HPAS 2016 Mathematics Optional Paper-2: Complete Solutions
Welcome to the comprehensive solution guide for the Himachal Pradesh Administrative Service (HPAS) 2016 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): Limit of a Sum as an Integral
- Question 1(b): Derivative of a Complex Function
- Question 1(c): Inverse Laplace Transform
- Question 1(d): Centre of General Linear Group
- Question 1(e): Formation of PDE
- Question 2(a): Extremal of a Functional
- Question 2(b): Complex Contour Integral
- Question 3(a): Converse of Lagrange’s Theorem
- Question 3(b): Product of Non-normal Subgroups
- Question 4(a): Convergence of Series
- Question 4(b): Real Integral using Cauchy Residue Theorem
- Question 5(a): Bilinear Transformation
- Question 5(b): Riemann Integrability of Continuous Function
- Question 6(a): Fundamental Theorem of Calculus Counterexample
- Question 6(b): Metric Space (Bounded Metric)
- Question 7(a): Newton-Raphson Method
- Question 7(b): Cubic Polynomial Interpolation
- Question 8(a): Flowchart for Quadratic Roots
- Question 8(b): Complete Integral of PDE
HPAS 2016 Maths Optional Paper-2 Question 1(a)
Show that
\lim_{n\to\infty}\left\{\frac{1}{\sqrt{n^{2}+1}}+\frac{1}{\sqrt{n^{2}+2}}+\frac{1}{\sqrt{n^{2}+3}}+\dots+\frac{1}{\sqrt{n^{2}+n}}\right\}=1Solution:
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HPAS 2016 Maths Optional Paper-2 Question 1(b)
Show that :
f(z)=|z|^{2}=x^{2}+y^{2}
has a derivative at the origin.
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 1(c)
Determine the inverse Laplace transform of \frac{e^{-1/s}}{s}
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 1(d)
Let G be a group of all 2\times2 non-singular matrices with real entries. Determine the centre of G.
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 1(e)
Form a partial differential equation by eliminating the arbitrary function from the equation :
lx+my+nz=\phi(x^{2}+y^{2}+z^{2}).
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 2(a)
On which curve can the functional:
\int_{0}^{\pi/2}(y^{\prime2}-y^{2}+2xy)dywith y(0)=0 and y(\pi/2)=0 be extremized?
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 2(b)
Evaluate the integral :
\int_{C}\sqrt{z}dz \text{, where C: } z=z(t)=e^{it}, 0\le t\le2\piSolution:
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HPAS 2016 Maths Optional Paper-2 Question 3(a)
Show that the converse of Lagrange’s theorem holds in a finite cyclic group.
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 3(b)
Give an example of two subgroups H and K which are not normal, but HK is a subgroup.
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 4(a)
Find out whether the series:
1+\frac{x}{1!}+\frac{2^{2}x^{2}}{2!}+\frac{3^{3}x^{3}}{3!}+\dotsis convergent or divergent for x\in R^{+}?
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 4(b)
By using Cauchy Residue theorem, evaluate the integral:
\int_{0}^{\infty}\frac{x^{2}}{x^{6}+1}dxSolution:
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HPAS 2016 Maths Optional Paper-2 Question 5(a)
Determine the bilinear transformation that maps the points z=0, -i, 2i into the points w=5i, \infty, -i/3 respectively. What is the image of |z| < 1 under this transformation?
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 5(b)
If f:[a,b]\to R is continuous on [a, b] then show that the function is Riemann-integrable on [a, b].
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 6(a)
Show that the identity:
\int_{a}^{b}f'(x)dx=f(b)-f(a)is not always valid, with the help of an example.
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 6(b)
Let (X, d) be a metric space. Show that the function d^{*} defined by:
d^{*}(x,y)=\frac{d(x,y)}{1+d(x,y)}for all x, y\in X is a metric on X.
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 7(a)
By using the Newton-Raphson method, find a root of the equation:
x \sin x+\cos x=0.
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 7(b)
Determine the cubic polynomial which takes the following values :
y(0)=1, y(1)=0, y(2)=1, y(3)=10
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 8(a)
Draw a flowchart to find the roots of a quadratic equation ax^{2}+bx+c=0.
Solution:
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HPAS 2016 Maths Optional Paper-2 Question 8(b)
Find a complete integral of the partial differential equation (p + q) (px+ qy) = 1.
Solution:
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