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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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:
Q.2
Let \(A\in M_{2\times2}(\mathbb{R})\) such that \(A^2=0\). Then:
- \(A\) is similar to \(\begin{bmatrix}0 & 0\\ 1 & 0\end{bmatrix}\).
- \(A\) is similar to \(\begin{bmatrix}0 & 0\\ 0 & 1\end{bmatrix}\) where \(A\neq 0\).
- \(A\) is similar to \(\begin{bmatrix}0 & 0\\ 1 & 0\end{bmatrix}\) only if \(A\neq 0\).
Choose the correct answer:
Q.3
Consider the following statements:
- 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.
- There exists a continuous function \(f:\mathbb{R}\to \mathbb{R}\) that takes every value thrice.
- 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?
Q.4
Consider the following statements:
- If \(A\) and \(B\) are closed subsets of \(\mathbb{R}\), then the Minkowski sum \(A+B\) is also closed in \(\mathbb{R}\).
- If \(A\) and \(B\) are connected subsets of \(\mathbb{R}\), then the Minkowski sum \(A+B\) is also connected.
- 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?
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:
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:
- For any choices of \(F_1\) and \(F_2\).
- If one of them contains the other.
- If \(F_1 \subseteq F_2\).
- If \(F_1 = F_2\).
How many of the options given above are correct?
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:
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:
Q.9
Consider the following pairs:
- \(C[0,1]\) with \(\|f\|_\infty\): It is complete
- \(C[0,1]\) with the norm \(\|f\|_\infty + |f(1/2)|\): It is complete
- \(C[0,1]\) with \(\|f\|_2\): It is complete
How many of the above pairs are correctly matched?
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
- \(|\lambda|\leq 7+\sqrt{34}\)
- \(|\lambda|\leq 6+2\sqrt{5}\)
- \(|\lambda|\leq 1+3\sqrt{5}+\sqrt{34}\)
- \(|\lambda|\leq 13\)
How many of the options given above are correct?
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 \(c=\)?
- 2
- -2
- \(\dfrac{24}{\pi^2}\sum_{n=1}^{\infty}\dfrac{1}{2n^2}\)
- \(-\dfrac{24}{\pi^2}\sum_{n=1}^{\infty}\dfrac{1}{2n^2}\)
How many of the options given above are correct?
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.
- \(2\sum_{n=1}^{\infty}\dfrac{(-1)^{n-1}(2\pi)^{2n}}{(2n)!}, 3\)
- -2, 3
- \(2, 3\sum_{n=1}^{\infty}\dfrac{({\dfrac{\pi}{2})}^{2n-1}(-1)^{n-1}}{(2n-1)!}\)
- -1, 3
How many of the options given above are correct?
Q.13
Consider the following statements:
- \(A\) and \(B\) are positive definite matrices iff \(A+B\) is positive definite always.
- If \(A\) is a negative definite matrix, then \(A^k\) is negative definite iff \(k\) is odd.
- 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?
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:
Q.15
Consider the following pairs:
- \(P_2(\mathbb{R})/\mathbb{R}\) : dimension 2
- \(M_2(\mathbb{C})/\mathbb{R}\) : dimension 4
- \(M_2(\mathbb{R})/\mathbb{Q}\) : dimension infinite
How many of the above pairs are correctly matched?
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:
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:
- 7
- 10
- 12
- 24
How many of the options given above are correct?
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:
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:
- \(\dfrac{\pi^{\frac{3}{2}}}{\cdot16\sqrt{6}}\)
- \(\dfrac{\pi^2}{16\sqrt{6}}\)
- \(\dfrac{\pi^2}{8\sqrt{6}}\)
- \(\dfrac{\pi}{8\sqrt{6}}\)
How many of the options given above are correct?
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
- If \(\text{Nullity}(A)=\text{Nullity}(B)\), then \(A\) is similar to \(B\).
- If \(\text{Rank}(A)=\text{Rank}(B)\), then \(A\) is similar to \(B\).
- 1 and 2 both are correct.
How many of the options given above are correct?
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
- \(A\) is positive definite for some \(a \in \mathbb{R}\).
- \(A\) is negative definite for some \(a \in \mathbb{R}\).
- \(A\) is indefinite if \(7a < -341\).
How many of the options given above are correct?
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 \[ k=2025 \lim_{n\to\infty}\exp\left(\sum_{t=1}^n \frac{1}{n+t}\right). \]
- If \(A\) is diagonalizable, then \(\dim(E_{\lambda=1})=2025\), for \(g(1)\neq 0\).
- If \(A\) is diagonalizable, then \(\dim(E_{\lambda=1})=4050\), for \(g(1)\neq 0\).
- If \(A\) is diagonalizable, then \(\dim(E_{\lambda=1})=\dfrac{48600}{\pi^2}\sum_{n=1}^{\infty}\dfrac{(-1)^{n-1}}{n^2}\), for \(g(1)\neq 0\).
- If \(A\) is diagonalizable, then \(\dim(E_{\lambda=1})=6075\), for \(g(1)\neq 0\).
How many of the options given above are correct?
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, 2, 3
- 3, 2, 1
- 3, 1, -1
How many of the options given above are correct?
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, \(-4\pm \sqrt{2}\)
- 1, 1, 2, \(\pm \sqrt{2}\)
- 1, -1, -2, \(\pm \sqrt{2}\)
How many of the options given above are correct?
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:
- \(\rho(ABCD)+\rho(C)\geq \rho(ABC)+\rho(CD)\) , \( \rho(A)=\text{rank}(A)\)
- \(\rho(ABCD)+\rho(BC)\geq \rho(ABC)\)
\(+\rho(BCD)\) - \(\rho(ABCD)+q \leq \rho(ABC)+\rho(D)\)
How many of the options given above are correct?
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
- Eigenvalues of \(T\) are \(\pm i\).
- Eigenvalues of \(T\) are \(\pm 1\).
- \(T\) has no eigenvalues.
How many of the options given above are correct?
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:
Q.28
Consider the following statements:
Let \(U(n)=\{ m : 1 \leq m < n, \ (m,n)=1, \ n>1 \}.\)
- Number of elements of order 4 in \(U(32)\) are 2.
- \(U(8) \cong V_4\) (Klein 4-group).
- \(\varphi(U(128))=64.\)
How many of the statements given above are correct?
Q.29
Consider the following statements:
Locate the zeroes for \(f(z)=z^9-8z^2+5\).
- No zeroes lying inside circle \(|z|=\tfrac{1}{2}\).
- Two zeroes lie within \(\dfrac{1}{2}\leq |z| < 1 \)
- Seven zeroes lie within \(1 \leq |z| < \tfrac{3}{2}\).
How many of the statements given above are correct?
Q.30
Consider the following statements:
Let \(f(z)=u(x,y)+iv(x,y)\) be entire function.
- If \(u(x,y)\leq x \ \forall z=x+iy\) then \(f(z)\) is a polynomial of degree at most one.
- If \(v(x,y)\geq x \ \forall z=x+iy\) then \(f(z)\) is a polynomial of degree 1.
- If \(f(z)\) is non-constant then \(f(\mathbb{C})\) is dense in \(\mathbb{C}\).
How many of the statements given above are correct?
Q.31
Consider the following Statements:
- If all the zeroes of a polynomial lie in a half plane, then zeroes of derivative also lie in the same half plane.
- If all zeroes of a polynomial lie in the unit circle then zeroes of derivative lie in closed unit disk \(\{ z\in\mathbb{C} : |z|\leq 1\}\).
- If all zeroes of a polynomial are reals then zeroes of derivative are also reals.
- 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?
Q.32
Consider the following pairs:
- \(f_n(x)=xe^{-nx^2}, x\in\mathbb{R}\) : Uniformly convergent on \(\mathbb{R}\).
- \(f_n(x)=xe^{-nx^2}, x\in\mathbb{R}\) : Uniformly convergent only on compact subset of \(\mathbb{R}\).
- \(f_n(x)=xe^{-nx^2}, x\in\mathbb{R}\) : Bounded and not uniformly convergent on \(\mathbb{R}\).
How many of the above pairs are correctly matched?
Q.33
Consider the following statements:
- 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 \(\lim_{n\to\infty} f_n(1/2)=0\).
- \(\{(x,y)\in\mathbb{R}^2:xy=1\}\) is connected.
- \(\{(x,y)\in\mathbb{C}^2:xy=1\}\) is not connected.
- \(\sum_{n=1}^\infty x^2e^{-nx}\) converges uniformly on \((0,\infty)\).
How many of the statements given above are correct?
Q.34
Consider the following statements:
- Let \(\mathbb{N}\) be endowed with the co-finite topology, then \(\mathbb{N}\) is not path connected.
- \(GL(n,\mathbb{R})\) has two connected components.
- \(SL(n,\mathbb{R})\) is connected.
How many of the statements given above are correct?
Q.35
Consider the following statements:
- If \(f\) is continuous and \(-f\) is well defined, then \(-f\) is continuous.
- The union of two locally compact spaces remains locally compact.
- Total boundedness is a topological property.
How many of the statements given above are correct?
Q.36
Consider the following Statements:
- If \(V_n\) is open dense subset of \(\mathbb{R}\), \(\forall n\), then \(\bigcap_{n\geq 1} V_n\) is non-empty.
- For each \(n\geq 1\), \(V_n\) is open dense subset of \(\mathbb{R}\), then \(\bigcap_{n\geq 1} V_n\) is countable.
- For \(n\geq 1\), \(V_n\) is open dense subset of \(\mathbb{R}\), then \(\bigcap_{n\geq 1} V_n\) is uncountable.
- 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?
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:
Q.38
Consider the following pairs:
- \(x^3-2 : S_3\)
- \(x^4-7 : D_8\)
- \(x^8-24x^6+144x^4-288x^2+144 : Q_8\)
How many of the above pairs are correctly matched?
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:
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:
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:
Q.42
Consider the following statements:
- \(\mathbb{Z}\left[\tfrac{1+\sqrt{-3}}{2}\right]\) is a Euclidean domain.
- \(\mathbb{Z}\left[\tfrac{1+\sqrt{-7}}{2}\right]\) is a Principal Ideal domain.
- \(\mathbb{Z}\left[\tfrac{1+\sqrt{-23}}{2}\right]\) is a Euclidean domain.
How many of the statements given above are correct?
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:
Q.44
Given below are two statements:
Statement I: Let \(f:D(0,2)\to\mathbb{C}\) be analytic function satisfying \(f(0)=1, |f(e^{i\theta})|>2 \ \forall \theta\in[-\pi,\pi]\). Then \(\exists z_0\in D(0,2)\) such that \(f'(z_0)=0\).
Statement II: If \(f\) is analytic at infinity then derivative of \(f\) need not to be analytic at infinity.
Choose the correct answer:
Q.45
Given below are two statements:
Statement I: Let \(f\) be a non-constant and analytic in an open set containing the closed unit disc. If \(|f(z)|=1\) whenever \(|z|=1\), then for every \(w_0\in \mathbb{C} : |w| < 1, f= w_0\) has a root.
Statement II: Every polynomial of degree \(n\) assumes each complex number exactly \(n\) times.
Choose the correct answer:
Q.46
Match the number of zeroes for the following functions inside \(|z|=1\):
- \(z^7-4z^3+z+1 : 3\)
- \(z^7-5z^2+12 : 0\)
- \(e^z-3z^2 : \infty\)
How many of the above pairs are correctly matched?
Q.47
Consider the following pairs:
- \(\displaystyle \sum_{n=1}^\infty \frac{e^{-n|x|}}{n^3}\) : The series converges uniformly on \((- \pi,\pi)\).
- \(\displaystyle \sum_{n=1}^\infty \left(\frac{x}{n}\right)^n\) : Converges uniformly on \((- \pi,\pi)\).
- \(\displaystyle \sum_{n=1}^\infty \frac{1}{[(x+\pi)n]^2}\) : Converges uniformly on \((- \pi,\pi)\).
How many of the above pairs are correctly matched?
Q.48
Consider the following pairs:
- \(s_1=\left\{\frac{\sin(\tfrac{n\pi}{2017})}{n}: n\in \mathbb{N}\right\},\)
\(s_1 \text{ is compact in} \; \mathbb{R}.\) - \(s_2=\left\{\frac{\cos(\tfrac{n\pi}{2017})}{n}: n\in \mathbb{N}\right\},\)
\(s_2 \text{ is compact in}\; \mathbb{R}.\) - \(s_3=\left\{\frac{\tan(\tfrac{n\pi}{2017})}{n}: n\in \mathbb{N}\right\},\)
\(s_3 \text{ is compact in} \; \mathbb{R}.\)
How many of the above pairs are correctly matched?
Q.49
Given below are two statements:
Statement I: If a rational function \(f(z)=\tfrac{p(z)}{q(z)}\) has a removable singularity at \(z=\infty\) then \(\deg p(z) \leq \deg q(z)\).
Statement II: If a rational function \(f(z)=\tfrac{p(z)}{q(z)}\) has a pole (simple pole) at \(z=\infty\) then \(\deg p(z) > \deg q(z)\).
Choose the correct answer:
Q.50
Consider the following pairs:
- \(|z-(1+2i)| = \text{Re}(z-2+3i) : \) Parabola
- \(|z-(1+i)|+|z-(-1+i)|=2.5 : \) Ellipse
- \(|z-(1+i)|=1 : \) Circle
How many of the above pairs are correctly matched?
Q.51
Consider the following pairs. Let \(C\) be unit circle \(|z|=1\) with counter clockwise orientation:
- \(\int_C z^m dz = 0\) : if \(m \neq -1\)
- \(\int_C \overline{z}^m dz = 0\) : if \(m \neq 1\)
- \(\int_C z^m |dz| = 0\) : if \(m \neq 0\)
How many of the above pairs are correctly matched?
Q.52
Given below are two statements:
Statement I: Let \(f:\mathbb{R}\to\mathbb{R}\) be a continuous function satisfying \(f(x)=f(x+\sqrt{2})=f(x+\sqrt{3}), \forall x\). Then \(f\) is constant.
Statement II: Let \(E\) be a Lebesgue measurable subset of \(\mathbb{R}\) with \(\mu_L(E)=1\). Then there exists a Lebesgue measurable set \(A \subset E\) such that \(\mu_L(A)=1/2\).
Choose the correct answer:
Q.53
Consider the following statements:
- Let \(f\) be a twice differentiable function on \(\mathbb{R}\) such that both \(f\) and \(f''\) are strictly positive on \(\mathbb{R}\). Then \(\lim_{x\to\infty} f(x)=\infty\).
- Let \(f:\mathbb{R}^2 \to \mathbb{R}\) by \(f(x,y)=x^[y]\), then \(f\) is continuous at a point of \(\mathbb{R}^2\).
- There exists a function \(f:\mathbb{R}\to\mathbb{R}\) satisfying \(f(-2)=-2, f(2)=2\) and \(|f(x)-f(y)|\leq |x-y|^{3/2}, \forall x,y\in\mathbb{R}\).
How many of the statements given above are correct?
Q.54
Given below are two statements:
Statement I: Let \(f:[-1,1]\to\mathbb{R}\) be a continuous function. Assume that \(f(-1)f(1) < 0.\) Then there exists some \(\alpha \in (-1, 1)\), such that \(f(\alpha ) = 0\).
Statement II: The pre-image of a connected set by a continuous function is connected.
Choose the correct answer:
Q.55
Given below are two statements:
Statement I: \(\mathbb{Q}^{\mathbb{N}}\) is first countable but not second countable.
Statement II: Let \begin{align} & x_1=\{(x_1,x_2)\in\mathbb{R}^2 \mid 0<x_1<\infty, \notag \\ &\quad x_2=\sin\tfrac{1}{x_1} \} \end{align} and \(x_2=\{(0,x_2)\in\mathbb{R}^2 \mid -1\leq x_2 \leq 1\}\). Then \(x_1\) and \(x_1\cup x_2\) both are connected, but \(x_1\cup x_2\) is not path connected.
Choose the correct answer:
Q.56
Given below are two statements:
Statement I: In a discrete probability weighted sum of all the probabilities can be 5.
Statement II: In a discrete probability weighted sum of all the probabilities always 1.
Choose the correct answer:
Q.57
Given below are two statements:
Statement I: The scatter in a series of values about the average is called central tendency.
Statement II: The scatter in a series of values about the average is called dispersion.
Choose the correct answer:
Q.58
Consider the following statements:
- The feasible solution of a LPP belongs to 1st and 2nd quadrants.
- Only 1st quadrant.
- Only 1st and 3rd quadrants.
How many of the statements given above are correct?
Q.59
Consider the following statements:
- The maximum ordinate of a normal curve is at \(X=\mu\).
- In a normal curve, \(\mu \pm 0.6745\sigma\) covers 50% area.
- The upper quartiles of a standard normal distribution is \(0.6745\sigma\).
How many of the statements given above are correct?
Q.60
Consider the following statements:
- Poisson distribution is highly skewed with high mean.
- If \(X\) in Poisson, then standard error of \(X\) is \(\sqrt{\frac{\lambda}{n}}\).
- In Poisson distribution, parameter \(\lambda > 0\) and taking finite value.
How many of the statements given above are correct?
Q.61
Consider the following statements:
- \(\text{Card}\left(\tfrac{\mathbb{Z}[i]}{\langle 1+2i \rangle}\right)=2\)
- \(\text{Card}\left(\tfrac{\mathbb{Z}[i]}{\langle 1+2i \rangle}\right)=5\)
- \(\tfrac{\mathbb{Z}[i]}{\langle 1+2i \rangle}\) is a field
- \(\tfrac{\mathbb{Z}[i]}{\langle 1+2i \rangle}\) has a non-zero proper ideal
How many of the statements given above are correct?
Q.62
Consider the following pairs: Let \(GL(n,\mathbb{Z}_p)\) be the general linear group of \(n \times n\) matrices, where \(\mathbb{Z}_p = \mathbb{F}_p\).
- \(o(GL(2,\mathbb{Z}_2)) : 4\)
- \(o(GL(2,\mathbb{Z}_2)) : 6\)
- \(o(GL(3,\mathbb{Z}_2)) : 160\)
How many of the above pairs are correctly matched?
Q.63
Consider the following statements:
- Group of units of the ring \(\mathbb{Z}[\sqrt{2}]\) is finite.
- Group of units of the ring \(\mathbb{Z}[\sqrt{2}]\) is infinite.
- Group of units of the ring \(\mathbb{Z}[\sqrt{-2}]\) is finite.
- Group of units of the ring \(\mathbb{Z}[\sqrt{-10}]\) has finite order.
How many of the statements given above are correct?
Q.64
Given below are two statements:
Statement I: If \(G_1\) and \(G_2\) are two groups such that \(G_1 \subseteq G_2\), then \(G_1\) is a subgroup of \(G_2\).
Statement II: If \(G_1\) and \(G_2\) are two subgroups of a group \(G\), then \(G_1 \cup G_2\) is a subgroup of \(G_1\).
Choose the correct answer:
Q.65
Given below are two statements:
Let \(P\) be a prime number and \(f(x) \in \mathbb{Z}[x]\) with \(\deg(f(x)) \geq 1\). Let \(\bar{f}(x) \in \mathbb{F}_p[x]\), then :.
Statement I: If \(\bar{f}(x)\) is irreducible over \(\mathbb{F}_p\) then \(f(x)\) is irreducible over \(\mathbb{Q}\).
Statement II: If \(\bar{f}(x)\) is reducible over \(\mathbb{F}_p\) then \(f(x)\) is reducible over \(\mathbb{Q}\).
Choose the correct answer:
Q.66
Consider the following statements:
- \(x^4 + 3x^4 + 6x + 3\) is irreducible over \(\mathbb{Q}\).
- \(10x^5 + 11x^3 + 121x + 11\) is reducible over \(\mathbb{Q}\).
- \(x^3 + 3x^2 + 2x + 1\) is irreducible over \(\mathbb{Q}\).
- \(x^3 + 4x^2 + x + 1\) is irreducible over \(\mathbb{Q}\).
How many of the statements given above are correct?
Q.67
Given below are two statements: Let \(I = \int_a^b x^2 dx\); \(a\) and \(b\) are given.
Statement I: The value of \(I\) obtained by using trapezoidal rule is always greater than or equal to the exact value of definite integral.
Statement II: The value of \(I\) obtained by using Simpson’s rule is always equal to exact value of the definite integral.
Choose the correct answer:
Q.68
Consider the following pairs: Match the order
- Bisection method : 1
- Secant method : Golden Ratio
- Newton Raphson method : 2
How many of the above pairs are correctly matched?
Q.69
Given below are two statements:
Statement I: Two steps of Euler’s method with step size 0.25 give same accuracy to one step of modified Euler’s method with step size 0.5.
Statement II: Two steps of Euler’s method with step size 0.1 give more accurate solution than one step of modified Euler’s method with step size 0.2.
Choose the correct answer:
Q.70
Given below are two statements:
Statement I: If \(o(G) = 175\), then there exists a subgroup \(H\) of \(G\) such that \(o(H) = 35\).
Statement II: The number of non-isomorphic abelian groups of order 24 is equal to 4.
Choose the correct answer:
Q.71
Consider the following statements: Let \(G\) be a \(p\)-group.
- \(G\) can not be simple.
- \(G\) may be simple.
- If \(o(G) = p^n\) then \(G\) is not simple for any \(n \in \mathbb{N} - \{1\}\).
How many of the statements given above are correct?
Q.72
Consider the following pairs: Let \(G\) be a group and \(p\) be a prime number.
- \(o(G) = p^2\) : Abelian group
- \(o(G) = p\) : Cyclic group
- \(o(G) = p^3\) : May be abelian
How many of the above pairs are correctly matched?
Q73.
Given below are two statements :Statement I : The \(p^{th}\) root of a given number \(N\) is root of equation \(f(x): x^p - N = 0\) then iteration formula by Newton Raphson method is: \[ x_{n+1} = \frac{(p-1)x_n^p + N}{p x_n^{p-1}} \]
Statement II : The iteration formula for \(f(x): x^p - N = 0\) by Secant method is:
\[ x_{n+1} = x_n - \frac{x_n^p - N}{x_n^p - x_{n-1}^p}(x_n - x_{n-1}) \]
In the light of the above statements, choose the Correct answer from the options given below :
(A) Both statement I and statement II are true
(B) Both statement I and statement II are false
(C) Statement I is correct but statement II is false
(D) Statement I is incorrect but statement II is true
(E) Question not attempted
Q74.
Consider the following pairs: Match the order of fitting polynomial1. Trapezoidal rule : 1
2. Simpson’s \(\tfrac{1}{3}\) rule : 2
3. Simpson’s \(\tfrac{3}{8}\) rule : 3
How many of the above pairs are correctly matched?
(A) Only one pair
(B) Only two pairs
(C) All three pairs
(D) None of the pairs
(E) Question not attempted
Q75.
Consider the following statements :Let \(R\) be the ring of all continuous functions \(f : \mathbb{R} \to \mathbb{R}\) and \(B = \{ f \in R \mid f(0) = \text{even integer} \}\). Then
1. \(B\) is a subring of \(R\).
2. \(B\) is an ideal of \(R\).
3. \(B\) is an integral domain.
How many of the statements given above are correct?
(A) Only one
(B) Only two
(C) All three
(D) None
(E) Question not attempted
Q76.
Given below are two statements :Statement I : Let \(G\) be an abelian group with \(o(G) = 122\), then there are 62 elements of order \(k\), if \(1 \leq k < 122\).
Statement II : If \(o(G) = 2809\), then \(G\) is abelian.
In the light of the above statements, choose the Correct answer from the options given below :
(A) Both Statement I and Statement II are true
(B) Both Statement I and Statement II are false
(C) Statement I is correct but Statement II is false
(D) Statement I is incorrect but Statement II is true
(E) Question not attempted
Q77.
Given below are two statements :Let \(G\) be a group and \(H\) be a subgroup of \(G\). Then
Statement I : \(G/H\) is a group of cosets of \(H\), where \(G/H = \{aH \mid aHbH = abH\}\) and \(H\) is any subgroup of \(G\).
Statement II : \(G/H\) is a group for some subgroup \(H\).
In the light of the above statements, choose the Correct answer from the options given below :
(A) Both Statement I and Statement II are true
(B) Both Statement I and Statement II are false
(C) Statement I is correct but Statement II is false
(D) Statement I is incorrect but Statement II is true
(E) Question not attempted
Q78.
Consider the following statements :1. A continuous random variable assume any possible value between two points.
2. A discrete random variable assume only finite values.
3. A discrete sample space has only finite points.
4. Any random value can be assumed by any random variable.
How many of the statements given above are correct?
(A) Only one statement
(B) Only two statements
(C) Only three statements
(D) All four statements
(E) Question not attempted
Q79.
Consider the following statements and the options given below :Let \(x_i\) represent equally spaced points with \(x_i - x_{i-1} = n\). Then difference formula for approximation of \(f’(x_i)\) is:
1. \(\frac{f(x_{i+1}) - f(x_i)}{n}\)
2. \(\frac{f(x_i) - f(x_{i-1})}{2n}\)
3. \(\frac{f(x_{i+1}) - f(x_{i-1})}{2n}\)
4. \(\frac{f(x_i) - f(x_{i-1})}{n}\)
How many of the options given above are correct?
(A) Only one
(B) Only two
(C) Only three
(D) All four
(E) Question not attempted
Q80.
Give below are two statements: Let \(f\) be differentiable on \([-1,1]\). Let \(f(-1)=1, f(1)=1, f(0)=-1\).Statement I : \(\exists c \in (-1,1)\) such that \(f(c) = 0\).
Statement II : \(\exists c \in (-1,1)\) such that \(f’(c) = 0\).
In the light of the above statements, choose the Correct answer from the options given below :
(A) Both statement I and statement II are true
(B) Both statement I and statement II are false
(C) Statement I is correct but statement II is false
(D) Statement I is incorrect but statement II is true
(E) Question not attempted
Q.81
For \(\lambda, \mu \in \mathbb{R}\), consider the following integral equation
\( f(x) = e^x + \lambda \int_0^1 e^{i\mu(x-r)} f(t) dt, \; x \in \mathbb{R} \).
Consider the following statements:
Statement I: There is a solution to the above integral equation for any choice of \(\lambda, \mu \in \mathbb{R}\).
Statement II: The solution to the above integral equation does not exist if \(\lambda = 1\) and \(\mu = 3\).
Choose the correct answer:
Q.82
For \(\lambda \in \mathbb{R}\) and \(K:[0,1]\times[0,1] \to \mathbb{R}\) be a continuously differentiable function, consider the following integral equation
\( f(x) = \sin(x^2) + \lambda \int_0^x K(x,y) f(y) dy, \;\)
\(x \in [0,1]. \)
Now consider the following statements:
Statement I: The above integral equation has a unique solution if and only if \(|\lambda| < 1\).
Statement II: The above integral equation has a unique solution for any choice of \(\lambda \in \mathbb{R}\) and a continuously differentiable function \(K:[0,1]\times[0,1] \to \mathbb{R}\).
Choose the correct answer:
Q.83
Assume that \(p,q:[a,b]\to \mathbb{R}\) be two continuous functions. Now for the following ODE
\( y''(x) + p(x) y'(x) + q(x) y(x) = 0, \)
\(\; x \in (a,b) \)
Consider the following statements:
Statement I: Any solution to the above ODE whose graph is tangent to x-axis at some point \(c \in (a,b)\) is identically zero.
Statement II: Any solution to the above ODE whose graph has tangent not equal to the x-axis but parallel to x-axis at some point \(d \in (a,b)\) is identically zero.
Choose the correct answer:
Q.84
Consider the following statements:
Statement I: The initial value problem \(\frac{dy}{dx} = y^{4/3}, \; y(0)=1\) has a unique solution which exists for each \(x \geq 0\).
Statement II: The maximum interval of existence for solution to the initial value problem \(\frac{dy}{dx} = y^{4/3}, \; y(0)=1\) is \(\mathbb{R}\).
Choose the correct answer:
Q.85
Consider the following functional
\begin{align} & J(y_1(x), y_2(x)) = \int_0^1 \bigg[ y_1^2(x) \notag \\ & - 2y_1(x)y_2(x) + \left(\frac{dy_1}{dx}\right)^2 + \left(\frac{dy_2}{dx}\right)^2 \bigg] dx. \end{align}
Statements:
- The Euler–Lagrange equation corresponding to the functional is a system of linear differential equations.
- If \(y_1(0)=y_2(0)=0\), then there exists a unique solution.
- If \(y_1(0)=y_2(0)=y_2(1)=0\), then there exists a unique solution.
- There is no solution satisfying \(y_1(0)=y_2(0)=0\) and \(y_1'(0)=y_2'(0)=0\).
How many of the statements given above are correct?
Q.86
For \(a,b \in \mathbb{R}\) with \(a < b\) and \(\lambda \in \mathbb{R}\), consider the following integral equation
\( f(x) = e^x + \lambda \int_a^b \sin(2y) f(y) dy,\)
\(\; x \in [a,b]. \)
Statements:
- There exist a unique pair \(a,b \in \mathbb{R}\) with \(a<b\) such that the above integral equation has a unique solution for any choice of \(\lambda\).
- Given \(a,b \in \mathbb{R}\), the above equation cannot have a unique solution for any choice of \(\lambda\).
- Given \(a,b \in \mathbb{R}, a<b\), there exists a unique \(\lambda \in \mathbb{R}\) such that the above equation has a unique solution.
- There always exists a choice of \(a<b\) and \(\lambda \in \mathbb{R}\setminus\{0\}\) for which the above integral equation has a unique solution.
How many of the statements given above are correct?
Q.87
Consider the following statements:
Statement I: The initial value problem \[ y' = 1 + y^{2/3}, y(0) = 0 \] has a unique solution in some open interval around 0.
Statement II: There exists a solution of \( 3|y'| + 2|y| + 1 = 0 \).
Statement III: There exists a solution to the initial value problem \( y' = -y^2, y(1) = 1 \) whose interval of existence contains some open interval around 0.
Statement IV: There exists a nonzero solution to the initial value problem \( y' = y^2, y(0) = 0 \) whose interval of existence is \(\mathbb{R}\).
How many of the statements given above are correct?
Q.88
For the following partial differential equation (PDE):
\[\dfrac{\partial^2 u}{\partial x^2} - 4 \dfrac{\partial^2 u}{\partial x \partial y} + 4 \dfrac{\partial^2 u}{\partial y^2} = 0\]
Consider the following statements:
Statement I: The above PDE is elliptic.
Statement II: For any two times differentiable functions \(f, g: \mathbb{R} \to \mathbb{R}\), the function \(v(x,y) := f(y+2x) + x g(x+2y),\)
\((x,y) \in \mathbb{R}^2\) is a solution to the above PDE.
Choose the correct answer:
Q.89
Consider the following statements:
Statement I: There exists a unique solution to the Euler–Lagrange equation corresponding to the functional \(J(y) := \int_0^{100} (y^2 + \frac{dy}{dx}) dx, y(0) = 0,\)
\(y(100) = 1.\)
Statement II: There exists a unique solution to the Euler–Lagrange equation corresponding to the functional \(J(y) := \int_0^{100} y^2(x) dx, y(0) = 0,\)
\(y(100) = 1.\)
Statement III: The solution \(y\) to the Euler–Lagrange equation corresponding to the functional \(J(y) := \int_0^{100} (y^2 + \frac{dy}{dx}) dx, y(0) = 0,\)
\(y(100) = 1\) is a polynomial of degree 2.
How many of the statements given above are correct?
Q.90
Consider the following statements:
Statement I: There exists a nonzero characteristic \(\lambda \in \mathbb{R}\) for the integral equation \[f(x) = \tan x + \lambda \int_0^x \cos^2(y) f(y) dy\]
\[, x \in [0, \pi/4].\]
Statement II: For any \(\lambda \neq 0\), there exists a nonzero solution to the integral equation \[f(x) = \lambda \int_0^x \cos^2(y) f(y) dy, x \in [0, \pi/4]\] such that \(f(0) = 0.\)
Statement III: All continuous solutions to the above integral equations are Lipschitz continuous on \([0, \pi/4].\)
How many of the statements given above are correct?
Q.91
Assume that \(p, q\) are continuous functions on \([a,b]\) and \(y_1, y_2\) be two solutions of
\(\dfrac{d^2 y}{dx^2} + p(x) \dfrac{dy}{dx} + q(x) y = 0, \; x \in (a,b)\)
such that their Wronskian and the solution \(y_1\) at some \(on\; (a,b)\) are zero. Now consider the following statements:
Statement I: \(y_1(x) = 0, \; \forall x \in (a,b).\)
Statement II: \(y_2(x) = \left(\dfrac{y_2'(x_0)}{y_1'(x_0)}\right) y_1(x), \; \forall x \in (a,b).\)
Choose the correct option:
Q.92
Consider the following second order linear ODE:
\[x^2 y''(x) + y(x) = 0, \; 1 \leq x \leq a,\]
with boundary conditions \(y(1) = y(a) = 0.\)
Suppose \(A := \{ a < 1 : (1) \; \text{has a nonzero solution} \}.\)
Statement I: \(A = \varnothing.\)
Statement II: \(A\) is a countably infinite set.
Choose the correct answer:
Q.93
For the following partial differential equation (PDE)
\[\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = 0, \; \text{in } \mathbb{R}^2\]
Consider the following statements:
Statement I: There exists a nonzero solution \(u\) to the above PDE such that \(\lim_{\sqrt{x^2+y^2} \to \infty} u(x,y)\) exists finitely.
Statement II: If \(u\) is a solution to the above PDE, then \(v(x,y) = u(y, -3x),\;(x, y) \in \mathbb{R}^2\) is also a solution to the above PDE.
Statement III: There are infinitely many nonconstant smooth functions which solve the above PDE.
How many of the statements given above are correct?
Q.94
Consider the following Cauchy problem for first order partial differential equation (PDE)
\[y \dfrac{\partial u}{\partial x} + x \dfrac{\partial u}{\partial y} = 0,\] \[(x,y) \in \mathbb{R} \times [0,\infty), u(x,0) = 0, x \in \mathbb{R}.\]
Now consider the following statements:
Statement I: There exists a unique solution to the above problem which exists on \(\mathbb{R} \times [0,\infty).\)
Statement II: There exists a nonzero solution \(u\) to the above problem on \(\mathbb{R} \times [0,\infty)\) such that the set \(A := \{ (x,y) \in \mathbb{R} \times [0,\infty) : u(x,y) = 0 \}\) is uncountable.
Statement III: The solution to the above problem does not exist on \(\mathbb{R} \times [0,\infty).\)
How many of the statements given above are correct?
Q.95
Consider the following statements:
1. Angle between the incident and scattered direction is called solid angle.
2. Halley’s comet has a value of eccentricity 0.096.
3. The velocity of satellite is \(\sqrt{2gR^2\left(\dfrac{1}{r} - \dfrac{1}{2a}\right)}.\)
How many of the statements given above are correct?
Q.96
Consider the following pairs:
1. \(\dfrac{d}{dt}\left(\dfrac{\partial \vec{r}_i}{\partial q_j}\right) : \dfrac{-\partial}{\partial q_j}\left(\dfrac{d \vec{r}_i}{dt}\right)\)
2. \(\dfrac{d}{dt}\left(\dfrac{\partial \vec{r}_i}{\partial q_j}\right) : \dfrac{\partial}{\partial q_j}\left(\dfrac{d \vec{r}_i}{dt}\right)\)
3. \(\dfrac{d}{dt}\left(\dfrac{\partial \vec{r}_i}{\partial q_j}\right) : \dfrac{d}{dq_j}\left(\dfrac{d \vec{r}_i}{dt}\right)\)
How many of the above pairs are correctly matched?
Q.97
Consider the following statements:
Statement I: There exists a nonzero continuous function \(f:[a,b]\to \mathbb{R}\) such that \(\int_a^b f(x)g(x) dx = 0\), for any choice of continuous function \(g:[a,b]\to \mathbb{R}\) such that \(g(a)=g(b)=0.\)
Statement II: There exists at least one nonzero continuous function \(f_1\) or \(f_2:[a,b]\to \mathbb{R}\) such that \[\int_a^b [f_1(x)g_1(x) + f_2(x)g_2(x)] dx = 0\], for any choice of continuous functions \(g_1,g_2:[a,b]\to \mathbb{R}\) such that \(g_1(a)=g_1(b)=g_2(a)=g_2(b)=0.\)
Choose the correct answer:
Q.98
Consider the following statements:
Statement I: The Euler–Lagrange equation corresponding to the functional \(J(u) = \int_\Omega (1 + |\nabla u(x)|^2)^{1/2} dx\) is a nonlinear partial differential equation where \(\Omega \subset \mathbb{R}^n\) is an open set and \(u:\mathbb{R}^n \to \mathbb{R}\) is a two times continuously differentiable function.
Statement II: There exists a solution to the Euler–Lagrange equation corresponding to the functional \(J(u) = \int_\Omega (1 + |\nabla u(x)|^2)^{1/2} dx\) where \(\Omega \subset \mathbb{R}^n\) is an open set and \(u:\mathbb{R}^n \to \mathbb{R}\) is a two times continuously differentiable function.
Choose the correct answer:
Q.99
Suppose \(g:\mathbb{R}\to \mathbb{R}\) is a continuously differentiable function such that \(g'(t)\geq 0\) for all \(t\in \mathbb{R}.\) Now if \(u,v:[a,b]\to \mathbb{R}\) are continuous functions such that
\[u(t) > 10 + \int_a^t g(u(s))ds\;\; and \] \[v(t) > 10 + \int_a^t g(v(s))ds\]
hold for any \(t\in [a,b]\). Now consider the following statements:
Statement I: \(u(t) < v(t)\) for all \(t\in [a,b].\)
Statement II: \(u(t) > v(t)\) for all \(t\in [a,b].\)
Statement III: \(u(t) = v(t)\) for all \(t\in [a,b].\)
How many of the statements given above are correct?
Q.100
Consider the following statements:
Statement I: \(\lambda=0\) is an eigenvalue of \[y'' + \lambda y = 0, y(0)=y(\pi)+ky'(\pi)=0\] if and only if \(k=-\pi.\)
Statement II: There are finitely many solutions to \[y' = 5y, y(0)=0\] but infinitely many solutions to \[y' = y^{1/4}, y(0)=0.\]
Statement III: There are exactly two solutions to \[y'' - y = 0, y(0)=0, y(\pi)=1.\]
How many of the statements given above are correct?
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