hpas 2019 MATHS p2

by

HPAS 2019 Mathematics Optional Paper-2: Complete Solutions

Welcome to the comprehensive solution guide for the Himachal Pradesh Administrative Service (HPAS) 2019 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 2019 Maths Optional Paper-2 Question 1(a)

Discuss the convergence of the series:

1 + \frac{x}{2} + \frac{2!}{3}x^2 + \frac{3!}{4}x^3 + \dots

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 1(b)

Examine if the function u = e^{2xy}\sin(x^2 - y^2) is harmonic. Find the complex function f(z) in terms of z, where f(z) = u+iv.

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 1(c)

Obtain the Laplace transform of the function f(t) = e^{-3t}u(t-2).

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 1(d)

Obtain the smallest positive root of the equation x \log_{10}x - 1.2 = 0, correct to 2 decimal places, using the Newton-Raphson method.

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 2(a)

Find the transformation which maps the semi-infinite strip of width \pi, bounded by the lines v=0, v=\pi, and u=0, into the upper half of the z-plane.

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 2(b)

Evaluate

\int_C \frac{3z^2+z}{z^2-1} dz

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

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 3(a)

Show that a cyclic group is necessarily abelian. Show by an example that the converse may not be true.

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 3(b)

Examine if there exists a one-to-one correspondence between the right and left cosets of H in G, if H is any subgroup of G.

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 3(c)

If f:G \to G' is a homomorphism, then prove that Im(f) is a subgroup of G'.

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 4(a)

Define a compact metric space. Prove that a closed subset of a compact space is compact.

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 4(b)

Show that the integral

\int_{a}^{\infty} \frac{x}{1+x^4\sin^2x} dx

is divergent.

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 5(a)

Solve the following system of equations by the Gauss-Seidel method, correct to 2 decimal places:

\begin{cases} x_1 + 6x_2 + 2x_3 = 6 \\ 5x_1 + x_2 - x_3 = 12 \\ 3x_1 - 2x_2 + 8x_3 = -4 \end{cases}

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 5(b)

From the following table, obtain f(2.07) using the best formula:

x:     2.00,     2.05,     2.10,     2.15,     2.20,     2.25

f(x): 0.69315, 0.71784, 0.74194, 0.76547, 0.78846, 0.81093

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 6(a)

Obtain the inverse Laplace Transform of f(s) = \log\frac{s+1}{s-1}.

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 6(b)

Find the Laplace transform of the following periodic function f(t) with period 2c:

f(t) = \begin{cases} t & , \ 0 < t < c \\ 2c - t & , \ c < t < 2c \end{cases}

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 6(c)

Solve the following Initial Value Problem using the Laplace transform:

x'' + 2x' + 5x = e^{-t}\sin t

with initial conditions x(0)=0 and x'(0)=1.

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 7(a)

Find the series solution of the following differential equation:

(1+x^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx} - y = 0

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 7(b)

Solve the partial differential equation

(x^2 - y^2 - z^2)p + 2xyq = 2xz

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

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 8(a)

What do the following explain in the C Language:
(i) ++nc
(ii) else if (condition)
(iii) tolower(c)
(iv) Pointer

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 8(b)

Using the Euler equation, find the extremal of the following functional:

\int_{a}^{b} (12xy(x) + (y')^2) dx

Solution:

Enter solution here

HPAS 2019 Maths Optional Paper-2 Question 8(c)

Let f be a continuous function. Then show that f is Riemann integrable.

Solution:

Enter solution here

Leave a Comment