HPAS 2018 Mathematics Optional Paper-2: Complete Solutions
Welcome to the comprehensive solution guide for the Himachal Pradesh Administrative Service (HPAS) 2018 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
- Question 1(a): Group of Bijections
- Question 1(b): Closure in Metric Space
- Question 1(c): Complex Exponentiation
- Question 1(d): Limsup and Liminf
- Question 2(a): Order of a Group
- Question 2(b): Group Homomorphism Image
- Question 3(a): Convergence of Improper Integral
- Question 3(b): Laplace Transform of Fractional Power
- Question 4(a): PDE for Right Circular Cones
- Question 4(b): Wave Equation Verification
- Question 5(a): Integral of Piecewise Function
- Question 5(b): Darboux Sums Inequality
- Question 6(a): Open Sphere in Euclidean Metric
- Question 6(b): Closed Sphere in Taxicab Metric
- Question 7(a): Series Convergence (Factorial)
- Question 7(b): Series Convergence (Root Test)
- Question 8(a): Trapezoidal Rule Algorithm
- Question 8(b): Euler’s Method Algorithm
HPAS 2018 Maths Optional Paper-2 Question 1(a)
Let X be any non-empty set and S(X) be the set of all bijections of X onto itself. Then prove that (S(X), \circ) is an abelian group if and only if X is a set with one or two elements, where \circ is the operation of composition of functions.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 1(b)
Let (X, d) be a metric space and let A \subseteq X. Then prove that \overline{A} = \{x \in X : d(x,A) = 0\}.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 1(c)
If X = \sqrt{-1}, then find the value of X^X.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 1(d)
Find the \limsup and \liminf of the sequence (-1)^n + \frac{1}{n}.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 2(a)
Let G be a group that has two subgroups of orders 45 and 75. If |G| < 400, find |G|.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 2(b)
Let f:G \to H be a group homomorphism with kernel K. If the orders of G, H, and K are 75, 45, and 15 respectively, find the order of the image f(G).
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 3(a)
Determine the values of s for which the improper integral \int_{0}^{\infty} e^{-sx} dx converges.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 3(b)
Find the value of L\{x^{7/2}\}.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 4(a)
Obtain the partial differential equation for the set of all right circular cones whose axes coincide with the z-axis.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 4(b)
Prove that any sufficiently differentiable function of the form F(x+kt) satisfies the wave equation F_{xx} = (1/k^2)F_{tt}.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 5(a)
Let
f(x) = \begin{cases} c, & 0 \le x \le c \\ 2c, & c < x \le 1 \end{cases}If \int_{0}^{1} f(x) dx = \frac{7}{16}, find the value of c.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 5(b)
Let f:[0,1] \to \mathbb{R} be a continuous function. For any partition P of [0, 1], let L(P,f) and U(P,f) denote the lower and upper Darboux sums respectively. Let P = \{0, 0.01, 0.02, \dots, 1\} and Q = \{0, 0.001, 0.002, \dots, 1\} be two partitions of [0, 1]. Then prove that L(P,f) \le L(Q,f) \le U(Q,f) \le U(P,f).
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 6(a)
Define an open sphere in a metric space. Describe with a figure the open sphere of unit radius centered at (0,0) for the metric d(z_1, z_2) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2} defined on \mathbb{R}^2, where z_1 = (x_1, y_1) and z_2 = (x_2, y_2) are any two points of \mathbb{R}^2.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 6(b)
Define a closed sphere in a metric space. Describe with a figure the closed sphere of unit radius centered at (0,0) for the metric d(z_1, z_2) = |x_1-x_2| + |y_1-y_2| defined on \mathbb{R}^2, where z_1 = (x_1, y_1) and z_2 = (x_2, y_2) are any two points of \mathbb{R}^2.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 7(a)
Test the convergence of the series
\sum_{n=1}^{\infty} \frac{1}{\sqrt{n!}}Solution:
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HPAS 2018 Maths Optional Paper-2 Question 7(b)
Test the convergence of the series
\sum_{n=1}^{\infty} \left(1+\frac{1}{n}\right)^{-n^2}Solution:
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HPAS 2018 Maths Optional Paper-2 Question 8(a)
Write an algorithm and draw a flow chart for integrating \int_{a}^{b} f(x) dx by the Trapezoidal rule, taking a step size h.
Solution:
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HPAS 2018 Maths Optional Paper-2 Question 8(b)
Write an algorithm and draw a flow chart for finding the value of y at x=x_n for the differential equation \frac{dy}{dx}=f(x,y), taking a step size h, when the initial values of x and y are given, by Euler’s method.
Solution:
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