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HPAS 2020 Mathematics Optional Paper-2: Complete Solutions

Welcome to the comprehensive solution guide for the Himachal Pradesh Administrative Service (HPAS) 2020 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 2020 Maths Optional Paper-2 Question 1(a)

Show that

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

is a conditionally convergent series.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 1(b)

Determine all group homomorphisms from S_3 to \mathbb{Z}_3.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 1(c)

Solve:
x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = -xy

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 1(d)

Find the value of y(0.5) for the initial value problem \frac{dy}{dx}=y, y(0)=1, using Euler’s method with a step size of h=0.1.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 2(a)

Prove that any subgroup of a cyclic group is cyclic.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 2(b)

Define the order of an element in a group. Let G be a finite group of even order. Show that G has an element of order 2 and that the number of elements of order 2 in G is odd.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 2(c)

State Abel’s test for convergence of improper integrals. Test the convergence of

\int_{0}^{\infty} \frac{\sin x}{\log(x+2)} dx

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 3(a)

State the logarithmic test for convergence of a series. If \{a_n\} is a sequence of real numbers such that \{n^2a_n\} is a convergent sequence, show that \sum_{n=1}^{\infty} a_n is an absolutely convergent series.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 3(b)

State the second mean value theorem of integral calculus. Let f:[0,1] \to \mathbb{R} be a continuous function such that \int_{0}^{1} |f(x)| dx = 0. Show that f is identically zero on [0,1].

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 3(c)

Let f(z) = \frac{z-1}{z+1} and let L be the line in the z-plane through z=0 and z=1+i. Find the image of L under f.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 4(a)

Find the bilinear transformation which maps the points 0, i, 1+i onto 2i, -1, and 0, respectively.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 4(b)

Prove that every bounded entire function is constant. Is f(z) = \sin z bounded on \mathbb{C}? Support your answer.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 4(c)

Define a connected metric space. Prove that every closed interval of the real line \mathbb{R} is connected.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 5(a)

Define a continuous function between metric spaces. Prove that any real-valued continuous function on a compact metric space is bounded.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 5(b)

Obtain the interpolating polynomial by Newton’s divided difference formula for the following data. Also, find the value of f(3.5).

x:     -3,   -1,   0,   3,       5

f(x): -30, -22, -12, 330, 3458

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 6(a)

Find the shortest distance between the parabola y=x^2 and the straight line x-y=5.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 6(b)

Solve the following initial value problem using the Laplace transform:

\frac{d^2y}{dx^2} - 3\frac{dy}{dx} - 4y = x^2

with y(0)=2 and y'(0)=1.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 6(c)

Find the Laplace transform of the following periodic function f(x) with period 2\pi:

f(x) = \begin{cases} x & , \ 0 \le x \le \pi \\ 2\pi - x & , \ \pi \le x \le 2\pi \end{cases}

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 7(a)

Find the general solution of the following partial differential equation:

D(D^2 - D' + 1)(D + 2D' + 1)^3 z = 0

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

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 7(b)

Apply Charpit’s method to find the complete integral of the partial differential equation xp^2 + yq^2 = z, where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 7(c)

Find a real root of the equation xe^x - \cos x = 0 using the Regula-Falsi method.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 8(a)

Evaluate the integral

I = \int_{0}^{1} \frac{dx}{1+x}

correct to three decimal places by using the trapezoidal rule and Simpson’s one-third rule, taking h=0.25.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 8(b)

Find the extremal of the following functional

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

that satisfies the conditions y(0)=0, y'(0)=1, y(1)=1, and y'(1)=1.

Solution:

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HPAS 2020 Maths Optional Paper-2 Question 8(c)

Write a flow chart in C language for the Runge-Kutta method of the fourth order.

Solution:

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