Paper II PYQs-2014

1.(a) Let \(G\) be the set of all real \(2 \times 2 \left( \begin{array}{cc}{x} & {y} \\ {0} & {z}\end{array}\right)\), where \(x z \neq 0\) matrices. Show that \(G\) is group under matrix multiplication. Let \(N\) denote the subset \(\left\{\left[ \begin{array}{cc}{1} & {a} \\ {0} & {1}\end{array}\right]: a \in R\right\}\). Is \(N\) a normal subgroup of \(G\)? Justify your answer.

[10M]

1.(b) Test the convergence of the improper integral \(\int_{1}^{\infty} \dfrac{d x}{x^{2}\left(1+e^{-x}\right)}\).

[10M]

1.(c) Prove that the function \(f(z)=u+i v\), where \(f(z)=\dfrac{x^{3}(1+i)-y^{3}(1-i)}{x^{2}+y^{2}}\), \(z \neq 0\); \(f(0)=0\) satisfies Cauchy-Riemann equations at the origin, but the derivative of \(f\) at \(z=0\) does not exist.

[10M]

1.(d) Expand in Laurent series the function \(f(z)=\dfrac{1}{z^{2}(z-1)}\) about \(z=0\) and \(z=1\).

[10M]

1.(e) Solve graphically:

Maximize
\(Z=6 x_{1}+5 x_{2}\),
subject to:
\(2 x_{1}+ x_{2} \leq 216\)
\(x_{1}+x_{2} \leq 11\)
\(x_{1} + 2x_{2} \geq 6\) \(5x_{1} + 6x_{2} \leq 90\)
\(x_{1}\), \(x_{2} \geq 0\)

[10M]

2.(a) Show that \(Z_{7}\) is a field. Then find \(([5]+[6])^{-1}\) and \((-[4])^{-1}\) in \(Z_{7}\).

[15M]

2.(b) Integrate \(\int_{1}^{0} f(x) d x\), where \(f(x)=\left\{\begin{array}{ll}{2 x \sin \dfrac{1}{x} - \cos \dfrac{1}{x}, x} & {\in[0,1]} \\ {0,} & {x=0}\end{array}\right.\)

[15M]

2.(c) Find the initial basic feasible solution to the following transportation problem by Vogel’s approximation method. Also, find its optimal solution and the minimum transportation cost.

\[\begin{array}{|c|c|c|c|c|}\hline & {D_{1}} & {D_{2}} & {D_{3}} & {D_{4}} & {\text { Supply }} \\ \hline O_{1} & {6} & {4} & {1} & {5} & {14} \\ \hline O_{2} & {8} & {9} & {2} & {7} & {16} \\ \hline O_{3} & {4} & {3} & {6} & {2} & {5} \\ \hline {Demand} & {6} & {10} & {15} & {4} & {} \\ \hline\end{array}\]

[20M]

3.(a) Show that the set \(\left\{a+b \omega : \omega^{3}=1\right\}\), where \(a\) and \(b\) are real numbers, is a field with respect to usual addition and multiplication.

[15M]

3.(b) Obtain \(\dfrac{\partial^{2} f(0,0)}{\partial x \partial y}\) and \(\dfrac{\partial^{2} f(0,0)}{\partial y \partial x}\) for the function

\[f(x, y)=\left\{\begin{array}{c}{\dfrac{x y\left(3 x^{2}-2 y^{2}\right)}{x^{2}+y^{2}},(x, y) \neq(0,0)} \\ {0 \quad,(x, y) = (0,0)}\end{array}\right.\]

Also, discuss the continuity \(\dfrac{\partial^{2} f}{\partial x \partial y}\) and \(\dfrac{\partial^{2} f}{\partial y \partial x}\) of \(f\) at \((0,0)\).

[15M]

3.(c) Evaluate the integral \(\int_{0}^{\pi} \dfrac{d \theta}{\left(1+\dfrac{1}{2} \cos \theta\right)^{2}}\) using residues.

[20M]

4.(a) Prove that the set \(Q(\sqrt{5})=\{a+b \sqrt{5} : a, b \in Q\}\) is commutative ring with identity.

[15M]

4.(b) Find the minimum value of \(x^{2}+y^{2}+z^{2}\) subject to the condition \(x y z=a^{3}\) by the method of Lagrange multipliers.

[15M]

4.(c) Find all optimal solutions of the following linear programming problem by the simplex method.
Maximize
\(Z=30 x_{1}+24 x_{2}\),
subject to:
\(5 x_{1}+4 x_{2} \leq 200\)
\(x_{1} \leq 32\)
\(x_{2} \leq 40\)
\(x_{1} x_{2} \geq 0\)

[20M]

5.(a) Solve the partial differential equation \(\left(2 D^{2}-5 D D^{\prime}+2 D^{\prime 2}\right) z=24(y-x)\).

[10M]

5.(b) Apply Newton-Raphson method to determine a root of the equation \(\cos x-x e^{x}=0\) correct up to four decimal places.

[10M]

5.(c) Use five subintervals to integrate \(\int_{0}^{1} \dfrac{d x}{1+x^{2}}\) using trapezoidal rule.

[10M]

5.(d) Use only AND and OR logic gates to construct a logic circuit for the Boolean expression \(z=x y+u v\).

[10M]

5.(e) Find the equation of motion of a compound pendulum using Hamilton’s equations.

[10M]

6.(a) Reduce the equation \(\dfrac{\partial^{2} z}{\partial x^{2}}=x^{2} \dfrac{\partial^{2} z}{\partial y^{2}}\) to canonical form.

[15M]

6.(b) Solve the system of equations \(2 x_{1}-x_{2}=7\)
\(-x_{1}+2 x_{2}-x_{3}=1\)
\(-x_{2}+2 x_{3}=1\)
using Gauss-Seidel iteration method (perform three iterations).

[15M]

6.(c) Use Runge-Kutta formula of fourth order to find the value of \(y\) at \(x=0.8\), where \(\dfrac{d y}{d x}=\sqrt{x+y}\), \(y(0.4)=0.41\). Take the step length \(h=0.2\).

[20M]

7.(a) Find the deflection of a vibrating string (length \(=\pi\), ends fixed, \(\dfrac{\partial^{2} u}{\partial t^{2}}\)=\(\dfrac{\partial^{2} u}{\partial x^{2}} )\) corresponding to zero initial velocity and initial deflection. \(f(x)=k(\sin x-\sin 2 x)\).

[15M]

7.(b) Draw a flowchart for Simpson’s one-third rule.

[15M]

7.(c) Given the velocity potential \(\phi=\dfrac{1}{2} \log \left[\dfrac{(x+a)^{2}+y^{2}}{(x-a)^{2}+y^{2}}\right]\), determine the streamlines.

[20M]

8.(a) Solve \(\dfrac{\partial^{2} u}{\partial t^{2}}=\dfrac{\partial^{2} u}{\partial x^{2}}\), \(0< x< 1\), \(t > 0\), given that: i) \(u(x, 0)=0\), \(0 \leq x \leq 1\)
ii) \(\dfrac{\partial u}{\partial t}(x, 0)=x^{2}\), \(0 \leq x \leq 1\)
iii) \(u(0, t)=u(1, t)=0\), for all \(t\)

[15M]

8.(b) For any Boolean variables \(x\) and \(y\), show that \(x+x y=x\).

[15M]

8.(c) Find Navier-Stokes equation for steady laminar flow of a viscous incompressible fluid between two infinite parallel plates.

[20M]