n\). If \(\lim_{n\to\infty}\int_0^1 f_n(x)^2 dx =0\) then \(\lim_{n\to\infty} f_n(1/2)=0\).
2. \(\{(x,y)\in\mathbb{R}^2:xy=1\}\) is connected.
3. \(\{(x,y)\in\mathbb{C}^2:xy=1\}\) is not connected.
4. \(\sum_{n=1}^\infty x^2e^{-nx}\) converges uniformly on \((0,\infty)\).

How many of the statements given above are correct?

1. Only one statement
2. Only two statements
3. Only three statements
4. All four statements
5. Question not attempted

✅ Correct Answer: A.

Q.34

Consider the following statements:

1. Let \(\mathbb{N}\) be endowed with the co-finite topology, then \(\mathbb{N}\) is not path connected.
2. \(GL(n,\mathbb{R})\) has two connected components.
3. \(SL(n,\mathbb{R})\) is connected.

How many of the statements given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: C.

Q.35

Consider the following statements:

1. If \(f\) is continuous and \(-f\) is well defined, then \(-f\) is continuous.
2. The union of two locally compact spaces remains locally compact.
3. Total boundedness is a topological property.

How many of the statements given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: Question Removed.

Q.36

Consider the following Statements:

1. If \(V_n\) is open dense subset of \(\mathbb{R}\), \(\forall n\), then \(\bigcap_{n\geq 1} V_n\) is non-empty.
2. For each \(n\geq 1\), \(V_n\) is open dense subset of \(\mathbb{R}\), then \(\bigcap_{n\geq 1} V_n\) is countable.
3. For \(n\geq 1\), \(V_n\) is open dense subset of \(\mathbb{R}\), then \(\bigcap_{n\geq 1} V_n\) is uncountable.
4. For \(n\geq 1\), \(V_n\) is open dense subset of \(\mathbb{R}\), then \(\bigcap_{n\geq 1} V_n\) is dense in \(\mathbb{R}\).

How many of the statements given above are correct?

1. Only one statement
2. Only two statements
3. All three statements
4. All four statements
5. Question not attempted

✅ Correct Answer: C.

Q.37

Given below are two statements:

Statement I: \(f(z)=e^z\) maps vertical line segment \(x=a, -\pi < y\leq \pi\) onto circle \(|f(z)|=e^a\).

Statement II: \(f(z)=e^z\) maps the horizontal line \(y=b, -\infty < x < \infty\) onto the ray \(\arg(e^z)=b\).

Choose the correct answer:

1. Both Statement I and Statement II are true.
2. Both Statement I and Statement II are false.
3. Statement I is correct but Statement II is false.
4. Statement I is incorrect but Statement II is true.
5. Question not attempted

✅ Correct Answer: A.

Q.38

Consider the following pairs:

1. \(x^3-2 : S_3\)
2. \(x^4-7 : D_8\)
3. \(x^8-24x^6+144x^4-288x^2+144 : Q_8\)

How many of the above pairs are correctly matched?

1. Only one pair
2. Only two pairs
3. All three pairs
4. None
5. Question not attempted

✅ Correct Answer: Question Removed.

Q.39

Given below are two statements:

Statement I: Any order two normal subgroup of a group is a central subgroup.

Statement II: If a p-subgroup of a group is normal then it must be unique.

Choose the correct answer:

1. Both Statement I and Statement II are true.
2. Both Statement I and Statement II are false.
3. Statement I is correct but Statement II is false.
4. Statement I is incorrect but Statement II is true.
5. Question not attempted

✅ Correct Answer: C.

Q.40

Given below are two statements:

Statement I: Let \(f:\mathbb{R}\to\mathbb{R}\) satisfy \(f(x)\leq f(y)\) for \(x\leq y\), then the set where \(f\) is not continuous is finite or countably infinite.

Statement II: Let \(A\) be a countably infinite subset of \(\mathbb{R}\). Then there exists a non-zero continuous function \(g:\mathbb{R}\to\mathbb{R}\) such that \(\int_a^b g(x)dx=0\) if \(a,b\notin A\).

Choose the correct answer:

1. Both Statement I and Statement II are true.
2. Both Statement I and Statement II are false.
3. Statement I is correct but Statement II is false.
4. Statement I is incorrect but Statement II is true.
5. Question not attempted

✅ Correct Answer: C.

Q.41

Given below are two statements:

Statement I: Let \(P_n(x)\) be any polynomial with integer coefficients whose degree is greater than or equal to 1. Then \(\exists\) at least one point \(x\in[-2,2]\) satisfying \(|P_n(x)|\geq 2\).

Statement II: Let \(P_n(x)\) be any polynomial with real coefficients where degree is greater than or equal to 1. Then \(\exists\) at least one point \(x\in[-2,2]\) satisfying \(|P_n(x)|\geq 2\).

Choose the correct answer:

1. Both Statement I and Statement II are true.
2. Both Statement I and Statement II are false.
3. Statement I is correct but Statement II is false.
4. Statement I is incorrect but Statement II is true.
5. Question not attempted

✅ Correct Answer: C.

Q.42

Consider the following statements:

1. \(\mathbb{Z}\left[\tfrac{1+\sqrt{-3}}{2}\right]\) is a Euclidean domain.
2. \(\mathbb{Z}\left[\tfrac{1+\sqrt{-7}}{2}\right]\) is a Principal Ideal domain.
3. \(\mathbb{Z}\left[\tfrac{1+\sqrt{-23}}{2}\right]\) is a Euclidean domain.

How many of the statements given above are correct?

1. Only one
2. Only two
3. All three
4. None
5. Question not attempted

✅ Correct Answer: B.

Q.43

Given below are two statements:

Let \(\mathbb{Z}[i]\) be the ring of Gaussian integers.

Statement I: \(1+10i\) is a unit in \(\mathbb{Z}[i]\).

Statement II: \(3+2i\) is a unit in \(\mathbb{Z}[i]\).

Choose the correct answer:

1. Both Statement I and Statement II are true.
2. Both Statement I and Statement II are false.
3. Statement I is correct but Statement II is false.
4. Statement I is incorrect but Statement II is true.
5. Question not attempted

✅ Correct Answer: B.

Q.50

Consider the following pairs:

1. \(|z-(1+2i)| = \text{Re}(z-2+3i) : \) Parabola
2. \(|z-(1+i)|+|z-(-1+i)|=2.5 : \) Ellipse
3. \(|z-(1+i)|=1 : \) Circle

How many of the above pairs are correctly matched?

1. Only one pair
2. Only two pairs
3. All three pairs
4. None of the pairs
5. Question not attempted

✅ Correct Answer: C.