HPAS 2014 Maths Optional Paper-2 Question 1(a)
If
A=\begin{pmatrix}1&2&3&4&5\\ 2&3&1&5&4\end{pmatrix}is a permutation on five symbols, then find A^{3} and order of A.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 1(b)
Test the convergence of the integral :
\int_{0}^{\infty}e^{-x^{2}}dxSolution:
HPAS 2014 Maths Optional Paper-2 Question 1(c)
Prove that every Cauchy sequence is bounded.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 1(d)
Prove that the set of real numbers R and the function defined as follows :
d:d(x,y)=|x-y| \quad \forall x,y\in R
form a metric space.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 1(e)
Solve the following partial differential equation:
p^{2}+q^{2}-2px-2qy+1=0
Solution:
HPAS 2014 Maths Optional Paper-2 Question 1(f)
Draw a flow chart to print all even numbers between 1 and 50.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 2(a)
State and prove Cayley’s Theorem.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 2(b)
Let I be the additive group of Integers. Let H be the subgroup of I such that :
H=\{mx:x\in I\} ,
where m is a fixed integer. Write the element of the quotient group \frac{I}{H}.
Also prepare a composition table for \frac{I}{H} when m=5.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 3(a)
Let f be bounded on [a, b]. Then prove that f is R-integrable over [a, b] iff given \epsilon > 0 there exists a partition P of [a, b] such that :
0\le U(f,P)-L(f,P) < \epsilon
Solution:
HPAS 2014 Maths Optional Paper-2 Question 3(b)
For the functions:
f(x)=x , g(x)=e^{x} ,
then verify the second mean value theorem in the interval [-1, 1].
Solution:
HPAS 2014 Maths Optional Paper-2 Question 4(a)
Find whether the following series is convergent or divergent :
x^{2}+\frac{2^{2}\cdot x^{4}}{3 \cdot 4}+\frac{2^{2}\cdot 4^{2}}{3 \cdot 4 \cdot 5 \cdot 6}\cdot x^{6}+\frac{2^{2}\cdot 4^{2}\cdot 6^{2}}{3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}\cdot x^{8}+\dots
Solution:
HPAS 2014 Maths Optional Paper-2 Question 4(b)
Prove that the metric space (R, d) is complete, where d is the usual metric for the set of real numbers R.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 5(a)
If f(z)=u+iv is an analytic function of z=x+iy and
u-v=e^{x}(\cos y-\sin y)
find f(z) in terms of z.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 5(b)
If w=f(z) represents a conformal transformation of a domain D in the z-plane into a domain D’ of the w-plane, then prove that f(z) is an analytic function of z in D.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 6(a)
Solve :
pxy+pq+qy=yz
Solution:
HPAS 2014 Maths Optional Paper-2 Question 6(b)
Solve by Monge’s method :
pq=x(ps-qr)
Solution:
HPAS 2014 Maths Optional Paper-2 Question 7(a)
If
L_{n}(x)=\frac{e^{x}}{n!}\cdot\frac{d^{n}}{dx^{n}}(e^{-x}\cdot x^{n})
then find Laplace transform L\{L_{n}(x); p\}, p > 1.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 7(b)
Find the extremal curve of the functional :
I[y(x),z(x)]=\int_{0}^{\pi/2}\{(y^{\prime})^{2}+(z^{\prime})^{2}+2yz\}dx
given that :
y(0)=0 , y(\pi/2)=-1; z(0)=0, z(\pi/2)=1.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 8(a)
Find the roots of the quadratic equation :
x^{2}-5x+2=0
correct to four decimal places by the Newton-Raphson method.
Solution:
HPAS 2014 Maths Optional Paper-2 Question 8(b)
Evaluate :
\int_{0}^{1}\frac{dx}{1+x^{2}}
using Simpson’s \frac{1}{3} and \frac{3}{8} rule. Hence obtain the approximate value of \pi in each case.