HPAS 2014 Maths Optional Paper-2 Question 1(a)
If \[ A=(\begin{matrix}1&2&3&4&5\\ 2&3&1&5&4\end{matrix}) \] is a permutation on five symbols, then find \(A^{3}\) and order of A.For solution: Click here
HPAS 2014 Maths Optional Paper-2 Question 1(b)
Test the convergence of the integral : \[ \int_{0}^{\infty}e^{-x^{2}}dx \]For solution: Click here
HPAS 2014 Maths Optional Paper-2 Question 1(c)
Prove that every Cauchy sequence is bounded.For solution: Click here
HPAS 2014 Maths Optional Paper-2 Question 1(d)
Prove that the set of real numbers R and the function defined as follows : \( d:d(x,y)=|x-y|\forall x,y\in R \) form a metric space.For solution: Click here
HPAS 2014 Maths Optional Paper-2 Question 1(e)
Solve the following partial differential equation: \( p^{2}+q^{2}-2px-2qy+1=0 \)For solution: Click here
HPAS 2014 Maths Optional Paper-2 Question 1(f)
Draw a flow chart to print all even numbers between 1 and 50.For solution: Click here
HPAS 2014 Maths Optional Paper-2 Question 2(a)
State and prove Cayley’s Theorem.For solution: Click here
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 \(m=5\).For solution: Click here
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 : \( 0\le U(f,P)-L(f,P)<\epsilon \).For solution: Click here
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].For solution: Click here
HPAS 2014 Maths Optional Paper-2 Question 4(a)
Find whether the following series is convergent or divergent : \( x^{2}+\frac{2^{2}.x^{4}}{3.4}+\frac{2^{2}.4^{2}}{3.4.5.6}.x^{6}+\frac{2^{2}.4^{2}.6^{2}}{3.4.5.6.7.8}.x^{8}+… \)For solution: Click here
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.For solution: Click here
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.For solution: Click here
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.For solution: Click here
HPAS 2014 Maths Optional Paper-2 Question 6(a)
Solve : \( pxy+pq+qy=yz \).For solution: Click here
HPAS 2014 Maths Optional Paper-2 Question 6(b)
Solve by Monge’s method : \( pq=x(ps-qr) \).For solution: Click here
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 \(L\{L_{n}(x); p\}\), \(p>1\).For solution: Click here
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 : \( y(0)=0 \), \(y(\frac{\pi}{2})=-1 \); \(z(0)=0 \), \(z(\frac{\pi}{2})=1 \).For solution: Click here
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.For solution: Click here
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.For solution: Click here