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HPAS 2017 Mathematics Optional Paper-2: Complete Solutions

Welcome to the comprehensive solution guide for the Himachal Pradesh Administrative Service (HPAS) 2017 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 2017 Maths Optional Paper-2 Question 1(a)

If a \ne e is the only element of order 2 in a group G, then prove that ax = xa for all x \in G.

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 1(b)

Let p_1 and p_2 be two partitions of [a, b]. Show that L(f, p_1) \le U(f, p_2) and L(f, p_2) \le U(f, p_1).

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 1(c)

Evaluate the Laplace transform of the function f(x) = (x+2)^2 e^x.

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 1(d)

On which curves can the functional

H[y(x)] = \int_{0}^{1} [(y')^2 + 12xy] dx

with boundary conditions y(0)=0 and y(1)=1, be extremized?

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 2(a)

Prove that a finite group of prime order does not have any proper subgroup.

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 2(b)

Prove that the kernel of a homomorphism f from a group G to a group G’ is a normal subgroup of G.

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 3(a)

Let a real-valued function f be defined on [0, 1] by f(x)=x for all x \in [0, 1]. Show that f is Riemann-integrable over [0, 1] and that \int_{0}^{1} f(x) dx = \frac{1}{2}.

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 3(b)

Test the convergence of the following integral:

\int_{0}^{\infty} e^{-ax} \frac{\sin x}{x} dx \quad (a \ge 0)

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 4(a)

If (A, d) is a metric space, then show that:

|d(x_1, y_1) - d(x_2, y_2)| \le d(x_1, x_2) + d(y_1, y_2)

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 4(b)

Test the convergence of the series:

x^2(\log 2)^q + x^3(\log 3)^q + x^4(\log 4)^q + \dots

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 5(a)

Prove that the function u(x,y) = x^3 - 3xy^2 is harmonic and obtain its conjugate.

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 5(b)

Evaluate:

\int_C \frac{e^{3z}}{z-\pi i} dz

where C is the circle |z-1|=4.

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 6(a)

Solve the following equation by Charpit’s method:
(p^2+q^2)y = qz

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 6(b)

Solve:
x^2r + 2xys + y^2t = 0

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 7(a)

Show that:

L\left\{ \int_{0}^{x} \frac{1-e^{-u}}{u} du \right\}(p) = \frac{1}{p}\log\left(1+\frac{1}{p}\right)

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 7(b)

Apply the convolution theorem to find:

L^{-1}\left\{ \frac{p^2}{(p^2+a^2)^2} \right\}(x)

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 8(a)

Find the path on which a particle, in the absence of friction, will slide from one fixed point to another point not in the same vertical line in the shortest time under the action of gravity.

Solution:

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HPAS 2017 Maths Optional Paper-2 Question 8(b)

Find a root of the equation x \log_{10}x - 1.2 = 0 correct to four places of decimals.

Solution:

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