hpas 2023 MATHS p2

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HPAS 2023 Mathematics Optional Paper-2: Complete Solutions

Welcome to the comprehensive solution guide for the Himachal Pradesh Administrative Service (HPAS) 2023 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 2023 Maths Optional Paper-2 Question 1(a)

Suppose G is a finite group of order pq, where p and q are prime numbers such that p > q. Show that G has at most one subgroup of order p.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 1(b)

If f is a Riemann integrable function on the interval [a, b], then show that f^2 is also a Riemann integrable function.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 1(c)

Find the equation whose roots are 2\cos\frac{\pi}{7}, 2\cos\frac{3\pi}{7}, and 2\cos\frac{5\pi}{7}.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 1(d)

Determine the inverse Laplace Transform of the function:

\tan^{-1}\left(\frac{2}{s^2}\right)

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 1(e)

Show that every closed sphere is a closed set.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 2(a)

Show that every finitely generated subgroup of \langle \mathbb{Q}, +\rangle is cyclic, where \mathbb{Q} is the set of rational numbers.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 2(b)

Show that a subgroup of an infinite cyclic group is infinite.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 2(c)

Give an example of an infinite group in which every element is of finite order. Justify your answer.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 3(a)

Show that the function

f(x) = \begin{cases} x & \text{when x is rational} \\ -x & \text{when x is not rational} \end{cases}

is not Riemann integrable in the interval [a, b], but |f| is Riemann integrable.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 3(b)

For what values of m and n is the integral

\int_{0}^{1} x^{m-1}(1-x)^{n-1}\log x \,dx

convergent?

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 4(a)

Let \langle a_n \rangle be a sequence such that \lim_{n\to\infty} a_n = l. Then show that

\lim_{n\to\infty} \frac{a_1+a_2+a_3+\dots+a_n}{n} = l

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 4(b)

Show that every compact subset F of a metric space (X, d) is closed.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 4(c)

Let (X, d) be a metric space. Then show that any disjoint pair of closed sets in X can be separated by disjoint open sets in X.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 5(a)

Determine the analytic function f(z) = u+iv if

u-v = \frac{\cos x + \sin x - e^{-y}}{2(\cos x - \cosh y)}

and f\left(\frac{\pi}{2}\right) = 0.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 5(b)

Show that the transformation w = z + \frac{1}{z} converts the straight line \text{arg}(z) = \alpha (where |\alpha| < \frac{\pi}{2}) into a branch of a hyperbola with eccentricity \sec\alpha.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 6(a)

Solve the partial differential equation

\left(\frac{y-z}{yz}\right)p + \left(\frac{z-x}{zx}\right)q = \frac{x-y}{xy}

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

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 6(b)

Using Charpit’s method, find the solution of the partial differential equation p^2x + q^2y = z, where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 6(c)

Solve the partial differential equation (D^2 - DD' + D' - 1)z = \cos(x+2y) + e^y, where D = \frac{\partial}{\partial x} and D' = \frac{\partial}{\partial y}.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 7(a)

On which curve can the functional

\int_{0}^{\pi/2} (y'^2 - y^2 + 2xy) dx

with boundary conditions y(0)=0 and y(\frac{\pi}{2})=0, be extremized? (where y' = \frac{dy}{dx})

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 7(b)

By applying Gauss’s quadrature formula, compute the integral

\int_{5}^{12} \frac{1}{x} dx

Also find the error.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 7(c)

Show that the rate of convergence of the Newton-Raphson method is quadratic, and determine a root of the equation x^{10}-1=0 with the initial point x_0 = 0.5.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 8(a)

Let the polynomial \phi(x) be of the form \phi(x) = \sum_{i=0}^{n} L_i(x)y_i, where each Lagrangian function L_i(x) is a polynomial in x of degree less than or equal to n. Then show that \sum_{i=0}^{n} L_i(x) = 1.

Solution:

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HPAS 2023 Maths Optional Paper-2 Question 8(b)

Using the 4th order Runge-Kutta method, solve the differential equation \frac{dy}{dx} = -xy^2 with y(0)=1. Taking a step size of h=0.2, determine y(0.4).

Solution:

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