Paper II PYQs-2019
Section A
1.(a) Let \(G\) be a finite group, \(H\) and \(K\) subgroups of \(G\) such that \(K \subset H\). Show that \((G:K)=(G:H)(H:K)\).
[10M]
1.(b) Show that the function
\[f(x, y)=\left\{\begin{array}{ll}{\dfrac{x^2-y^2}{x-y},} & {(x, y ) \neq(1,-1),(1,1)} \\ {0} & {, (x, y )=(1,1), (1,-1) }\end{array}\right.\]is continuous and differentiable at \((1,-1)\).
[10M]
1.(c) Evaluate
\[\int^{\infty}_0 \dfrac{tan^{-1}(ax)}{x(1+x^2)}\,dx\;a>0,\;a\neq1\][10M]
1.(d) Suppose \(f(z)\) is analytic function on a domain \(D \in C\) and satisfies the equation \(Imf(z)=(Ref(z))^2\), \(Z\in D\). Show that \(f(z)\) is constant in \(D\).
[10M]
1.(e) Use graphical method to solve the linear programming problem.
Maximize \(Z=3x_1+2x_2\)
subject to
\(x_1-x_2\geq 1\),
\(x_1+x_2\geq 3\) and
\(x_1,x_2,x_3\geq 0\)
[10M]
2.(a) If \(G\) and \(H\) are finite groups whose orders are relatively prime, then prove that there is only one homomorphism from \(G\) to \(H\), the trivial one.
[10M]
2.(b) Write down all quotient groups of the group \(Z_{12}\).
[10M]
2.(c) Using differentials, find an approximate value of \(f(4.1,4.9)\) where
\[f(x,y)=(x^3+x^2y)^{\dfrac{1}{2}}\][15M]
2.(d) Show that an isolated singular point \(z_o\) of a function \(f(z)\) is a pole of order \(m\) if and only if \(f(z)\) can be written in the form \(f(z)=\dfrac{\phi^{m-1}(z_o)}{(m-1)!}\) if \(m \geq 1\).
Moreover \(Res_{z=z_0}\) \(f(z) = \dfrac{ \phi^{(m-1)} (z_0) }{(m-1)!}\) if \(m \geq 1\).
[15M]
3.(a) Discuss the uniform convergence of
\(f_n(x)=\dfrac{nx}{1+n^2x^2},\forall x\in R(-\infty,\infty)\), \(n=1,2,3..\)
[15M]
3.(b) Solve the linear programming problem using Simplex method.
Maximize \(Z=x_1+2x_2-3x_3-2x_4\)
subject to
\(x_1+2x_2-3x_3+x_4=4\),
\(x_1+2x_2+x_3+2x_4=4\) and
\(x_1\), \(x_2\), \(x_3\), \(x_4\geq 0\)
[15M]
3.(c)(i) Evalute the integral \(\int_C Re(z^2) dz\) from 0 to \(2+4i\) along the curve \(C\) where \(C\) is a parabola \(y=x^2\).
[10M]
3.(c)(ii) Let \(a\) be an irreducible element of the Euclidean ring \(R\), then prove that \(R/(a)\) is a field.
[10M]
4.(a) Find the maximum value of \(f(x,y,z)=x^2y^2z^2\) subject to the subsidiary condition \(x^2+y^2+z^2=c^2\), \((x,y,z>0)\).
[15M]
4.(b) Obtain the first terms of the Laurent series expansion of the function \(f(z)=\dfrac{1}{(e^x-1)}\) about the point \(z=0\) valid in the region \(0< \vert z \vert<2 \pi\).
[10M]
4.(c) Discuss the convergence of \(\int^2_1 \dfrac{\sqrt{x}}{\ln x} dx\).
[15M]
4.(d) Consider the following LPP
Maximize \(Z=2x_1+4x_2+4x_3-3x_4\) subject to \(x_1+x_2+x_3=4\), \(x_1+4x_2+x_4=8\) and \(x_1,x_2,x_3,x_4\geq 0\)
Use the dual problem to verify that the basic solution \((x_1,x_2)\) is not optimal.
[10M]
Section B
5.(a) Form a partial differential equation of the family of surfaces given by the following expression:
\(\psi(x^2+y^2+2z^2,y^2-2zx)=0\).
[10M]
5.(b) Apply Newton-Raphson method, to find a real root of transcendental equation \(xlog_{10} x=1.2\), correct to three decimal places.
[10M]
5.(c) A uniform rod \(OA\), of length \(2a\), free to turn about its end \(O\), revolves with angular velocity \(\omega\) about the vertical \(OZ\) through \(O\), and is inclined at a constant angle \(\alpha\) to \(OZ\); find the value of \(\alpha\).
[10M]
5.(d) Using Runge-Kutta method of fourth order, solve \(\dfrac{dy}{dx}=\dfrac{y^2-x^2}{y^2+x^2}\) with \(y(0)=1\) at \(x=0.2\). Use four decimal places for calculation and step length 0.2.
[10M]
5.(e) Draw a flow chart and write a basic algorithm (in FORTRAN/C/C++) for evaluating \(y=\int^6_0 \dfrac{\,dx}{1+x^2}\) using Trapezoidal rule.
[10M]
6.(a) Solve the first order quasilinear partial differential equation by the method of characteristics:
\(x\dfrac{\partial u}{\partial x}+(u-x-y)\dfrac{\partial u}{\partial y}=x+2y\) in \(x>0,-\infty<u<\infty\) with \(u=1+y\) on \(x=1\).
[15M]
6.(b) Find the equivalent numbers given in a specified number to the system mentioned against them:
(i) Integer 524 in binary system.
(ii) 101010110101.101101011 to octal system.
(iii) Decimal number 5280 to hexadecimal system
(iv) Find the unknown number \((1101.101)_8\) to \((?)_{10}\).
[15M]
6.(c) A circular cylinder of radius \(a\) and radius of gyration \(k\) rolls without slipping inside a fixed hollow cylinder of radius \(b\). Show Show that the plane through axes moves in a circular pendulum of length \((b-a)(1+\dfrac{k^2}{a^2})\).
[20M]
7.(a) Using Hamilton’s equation, find the acceleration for a sphere rolling down a rough inclined plane, if \(x\) be a distance of the point of contact of the sphere from a fixed point of the plane.
[15M]
7.(b) Apply Gauss-Seidel iteration method to solve the following system of equation:
\(2x+y-2z=17\)
\(3x+20y-z=-18\)
\(2x-3y+20z=25\), correct to three decimal places.
[15M]
7.(c) Reduce the following second order partial differential equation to canonical from and find the general solution:
\(\dfrac{\partial^2u}{\partial x^2}-2x\dfrac{\partial^2u}{\partial x\partial y}+x^2\dfrac{\partial^2u}{\partial y^2}\)=\(\dfrac{\partial u}{\partial y}+12x\)
[20M]
8.(a) Given the Boolean expression
\[X=AB+ABC+A\bar{B}\bar{C}+A\bar{C}\](i) Draw the logical diagram for the expression.
(ii) Minimize the expression.
(iii) Draw the logical diagram for the reduced expression.
[15M]
8.(b) A sphere of radius \(R\), whose centre is at rest, vibrates radially in an infinte incompressible fluid of density \(\rho\), which is at rest at infinity. If the pressure at infinity is \(\Pi\), show that the pressure at the surface of the sphere at time \(t\) is
\(\Pi +\dfrac{1}{2}\rho \left\{ \dfrac{d^2R^2}{dt^2}+(\dfrac{dR}{dt})^2 \right\}\).
[15M]
Here the motion of the fluid will take place in such a manner so that each element of the fluid moves towards the centre. Hence the free surface would be spherical. Thus the fluid velocity \(v^{\prime}\) will be radial and hence \(v^{\prime}\) will be function of \(r^{\prime}\) (the radial distance from the centre of the sphere which is taken as origin), and time \(t\) only. Let \(p\) be pressure at a distance \(r^{\prime}.\)
Let \(P\) be the pressure on the surface of the sphere of radius \(R\) and \(V\) be the velocity there. Then the equation of continuity is:
\(r^{\prime 2} v^{\prime}=R^{2} V=F(t) … (1)\) From (1) \(\frac{\partial v^{\prime}}{\partial t}=\frac{F^{\prime}(t)}{r^{\prime 2}} … (2)\) Again equation of motion is: \(\frac{\partial v^{\prime}}{\partial t}+v^{\prime} \frac{\partial v^{\prime}}{\partial r^{\prime}}=-\frac{1}{\rho} \frac{\partial p}{\partial r^{\prime}}\) \(\implies\) \(-\frac{F^{\prime}(t)}{r^{\prime 2}}+\frac{\partial}{\partial r^{\prime}}\left(\frac{1}{2} v^{\prime 2}\right)=-\frac{1}{\rho} \frac{\partial p}{\partial r^{\prime}},\) (using (2)) … (3)
Integrating with respect to \(r^{\prime}\), (3) reduces to: \(-\frac{F^{\prime}(t)}{r^{\prime}}+\frac{1}{2} v^{\prime 2}=-\frac{p}{\rho}+C\), \(C\) being an arbitrary constant When \(r^{\prime}=\infty,\) then \(v^{\prime}=0\) and \(p=\Pi\) so that \(C=\Pi / \rho .\) Then, we get \(-\frac{F^{\prime}(t)}{r^{\prime}}+\frac{1}{2} v^{\prime 2}=\frac{\Pi-p}{\rho}\) \(\implies p=\Pi+\frac{1}{2} \rho\left[2 \frac{F^{\prime}(t)}{r^{\prime}}-v^{\prime 2}\right] \ldots(4)\) But \(p=P\) and \(v^{\prime}=V\) when \(r^{\prime}=\mathrm{R}\). Hence (4) gives:
\(\mathrm{P}=\Pi+\frac{1}{2} \rho\left[\frac{2}{\mathrm{R}}\left\{F^{\prime}(t)\right\}_{r^{\prime}=R}-\mathrm{V}^{2}\right] \ldots (5)\) Also \(V=d R / d t .\) Hence using \((1),\) we have
\[\begin{aligned} \left\{F^{\prime}(t)\right\}_{r^{\prime}=R} &=\frac{d}{d t}\left(R^{2} V\right)=\frac{d}{d t}\left(R^{2} \frac{d \mathrm{R}}{d t}\right)=\frac{d}{d t}\left(\frac{R}{2} \cdot \frac{d R^{2}}{d t}\right) \\ &=\frac{R}{2} \frac{d^{2} R^{2}}{d t^{2}}+\frac{1}{2} \frac{d R^{2}}{d t} \frac{d R}{d t}=\frac{R}{2} \frac{d^{2} R^{2}}{d t^{2}}+R\left(\frac{d R}{d t}\right)^{2} \end{aligned}\]Using the above values of \(V\) and \(\left\{F^{\prime}(t)\right\}_{r^{\prime}=R},(5)\) reduces to
\[\begin{aligned} P&=\Pi+\frac{1}{2} \rho\left[\frac{2}{R}\left\{\frac{R}{2} \frac{d^{2} R^{2}}{d t^{2}}+R\left(\frac{d R}{d t}\right)^{2}\right\}-\left(\frac{d R}{d t}\right)^{2}\right] \\ &=\Pi+\frac{1}{2} \rho\left[\frac{d^{2} R^{2}}{d t^{2}}+\left(\frac{d R}{d t}\right)^{2}\right] \end{aligned}\]Hence, proved.
8.(c) Two sources, each of strength \(m\), are placed at the point at the points \((-a,0)\), \((a,0)\) and a sink of strength \(2m\) at origin. Show that the stream lines are the curves \((x^2+y^2)^2\)=\(a^2(x^2-y^2+\lambda xy)\), where \(\lambda\) is variable parameter.
Show also that the fluid speed at any point is \((2ma^2)/(r_1r_2r_3)\), where \(r_1\), \(r_2\) and \(r_3\) are the distances of the points from the sources and the sink, respectively.
[20M]
First Part: The complex potential \(w\) at any point \(P(z)\) is given by \(w=-m \log (z-a)-m \log (z+a)+2 m \log z \ldots (1)\)
\[\implies w=m\left[\log z^{2}-\log \left(z^{2}-a^{2}\right)\right]\]\(\implies \phi+i \psi=m\left[\log \left(x^{2}-y^{2}+2 b y\right)-\log \left(x^{2}-y^{2}-a^{2}+2 b y\right)\right],\) as \(z=x+i y\)
Equating the imaginary parts, we have \(\begin{array}{c} \psi=m\left[\tan ^{-1}\left\{2 x y /\left(x^{2}-y^{2}\right)\right\}-\tan ^{-1}\left\{2 x y /\left(x^{2}-y^{2}-a^{2}\right)\right\}\right] \\ \psi=m \tan ^{-1}\left[\frac{-2 a^{2} x y}{\left(x^{2}+y^{2}\right)^{2}-a^{2}\left(x^{2}-y^{2}\right)}\right], \text { on simplification } \end{array}\)
The desired streamlines are given by \(\psi=\) constant \(=m \tan ^{-1}(-2 / \lambda) .\) Then we obtain
\[(-2 / \lambda)=\left(-2 a^{2} x y\right) /\left[\left(x^{2}+y^{2}\right)^{2}-a^{2}\left(x^{2}-y^{2}\right)\right] \quad \text { or } \quad\left(x^{2}+y^{2}\right)^{2}=a^{2}\left(x^{2}-y^{2}+\lambda x y\right)\]Second Part: We have,
\[\begin{aligned} &\begin{aligned} \frac{d w}{d z} &=-\frac{m}{z-a}-\frac{m}{z+a}+\frac{2 m}{z}=-\frac{2 a^{2} m}{z(z-a)(z+a)} \\ \therefore~ &q=\left\vert\frac{d w}{d z}\right\vert=\frac{2 a^{2} m}{\vert z \\vert z-a\vert\vert z+a\vert}=\frac{2 a^{2} m}{r_{1} r_{2} r_{3}} \end{aligned}\\ &\begin{array}{llll} \text { where } & r_{1}=\vert z-a\vert, & r_{2}=\vert z+a\vert & \text { and } & r_{3}=\vert z\vert \end{array} \end{aligned}\]