Paper II PYQs-2014

1.(a) If $G$ is a group in which $(a\cdot (b{)}^4=a^4\cdot b^4,(a\cdot (b{)}^5=a^5\cdot b^5$ and $(a\cdot (b{)}^6=a^6\cdot b^6,$ for all $a,b\in G,$ then prove that $G$ is Abelian.

[10M]

1.(b) Let $f$ be defined on [0,1] as $f(x)=\{\begin{array}{l}\sqrt{1-x^2},\text{ if }x\text{ is rational }\\ 1-x,\text{ if }x\text{ is irrational }\end{array}$ Find the upper and lower Riemann integrals of $f$ over [0,1] .

[10M]

1.(c) Using Cauchy integral formula, evaluate ${\int }_C{\displaystyle \frac{z+2}{(z+1{)}^2(z-2)}}dz$ where $C$ is the circle $|z-i|=2$

[10M]

1.(d) Obtain the initial basic feasible solution for the transportation problem by North-West corner rule:

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1.(e)Find the constants $a,b,c$ such that the function $f(z)=2x^2-2xy-y^2+i(a{x}^2-bxy+c{y}^2)$ is analytic for all $z(=x+iy)$ and express $f(z)$ in terms of $z$.

[10M]

2.(a) Let $J_n$ be the set of integers mod $n.$ Then prove that $J_n$ is a ring under the operations of addition and multiplication mod $n.$ Under what conditions on $n$, $J_n$ is a field? Justify your answer.

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2.(b) Show that the function $f(x)=\sin {\displaystyle \frac{1}{x}}$ is continuous but not uniformly continuous on $(0,\pi )$

[10M]

5.(a) Use Lagrange’s formula to find the form of $f(x)$ from the following table:
| $x$ | 0 | 2 | 3 | 6 | |——|—|—|—|—| |$f(x)$|648|704|729|792|

[10M]

5.(b) Write a program in BASIC to integrate ${\int }_0^1{e}^{-2x}\sin xdx$ by Simpson’s ${\displaystyle \frac{1}{3}}rd$ rule with 20 subintervals.

[10M]

5.(c) Show that the general solution of the pde ${\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{x}^2}}={\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{t}^2}}$ is of the form $Z(x,y)=F(x+ct)+G(x-ct),$ where $F$ and $G$ are arbitrary functions.

[10M]

5.(d) Prove that the vorticity vector $\overrightarrow{\mathrm{\Omega }}$ of an incompressible viscous fluid moving in the absence of an external force satisfies the differential equation ${\displaystyle \frac{D\overrightarrow{\mathrm{\Omega }}}{Dt}}=\overrightarrow{\mathrm{\Omega }}\cdot \mathrm{\nabla }\mid \overrightarrow{q}+v{\mathrm{\nabla }}^2\overrightarrow{\mathrm{\Omega }}$ where $\overrightarrow{q}$ is the velocity vector with $\overrightarrow{\mathrm{\Omega }}=\mathrm{\nabla }\times \overrightarrow{q}$

[10M]

5.(e) Find the condition that $f(x,y,\text{lambd}(a)=0$ should be a possible system of streamlines for steady irrotational motion in two dimensions, where $\lambda$ is a variable parameter.

[10M]

6.(a) Verify that the differential equation $(y^2+yz)dx+(xz+z^2)dy+(y^2-xy)dz=0$ is integrable and find its primitive.

[10M]

6.(b) Show that the moment of inertia of a uniform rectangular mass $M$ and sides $2a$ and $2b$ about a diagonal is ${\displaystyle \frac{2M{a}^2{b}^2}{3(a^2+b^2)}}$.

[10M]

6.(c) The values of $f(x)$ for different values of $x$ are given as $f(1)=4,f(2)=5,f(7)=5$ and $f(8)=4.$ Using Lagrange’s interpolation formula, find the value of $f(6).$ Also find the value of $x$ for which $f(x)$ is optimum.

[10M]

6.(d) TBC Write a BASIC program to sum the series $S=1+x+x^2+\dots +x^n,$ for $n=30,60$ and 90 for the values of $x=0\cdot 1(0\cdot 1)0\cdot 3$

[10M]

7.(a) Solve: ${(D-3D^{\mathrm{\prime }}-2)}^{2}z=2e^{2x}\cot (y+3x)$

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7.(b) Solve the following system of equations: $\begin{array}{rl}2x_1+x_2+x_3-2x_4& =-10\\ 4x_1+2x_3+x_4& =8\\ 3x_1+2x_2+2x_3& =7\\ x_1+3x_2+2x_3-x_4& =-5\end{array}$

[10M]

7.(c) A uniform rod $OA$ of length $2a$ is free to turn about its end $O,$ revolves with uniform angular velocity $\omega$ about a vertical axis $OZ$ through $O$ and is inclined at a constant angle $\alpha$ to $OZ$. Show that the value of $\alpha$ is either zero or ${\cos }^{-1}\left({\displaystyle \frac{3g}{4a{\omega }^2}}\right)$

[10M]

8.(a) Using Runge-Kutta 4-th order method, find $y$ from ${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{y^2-x^2}{y^2+x^2}}$ with $y(0)=1$ at $x=0\cdot 2,0\cdot 4$

[10M]

8.(b) A plank of mass $M$ is initially at rest along a straight line of greatest slope of a smooth plane inclined at an angle $\alpha$ to the horizon and a man of mass $M^{\mathrm{\prime }}$ starting from the upper end walks down the plank so that it does not move. Show that he gets to the other end in time $\sqrt{{\displaystyle \frac{2M^{\mathrm{\prime }}a}{(M+M^{\mathrm{\prime }})g\sin \alpha }}}$ where $a$ is the length of the plank.

[10M]

8.(c) Prove that ${\displaystyle \frac{x^2}{a^2}}{\tan }^{2}t+{\displaystyle \frac{y^2}{b^2}}{\cot }^{2}t=1$ is a possible form for the bounding surface of a liquid and find the velocity components.

[10M]