HPAS 2018 Mathematics Optional Paper-1: Complete Solutions
Welcome to the comprehensive solution guide for the Himachal Pradesh Administrative Service (HPAS) 2018 Mathematics Optional Paper-1. 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 analytical geometry and calculus, 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): Order and degree of differential equation
- Question 1(b): System of linear equations
- Question 1(c): Intervals of increasing/decreasing function
- Question 1(d): Unit outward normal vector
- Question 2(a): General solution of ODE
- Question 2(b): Integrating factor of differential equation
- Question 3(a): Divergence of a vector field
- Question 3(b): Distance from point to plane
- Question 4(a): Determination of function and constants
- Question 4(b): Change of order of integration
- Question 5(a): Dimension of Kernel (Linear Mapping)
- Question 5(b): Rank of linear transformation
- Question 6(a): Determinant involving eigenvalues
- Question 6(b): Dimension of quotient space
- Question 7(a): Basis of inner product space
- Question 7(c): Orthonormal basis verification
- Question 8(a): Angular acceleration in plane motion
- Question 8(b): Line of action of resultant force
HPAS 2018 Maths Optional Paper-1 Question 1(a)
Find the order and degree of the differential equation whose general solution is y^2 = 2c(x+\sqrt{c}), where c is a positive parameter.
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 1(b)
How many solutions does the following system of linear equations have?
\begin{cases} -x+5y = -1 \\ x-y = 2 \\ x+3y = 3 \end{cases}Solution:
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HPAS 2018 Maths Optional Paper-1 Question 1(c)
Find the intervals in which the function f(x) = 10 - 6x - 2x^2 is strictly increasing or strictly decreasing.
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 1(d)
Find the unit outward normal vector at the point (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0) for the surface x^2+y^2+z^2=1.
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 2(a)
Two solutions of the ordinary differential equation y'' - 2y' + y = 0 are e^x and 5e^x. Is y = Ae^x + B(5e^x) the general solution of the differential equation?
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 2(b)
If the integrating factor of the differential equation (x^7y^2+3y)dx + (3x^8y-x)dy=0 is x^m y^n, then find the values of m and n.
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 3(a)
The direction of a vector A is radially outward from the origin, and its magnitude is |\vec{A}| = kr^n, where r^2 = x^2+y^2+z^2 and k is a constant. Find the value of n for which \nabla \cdot \vec{A} = 0.
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 3(b)
If P, Q, and R are three points having Cartesian coordinates (3, -2, -1), (1, 3, 4), and (2, 1, -2) respectively in the XYZ Cartesian plane, then find the distance from point P to the plane OQR, where O is the origin.
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 4(a)
Suppose a function f(x) satisfies the conditions: (i) f(0)=2, f(1)=1; (ii) f has a minimum value at x=5/2; and (iii) f'(x) = 2ax+b for all x. Determine the constants a, b and the function f(x).
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 4(b)
Change the order of integration in the integral \iint f(x,y)dxdy. The area of integration is enclosed by the curves y = x \tan\alpha, y = \sqrt{a^2-x^2}, x=0 and x = a\cos\alpha.
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 5(a)
Let U and V be vector spaces and T:U \to V be a surjective linear mapping. If \dim U=6 and \dim V=3, find \dim(\text{Ker } T).
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 5(b)
Let \mathbb{R}^3(\mathbb{R}) be a vector space with respect to ordinary addition and scalar multiplication. Find the rank of the linear transformation T: \mathbb{R}^3 \to \mathbb{R}^3 defined by T(x,y,z)=(y,0,z).
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 6(a)
Let A be a 3\times3 square matrix with eigenvalues 1, -1, and 0. Find the value of \det(I+A^{100}).
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 6(b)
Let \mathbb{R}^4(\mathbb{R}) be a vector space and let S be its subspace spanned by the vectors (1,2,3,0), (2,3,0,1), and (3,0,1,2). Find the dimension of the quotient space \mathbb{R}^4/S.
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 7(a)
In a finite-dimensional inner product space V, let \{w_1, w_2, \dots, w_n\} be an orthonormal subset of V such that
\sum_{i=1}^{n} |\langle w_i, v \rangle|^2 = ||v||^2for all v \in V. Find a basis of V.
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 7(c)
Let V=\mathbb{R}^2 be a finite-dimensional standard inner product space. Prove that \{(-1, 0), (0, -1)\} forms an orthonormal basis of V.
Solution:
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HPAS 2018 Maths Optional Paper-1 Question 8(a)
With usual notations, prove that the angular acceleration in the direction of motion of a point moving in a plane is
\frac{v}{\rho}\frac{dv}{ds} - \frac{v^2}{\rho^2}\frac{d\rho}{ds}Solution:
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HPAS 2018 Maths Optional Paper-1 Question 8(b)
Three forces P, Q, and R act along the sides of the triangle taken in order, formed by the lines x+y=1, y-x=1, and y=2. Find the equation of the line of action of their resultant.
Solution:
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