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HPAS 2024 Mathematics Optional Paper-2: Complete Solutions

Welcome to the comprehensive solution guide for the Himachal Pradesh Administrative Service (HPAS) 2024 Mathematics Optional Paper-2. This resource provides detailed, step-by-step solutions designed specifically for civil service aspirants to master the core mathematical concepts and methodologies required for the exam.

Whether you are revising key theorems, practicing previous year questions, or mastering advanced abstract algebra, real analysis, and differential equations, these carefully structured solutions will help streamline your preparation. Use the index below to jump directly to specific questions and topics.

Table of Contents

HPAS 2024 Maths Optional Paper-2 Question 1(a)

If the sequences \langle a_n \rangle and \langle b_n \rangle converge to finite limits a and b, respectively, then show that

\lim_{n\to\infty} \frac{a_1b_n + a_2b_{n-1} + \dots + a_nb_1}{n} = ab

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 1(b)

Evaluate the contour integral

\oint_{C} \frac{e^{3z}}{(z-\log 2)^4} dz

Where C is the square with vertices at \pm 1, \pm i.

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 1(c)

Find the Laplace transform of:

\frac{1}{t} \int_{0}^{t} e^{u} \sin u \,du

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 1(d)

Show that the Newton-Raphson process has a quadratic convergence.

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 2(a)

Let G be a group. Let Aut(G) denote the set of all automorphisms of G and let A(G) be the group of all permutations of G. Show that Aut(G) is a subgroup of A(G).

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 2(b)

Let \mathcal{R}[a, b] be the set of all Riemann integrable functions on the interval [a, b]. If f:[a,b] \to \mathbb{R} is a monotone function on [a, b], then show that f \in \mathcal{R}[a, b].

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 3(a)

If \langle S_n \rangle is a sequence of positive real numbers such that S_n = \frac{1}{2}(S_{n-1} + S_{n-2}) for all n > 2, then show that \langle S_n \rangle converges and find \lim_{n\to\infty} S_n.

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 3(b)

Let R_\infty be the extended set of real numbers. The function d is defined by d(x,y) = |f(x) - f(y)| for all x, y \in R_\infty, where f(x) = \frac{x}{1+|x|} when -\infty < x < \infty, f(x)=1 when x=\infty, and f(x)=-1 when x=-\infty. Show that (R_\infty, d) is a bounded metric space.

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 4(a)

Let (X, d) be a complete metric space and Y be a subspace of X. Show that Y is complete if and only if it is closed in (X, d).

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 4(b)

Let G be a region and suppose that f:G \to \mathbb{C} is analytic such that f(G) is a subset of a circle. Then show that f is constant.

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 5(a)

Solve the partial differential equation:

(x^2 - yz)p + (y^2 - zx)q = z^2 - xy

where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 5(b)

Using Monge’s method, solve the wave equation r = a^2t (for a > 0), where r = \frac{\partial^2 z}{\partial x^2} and t = \frac{\partial^2 z}{\partial y^2}.

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 6(a)

Is the Jacobi condition fulfilled for the extremal of the functional

\int_{0}^{a} (y'^2 - 4y^2 - e^{-x^2}) dx, \quad a \ne \frac{n\pi}{2}

with fixed boundaries A(0,0) and B(a,0)? (where y' = dy/dx)

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 6(b)

Using Euler’s method, solve the initial value problem \frac{dy}{dt} = 1 - t + 4y with y(0)=1, in the interval 0 \le t \le 0.5 with h=0.1. If the exact solution is y = -\frac{9}{16} + \frac{1}{4}t + \frac{19}{16}e^{4t}, then compute the error and the percentage error.

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 7(a)

If C(t) = \int_{t}^{\infty} \frac{\cos x}{x} dx , show that the Laplace transform of C(t) is \log\frac{\sqrt{s^2+1}}{s} .

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 7(b)

Write the algorithm for the Bisection method for finding a real root of the equation f(x)=0 which lies in the interval [a, b]. Further, develop a simple program in C language for finding a real root of the equation x^3 - 2x - 1 = 0 using the Bisection method.

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 8(a)

Let C be the unit circle z = e^{i\theta}, for -\pi \le \theta \le \pi. First, show that for any real constant a,

\int_{C} \frac{e^{az}}{z} dz = 2\pi i

Then, write this integral in terms of \theta to derive the integration formula:

\int_{0}^{\pi} e^{a \cos\theta} \cos(a \sin\theta) d\theta = \pi

Solution:

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HPAS 2024 Maths Optional Paper-2 Question 8(b)

Let f and g be integrable functions on [a, b]. Then, show that:
(i) f \cdot g is integrable on [a, b].
(ii) \max(f,g) and \min(f,g) are integrable on [a, b].

Solution:

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