The HPSC Assistant Professor Mathematics Exam 2025 held on 7 December 2025 included a detailed subjective paper that focused on testing deep conceptual understanding rather than direct formula-based problem solving. The questions covered a wide range of advanced mathematical topics such as Real Analysis, Partial Differential Equations, Calculus of Variations, Group Theory, Linear Algebra, Probability Theory, and Functional Analysis. Many problems required multi-step reasoning, rigorous proofs, and clear mathematical explanations, reflecting the standard expected at the assistant professor level. Tasks like solving boundary-value problems, finding extremals of functionals, analyzing algebraic structures, and computing probability distributions highlighted the exam’s emphasis on theoretical depth and analytical skill. Aspirants needed strong problem-solving ability, practice with formal proofs, and confidence in applying core methods across topics. This post reproduces the questions as asked in the exam to help future candidates understand the pattern, difficulty, and important themes for focused preparation.
Disclaimer: All the questions mentioned in this post belong to the Haryana Public Service Commission (HPSC). They are reproduced here strictly for educational and informational purposes only.
1). Let \( f \) be a real-valued continuous function on \([0,\infty)\) such that \[ \lim_{x\to\infty} \left[ f(x) + \int_0^x f(t)\,dt \right] \text{ exists.} \] Prove that \[ \lim_{x\to\infty} f(x) = 0. \]
2).
Let \( f \) be a real-valued continuous function on \(\mathbb{R}\) satisfying
\[
|f(x)| \le \frac{c}{1+x^2},
\]
where \( c \) is a positive constant. Define \(F : \mathbb{R} \to \mathbb{R}\) by
\[
F(x) = \sum_{n=-\infty}^{\infty} f(x+n).
\]
(a) Prove that \(F\) is continuous and periodic with period 1.
(b) Show that if \(G\) is continuous and periodic with period 1, then
\[
\int_0^1 F(x)G(x)\,dx = \int_{-\infty}^{\infty} f(x)G(x)\,dx.
\]
3).
Let \( f:\mathbb{R}^n \setminus \{0\} \to \mathbb{R} \) be differentiable functio. Suppose
\[
\lim_{x\to0} \frac{\partial f}{\partial x_j}(x)
\]
exists for each \( j=1,\dots,n \).
(a) Can \( f \) be extended to a continuous function from \(\mathbb{R}^n\) to \(\mathbb{R}\)?
(b) Assuming the continuity of \(f\) at the origin, is \(f\) differentiable from \(\mathbb{R}^n\) to \(\mathbb{R}\) ?
4).
Let \( T \) be a linear operator on a finite-dimensional vector space \(V\) over the field of real numbers and
\[
f(x)=x^4 - 9x^3 + 27x^2 - 31x + 12
\]
be an annihilating polynomial for \(T\).
Then Prove or disprove that
\(
V=\)
\[ \ker(T^2 - 2T + I)\]
\[\oplus\]
\[\ker(T^2 - 7T + 12I).
\]
5).
Let \( w_1,\dots,w_n \) be linearly independent vectors in \(\mathbb{C}^n\) with standered inner product \(\langle \rangle\),and Let \(v_1,\dots,v_n\) be the orthogonal vectors obtained from \( w_1,\dots,w_n \) by the Gram–Schmidt Orthogonalization process.
Let \(u_1,\dots,u_n\) be the orthonormal basis obtained by normalizing the \(v_{i}\)'s.
(a) Show
\[
w_k = \|v_k\| u_k + \sum_{i=1}^{k-1} \langle w_k, u_i\rangle u_i.\]
\[
\;\; \text{for all}\; 1 \leq k \leq n\]
(b) Prove that every non- singular matrix can be written as a product of a unitry matrix and an upper triangular matrix. This decomposition is called QR- decomposition of any non- singular matrix. Is this decomposition is unique?.
6). Let \(a\) be complex number with \(|a|\le1\). Consider the polynomial \[ P(z)=\frac{a}{2} + (1-|a|^2)z - \frac{\overline{a}}{2} z^2. \] Show that \(|P(z)| \le 1\) when \(|z|\le1\).
7). Let \(G\) be an abelian group and \(H\)be the subset of \(G\) contaning all the finite-order elements. Then show that \(H\) is a subgroup of \(G\). What happens when \(G\) is non-abelian? Justify your answer.
8). Compute the group of units of the ring \[ \mathbb{Z}\left[ \frac{1+\sqrt{-3}}{2} \right]. \]
9). Let \(M_{n\times n}(\mathbb{R})\) be the vector space of real \(n \times n\) matrices, identified with \(\mathbb{R}^{n^2}\) with usal topology on \(\mathbb{R}^{n^2}\) . Let \(X \subset M_{n\times n}(\mathbb{R})\) be compact set. Let \(S\subset C\) be the set of all complex numbers that are eigenvalues of at least one element of \(X\). Is \(S\) compact? Justify your answer.
10). Let \(SO(3)\) be the group of \(3\times3\) orthogonal matrices with determinant 1 over \(\mathbb{R}\). Is it compact under the subspace topology of \(\mathbb{R}^9\)? Justify your answer.
11).
Consider the following initial value problem (IVP)
\[
\begin{cases}
\dfrac{\partial u}{\partial t} - \Delta u = 0, \; (x,t)\in \mathbb{R}^2 \times (0,\infty) \\
u(x,0) = g(x),\; x\in \mathbb{R}^2
\end{cases}
\]
where \( g : \mathbb{R}^2 \to \mathbb{R} \) is a continuous, bounded function and
\[
\int_{\mathbb{R}^2} |g(x)|\,dx < \infty.
\]
Find an expression for solution to the above IVP and show that
\[
\lim_{t\to\infty} u(x,t)=0,
\]
uniformly for all \( x\in \mathbb{R}^2 \).
12). Find the extremals of \[ J(y,z)=\int_0^{\pi/2} (9y'^2 + 16z'^2 + 24yz)\,dx \] satisfying the boundary conditions \[ y(0)=z(0)=0,\] \[\; y(\pi/2)=z(\pi/2)=1. \]
13). Solve the system using Gaussian elimination: \[ x_1 + x_2 + x_3 = 6 \] \[ 3x_1 + (3+\varepsilon)x_2 + 4x_3 = 20 \] \[ 2x_1 + x_2 + 3x_3 = 13, \] with small \(\varepsilon\) such that \(1 \pm \varepsilon^2 \approx 1\).
14).
Let \(U_1, U_2, \ldots\) be independent and identically distributed continuous uniform random variables over \([0,1]\) and \(X_i = -\log_e U_i\).
(a) What is \(P(X_i > x)\)?
(b) What kind of random variable is \(X_i\)?
(c) Given a constant \(t>0\), let \(N\) denote the value of \(n\), such that
\[
\prod_{i=1}^{n} U_i > e^{-t} > \prod_{i=1}^{n+1} U_i .
\]
What is the probability mass function of \(N\)?
15).
The jointly continuous random variables \(X\) and \(Y\) have joint pdf
\(f_{X,Y}(x,y) = \begin{cases}
e^{-x-y}, & \text{if } 0 < x < \infty,\ 0 < y < \infty, \\
0, & \text{otherwise}.
\end{cases}\)
Determine the probability density function (PDF) and the variance of the random
variable \(Z = X + Y\).
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