HPAS 2016 Maths Optional Paper-2 Question 1(a)
Show that \[ \lim_{n\to\infty}\left\{\frac{1}{\sqrt{n^{2}+1}}+\frac{1}{\sqrt{n^{2}+2}}+\frac{1}{\sqrt{n^{2}+3}}+\dots+\frac{1}{\sqrt{n^{2}+n}}\right\}=1 \]For solution: Click here
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.For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 1(c)
Determine the inverse Laplace transform of \(\frac{e^{-1/s}}{s}\)For solution: Click here
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.For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 1(e)
Form a partial differential equation by eliminating the arbitrary function from the equation : \(lx+my+nz=\phi(x^{2}+y^{2}+z^{2})\).For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 2(a)
On which curve can the functional: \[ \int_{0}^{\pi/2}(y^{\prime2}-y^{2}+2xy)dy \] with \(y(0)=0\) and \(y(\pi/2)=0\) be extremized?For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 2(b)
Evaluate the integral : \[ \int_{C}\sqrt{z}dz \text{, where C: } z=z(t)=e^{it}, 0\le t\le2\pi \]For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 3(a)
Show that the converse of Lagrange’s theorem holds in a finite cyclic group.For solution: Click here
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.For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 4(a)
Find out whether the series: \[ 1+\frac{x}{1!}+\frac{2^{2}x^{2}}{2!}+\frac{3^{3}x^{3}}{3!}+\dots \] is convergent or divergent for \(x\in R^{+}\)?For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 4(b)
By using Cauchy Residue theorem, evaluate the integral: \[ \int_{0}^{\infty}\frac{x^{2}}{x^{6}+1}dx \]For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 5(a)
Determine the bilinear transformation that maps the points \(z=0\), -i, 2i into the points \(w=5i\), ∞, \(-i/3\) respectively. What is the image of \(|z|<1\) under this transformation ?For solution: Click here
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].For solution: Click here
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.For solution: Click here
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.For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 7(a)
By using the Newton-Raphson method, find a root of the equation: \(x \sin x+\cos x=0\).For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 7(b)
Determine the cubic polynomial which takes the following values : \(y(0)=1\), \(y(1)=0\), \(y(2)=1\), \(y(3)=10\)For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 8(a)
Draw a flowchart to find the roots of a quadratic equation \(ax^{2}+bx+c=0\).For solution: Click here
HPAS 2016 Maths Optional Paper-2 Question 8(b)
Find a complete integral of the partial differential equation \((p + q) (px+ qy) = 1\).For solution: Click here