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HPAS 2021 Mathematics Optional Paper-2: Complete Solutions

Welcome to the comprehensive solution guide for the Himachal Pradesh Administrative Service (HPAS) 2021 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 2021 Maths Optional Paper-2 Question 1(a)

Show that every closed subspace of a complete metric space (X, d) is complete.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 1(b)

Find a transformation w = f(z) which maps the real axis of the z-plane onto the real axis in the w-plane.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 1(c)

Give an example of a finite abelian group which is not cyclic.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 1(d)

Show by an example that if functions f and g are not Riemann integrable, then their product f \cdot g can be Riemann integrable.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 1(e)

Determine the inverse Laplace transform of:

\log\frac{s+c}{s+d}

where c and d are constants.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 2(a)

Show that a finite group having more than two elements has a non-trivial automorphism.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 2(b)

Prove that every quotient group of a cyclic group is cyclic. Does the converse of this statement hold? Justify your answer with an example.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 2(c)

If f: [a,b] \to \mathbb{R} is a step function, then show that f is Riemann integrable.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 3(a)

Let X be the set of all continuous real-valued functions on [0, 1], and let:

d(x,y) = \int_{0}^{1} |x(t) - y(t)| dt

Show that the metric space (X, d) is not complete.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 3(b)

Using the concept of residue, determine the value of the integral:

\int_{0}^{\infty} \frac{\sin(mx)}{x} dx

when m > 0.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 4(a)

Test the series for convergence:

\sum_{n=1}^{\infty} (-1)^{n+1} \frac{\log n}{n}

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 4(b)

Show that the sequence (s_n) defined by:

s_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}

is divergent.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 4(c)

If the partial sums of the series \sum a_n are bounded, show that the series

\sum_{n=1}^{\infty} a_n e^{-nt}

converges for t > 0.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 5(a)

Solve the following partial differential equation using the Lagrange method:

p(z+e^x) + q(z+e^y) = z^2 - e^{x+y}

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

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 5(b)

Find the root of the equation x \sin x + \cos x = 0 by using the Newton-Raphson method.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 5(c)

Find the complete integral of the partial differential equation:

2\sqrt{p} + 3\sqrt{q} = 6x + 2y

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

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 6(a)

Solve the partial differential equation:

(D^2 + DD' - 6D'^2)z = y \cos x

where D = \frac{\partial}{\partial x} and D' = \frac{\partial}{\partial y}.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 6(b)

Solve the partial differential equation using Monge’s method:

r - t \sin^2 x - p \cot x = 0

where p = \frac{\partial z}{\partial x}, r = \frac{\partial^2 z}{\partial x^2}, and t = \frac{\partial^2 z}{\partial y^2}.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 7(a)

Determine the inverse Laplace transform of:

\frac{1}{s^2 - e^{-as}}

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 7(b)

Using the concept of Laplace Transform, find the solution of the initial value problem:

t\frac{d^2y}{dt^2} + 2t\frac{dy}{dt} + 2y = 2

with y(0) = 1, and y'(0) is arbitrary.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 7(c)

Test for an extremum of the functional:

I[y(x)] = \int_{0}^{1} (xy + y^2 - 2y^2 y') dx

with boundary conditions y(0) = 1, y(1) = 2, where y' = \frac{dy}{dx}.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 8(a)

Determine the maximum error in evaluating the integral:

\int_{0}^{\pi/2} \cos x \,dx

by both the Trapezoidal and Simpson’s rules using four subintervals.

Solution:

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HPAS 2021 Maths Optional Paper-2 Question 8(b)

Apply Euler’s modified method to find the value of y at x = 0.1 correct to five decimal places, given:

\frac{dy}{dx} = x^2 + y

with the initial condition y(0) = 0.94.

Solution:

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