Paper II PYQs-2015

1.(a) If in a group G there is an element $a$ of order $360,$ what is the order of $a^{220}$ ? Show that if $G$ is a cyclic group of order $n$ and $m$ divides $n,$ then $G$ has a subgroup of order $m$.

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1.(b) Let $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ be an absolutely convergent series of real numbers. Suppose $\sum _{n=1}^{\mathrm{\infty }}{a}_{2n}={\displaystyle \frac{9}{8}}$ and $\sum _{n=0}^{\mathrm{\infty }}{a}_{2n+1}={\displaystyle \frac{-3}{8}}.$ What is $\sum _{n=1}^{\mathrm{\infty }}{a}_n?$ Justify your answer. (Majority of marks is for the correct justification).

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1.(c) Let $u(x,y)=\cos x\sinh y.$ Find the harmonic conjugate $v(x,y)$ of $u$ and express $u(x,y)+iv(x,y)$ as a function of $z=x+iy$

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1.(d) Solve graphically: Maximize $z=7x+4y$ subject to $2x+y\le 2,x+10y\le 10$ and $x\le 8$ (Draw your own graph without graph paper).

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2.(a) If $p$ is a prime number and $e$ a positive integer, what are the elements ${}^{\prime }{a}^{\prime }$ in the ring $Z_{p}e$ of integers modulo $p^e$ such that $a^2=a?$ Hence (or otherwise) determine the elements in $Z_{35}$ such that $a^2=a$.

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2.(b) Let $X=(a,b]$. Construct a continuous function $f:X\to R($ set of real numbers) which is unbounded and not uniformly continuous on X. Would your function be uniformly continuous on $[a+\epsilon ,b],a+\epsilon <b?$ Why?

[10M]

2.(c) Evaluate the integral ${\int }_r{\displaystyle \frac{z^2}{(z^2+1)(z-1{)}^2}}dz$, where $r$ is the circle $|z|=2$.

[10M]

3.(a) What is the maximum possible order of a permutation in $S_8,$ the group of permutations on the eight numbers $\{1,2,3,\dots ,8\}$? Justify your answer. (Majority of marks will be given for the justification).

[10M]

[10M]

3.(c) A manufacturer wants to maximise his daily output of bulbs which are made by two processes $P_1$ and $P_2$. If $x_1$ is the output by process $P_1$ and $x_2$ is the output by process $P_2$, then the total labour hours is given by $2x_1+3x_2$ and this cannot exceed $130,$ the total machine time is given by $3x_1+8x_2$ which cannot exceed 300 and the total raw material is given by $4x_1+2x_2$ and this cannot exceed $140.$ What should $x_1$ and $x_2$ be so that the total output $x_1+x_2$ is maximum $?$ Solve by the simplex method only.

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4.(a) Compute the double integral which will give the area of the region between the $y$ -axis, the circle $(x-2{)}^2+(y-4{)}^2=z^2$ and the parabola $2y=x^2$. Compute the integral and find the area.

[10M]

5.(a) Store the value of -1 in hexadecimal in a 32-bit computer.

[10M]

5.(b) Show that $\sum _{k=1}^n{l}_k(x)=1,$ where $l_k(x),k=1$ to $n,$ are Lagrange’s fundamental polynomials.

[10M]

5.(c) Derive the Hamiltonian and equation of motion for a simple pendulum.

[10M]

5.(d) Find the solution of the equation $u_{xx}-3u_{xy}+u_{yy}=\sin (x-2y)$.

[10M]

6.(a) Solve the following system of linear equations correct to two places by Gauss-Seidel method: $x+4y+z=-1,3x-y+z=6,x+y+2z=4$

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6.(b) Solve the differential equation $u_x^2-u_y^2$ by variable separation method.

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6.(c) In a steady fluid flow, the velocity components are $u=2kx,v=2ky$ and $w=-4kz$. Find the equation of a streamline passing through (1,0,1)

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7.(a) Solve the heat equation ${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}={\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}},0<x<1,t>0$ subject to the conditions $u(0,t)=u(1,t)=0$ for $t>0$ and $u(x,0)=\sin \pi x,0<x<1$

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7.(b) Find the moment of inertia of a uniform mass $M$ of a square shape with each side a about its one of the diagonals.

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7.(c) Use the classical fourth order Runge-Kutta methods to find solutions at $x=0\cdot 1$ and $x=0\cdot 2$ of the differential equation ${\displaystyle \frac{dy}{dx}}=x+y,y(0)=1$ with step size $h=0\cdot 1$.

[10M]

8.(a) Write a BASIC program to compute the product of two matrices.

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8.(b) Suppose $\overrightarrow{v}=(x-4y)\hat{i}+(4x-y)\hat{j}$ represents a velocity field of an incompressible and irrotational flow. Find the stream function of the flow.

[10M]

8.(c) Solve the wave equation ${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^2}}=c^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$ for a string of length $l$ fixed at both ends. The string is given initially a triangular deflection $u(x,0)=\{\begin{array}{cc}{\displaystyle \frac{2}{l}}x,& \text{ if }0<x<{\displaystyle \frac{l}{2}}\\ {\displaystyle \frac{2}{l}}(l-x),& \text{ if }& {\displaystyle \frac{l}{2}}\le x<l\end{array}$
with initial velocity $u_t(x,0)=0$.

[10M]