Paper II PYQs-2017
Section A
1.(a) Let \(x_{1}=2\) and \(x_{n+1}=\sqrt{x_{n}+20}, n=1,2,3, \ldots\), show that the sequence \(x_{1}, x_{2}, x_{3} . .\) is convergent.
[10M]
1.(b) Let \(G\) be a group of order \(n\). Show that \(G\) is isomorphic to a subgroup of the permutation group \(S_{n}\).
[10M]
1.(c) Find the supremum and infimum of \(\dfrac{x}{\sin x}\) on the interval \(\left( 0, \dfrac{\pi}{2} \right]\)
[10M]
1.(d) Determine all entire functions \(f(z)\) such that 0 is a removable singularity of \(f\left(\dfrac{1}{z}\right)\).
[10M]
1.(e) Using graphical method, find the maximum value of
\[2x+y\]subject to
\(4x+3y \leq 12\) \(4x+y \leq 8\) \(4x-y \leq 8\) \(x, y \geq 0\)
[10M]
2.(a) Let \(f(t)=\int_{0}^{t}[x] d x\) where \([x]\) denotes the largest integer less than or equal to \(x\).
i) Determine all the real numbers \(t\) at which \(f\) is differentiable.
ii) Determine all the real numbers \(t\) at which \(f\) is continuous but not differentiable.
[15M]
2.(b) Using contour integral method, prove that \(\int_{0}^{\infty} \dfrac{x \sin m x}{a^{2}+x^{2}} d x=\dfrac{\pi}{2} e^{-m a}\).
[15M]
2.(c) Let \(F\) be a field and \(F[x]\) denote the ring of polynomial over \(F\) in a single variable \(X\). For \(f(X), g(X) \in F[X]\) with \(g(X) \neq 0\), show that there exist \(q(X), r(X) \in F[X]\) such that degree \(r(X)\)< degree \(g(X)\) and \(f(X)=q(X) \cdot g(X)+r(X)\).
[20M]
3.(a) Show that the groups \(Z_{5} \times Z_{7}\) and \(Z_{35}\) are isomorphic.
[15M]
3.(b) Let \(f=u+i v\) be analytic function on the unit disc \(D=\{z \in C :\vert z \vert < 1\}\). Show that \(\dfrac{\partial^{2} u}{\partial x^{2}}+\dfrac{\partial^{2} u}{\partial y^{2}}\)=0=\(\dfrac{\partial^{2} v}{\partial x^{2}}+\dfrac{\partial^{2} v}{\partial y^{2}}\) at all points of \(\mathrm{D}\).
[15M]
3.(c) Solve the following linear programming problem by simplex method. Maximize \(z=3 x_{1}+5 x_{2}+4 x_{3}\), subject to:
\(2 x_{2}+3 x_{2} \leq 8\)
\(2 x_{1}+5 x_{2} \leq 10\)
\(3 x_{1}+2 x_{2}+4 x_{3} \leq 15 x_{3}\)
\(x_{1}, x_{2}, x_{3} \geq 0\)
[20M]
4.(a) For a function \(f: C \rightarrow C\) and \(n \geq 1\), let \(f^{(n)}\) denotes the \(n^{th}\) derivative of \(f\) and \(f^{(0)}=f\). Let \(f\) be an entire function such that for some \(n \geq 1\), \(f^{(n)}\left(\dfrac{1}{k}\right)=0\) for all \(k=1\), 2, 3, show that \(f\) is a polynomial.
[15M]
4.(b) Find the initial basic feasible solution of the following transportation problem using Vogel’s approximation methods and find the cost.
\(Destinations\)
\(\begin{array}{|c|c|c|c|c|c|} \hline { }&{D_{1}} & {D_{2}} & {D_{3}} & {D_{4}} & {D_{5}} \\ \hline O_{1} & {4} & {7} & {0} & {3} & {6}& {14}\\ \hline O_{2} & {1} & {2} & {-3} & {3} & {8}&{9}\\ \hline {O_{3}} & {3} & {-1} & {4} & {0} & {5} & {17} \\ \hline { }&{8} & {3} & {8} & {13} & {8} \\ \hline \end{array}\)
[15M]
4.(c) Let \(\sum_{n=1}^{\infty} x_{n}\) be a conditionally convergent series of real numbers. Show that there is a rearrangement \(\sum_{n=1}^{\infty} x_{\pi(n)}\) of the series \(\sum_{n=1}^{\infty} x_{n}\) that converges to 100.
[20M]
Section B
5.(a) Solve \(\left(D^{2}-2 D D^{\prime}-D^{2}\right) z\)=\(e^{x+2 y}+x^{3}+\sin 2 x\) where \(D \equiv \dfrac{\partial}{\partial x}\), \(D^{\prime} \equiv \dfrac{\partial}{\partial y}\), \(D^{2} \equiv \dfrac{\partial^{2}}{\partial x^{2}}\), \(D^{\prime 2} \equiv \dfrac{\partial^{2}}{\partial y^{2}}\).
[10M]
5.(b) Explain the main steps of the Gauss-Jordan method and apply this method to find the inverse of the matrix \(\begin{bmatrix}{2} & {6} & {6} \\ {2} & {8} & {6} \\ {2} & {6} & {8}\end{bmatrix}\).
[10M]
5.(c) Write the Boolean expression \(z(y+z)(x+y+z)\) in the simplest form using Boolean postulate rules. Mention the rules used during simplification. Verify your result by constructing the truth table for the given expression and for its simplest form.
[10M]
5.(d) Let \(\Gamma\) be a closed curve in \(xy-plane\) and let \(S\) denote the region bounded by the curve \(\Gamma\). Let \(\dfrac{\partial^{2} w}{\partial x^{2}}+\dfrac{\partial^{2} w}{\partial y^{2}}\)=\(f(x, y) \forall(x, y) \in S\). If \(f\) is prescribed at each point \((x, y)\) of \(S\) and \(w\) is prescribed on the boundary \(\Gamma\) of \(S\) then prove that any solution \(w=w(x, y)\), satisfying these conditions, is unique.
[10M]
5.(e) Show that the moment of inertia of an elliptic area of mass \(M\) and semi-axis \(a\) and \(b\) about a semi-diameter of length \(r\) is \(\dfrac{1}{4} M \dfrac{a^{2} b^{2}}{r^{2}}\). Further, prove that the moment of inertia about a tangent is \(\dfrac{5 M}{4} p^{2}\), where \(p\) is the perpendicular distance from the centre of the ellipse to the tangent.
[10M]
6.(a) Find a complete integral of the partial differential equation \(2(p q+y p+q x)+x^{2}+y^{2}=0\).
[15M]
6.(b) For given equidistant values \(u_{-1}\), \(u_{0}\), \(u_{1}\) and \(u_{2}\), a value is interpolated by Lagrange’s formula. Show that it may be written in the form \(u_{x}=y u_{0}+\dfrac{y\left(y^{2}-1\right)}{3 !} \Delta^{2} u_{-1}+\dfrac{x\left(x^{2}-1\right)}{3 !} \Delta^{2} u_{0}\), where \(x+y=1\).
[15M]
6.(c) Two uniform rods \(AB\) and \(AC\), each of mass \(m\) and length \(2 a\), are smoothly hinged together at \(A\) and move on a horizontal plane. At time \(t\), the mass centre of the road is at the point \((\xi, \eta)\) referred to fixed perpendicular axes \(Ox\), \(O y\) in the plane, and the rods make angles \(\theta \pm \phi\) with \(O x\). Prove that the kinetic energy of the system is \(m\left[\xi^{2}+\eta^{2}+\left(\dfrac{1}{3}+\sin ^{2} \phi\right) a^{2} \theta^{2}+\left(\dfrac{1}{3}+\cos ^{2} \phi\right) a^{2} \phi^{2}\right]\). Also derive Lagrange’s equation of motion for the system if an external force with components \([X, Y]\) along the axes acts at \(A\).
[20M]
7.(a) Reduce the equation \(y^{2} \dfrac{\partial^{2} z}{\partial x^{2}}-2 x y \dfrac{\partial^{2} z}{\partial x \partial y}+x^{2} \dfrac{\partial^{2} z}{\partial y^{2}}\)=\(\dfrac{y^{2}}{\partial x} \dfrac{\partial z}{\partial x}+\dfrac{x^{2}}{\partial y} \dfrac{\partial z}{\partial y}\) to canonical form and hence solve it.
[15M]
7.(b) Derive the formula \(\int_{a}^{b} y d x=\dfrac{3 h}{8}\left[\left(y_{0}+y_{n}\right)+3\left(y_{1}+y_{2}+y_{4}+y_{5}+\ldots+y_{n-1}\right)+2\left(y_{3}+y_{6}+y_{n-3}\right)\right]\). Is there any restriction on \(n\)? State that condition. What is the error bounded in the case of Simpson’s \(\dfrac{3}{8}\)th rule?
[20M]
7.(c) A stream is rushing from a boiler through a conical pipe, the diameters of the ends of which are \(D\) and \(d\). If \(V\) and \(v\) be the corresponding velocities of the stream and if the motion is assumed to steady and diverging from the vertex of the cone, then prove that \(\dfrac{v}{V}=\dfrac{D^{2}}{d^{2}} e^{\left(u^{2}-v^{2}\right) / 2 K}\), where \(K\) is the pressure divided by the density and is constant.
[15M]
Let \(A B\) and \(A^{\prime} B^{\prime}\) be the ends of the conical pipe such that \(A^{\prime} B^{\prime}=d\) and \(A B=D .\) Let \(\rho_{1}\) and \(\rho_{2}\) be densities of the stream at \(A^{\prime} B^{\prime}\) and \(A B\). By principle of conservation of mass, the mass of the stream that enters the end \(A B\) and leaves at the end \(A^{\prime} B^{\prime}\) must be the same. Hence the equation of continuity is \(\pi(d / 2)^{2} v \rho_{1}=\pi(D / 2)^{2} V \rho_{2}\)
so that \(\frac{v}{V}=\frac{D^{2}}{d^{2}} \times \frac{\rho_{2}}{\rho_{1}} \ldots (1)\)
By Bernoulli’s theorem (in absence of external forces like gravity), we have
\(\int \frac{d p}{\rho}+\frac{1}{2} q^{2}=C \ldots (2)\)
Given that \(\quad p / \rho=k \quad\) so that
\[d p=k d \rho \ldots (3)\]\(\therefore\) (2) reduces to
\(k \int \frac{d \rho}{\rho}+\frac{1}{2} q^{2}=C\) (using (3))
Integrating above, we get
\(k \log \rho+q^{2} / 2=C\), \(C\) being an arbitrary constant
\(\begin{array}{lllll}\text {When } & q=v, & \rho=\rho_{1} & \text { and } & \text { when } & q=V, & \rho=\rho_{2} .\end{array}\)
Hence (4) yields
\(k \log \rho_{1}+v^{2} / 2=C \quad\) and \(\quad k \log \rho_{2}+V^{2 /} 2=C\)
Subtracting, \(k\left(\log \rho_{2}-\log \rho_{1}\right)+\left(V^{2}-v^{2}\right) / 2=0\)
\(\implies \log \left(\rho_{2} / \rho_{1}\right)=\left(v^{2}-V^{2}\right) 2 k \quad\) or \(\quad \rho_{2} / \rho_{1}=e^{\left(v^{2}-V^{2}\right) 2 k}\)
Using (5), (1) reduces to \(v / V=\left(D^{2} / d^{2}\right) \times e^{\left(v^{2}-V^{2}\right) / 2 k}\)
Hence, proved.
8.(a) Given the one-dimensional wave equation \(\dfrac{\partial^{2} y}{\partial t^{2}}=c^{2} \dfrac{\partial^{2} y}{\partial x^{2}}\); \(t > 0\), where \(c^{2}=\dfrac{T}{m}\), \(T\) the constant tension in the string and \(m\) is the mass per unit length of the string.
(i) Find the appropriate solution of the wave equation.
(ii) Find also the solution under the conditions \(y(0, t)=0\), \(y(l, t)=0\) for all \(t\) and \(\left [ \dfrac{\partial y}{\partial t}\right]_{t=0}=0\), \(y(x, 0)=a \sin \dfrac{\pi x}{t}\), \(0< x< l\), \(a>0\).
[20M]
8.(b) Write an algorithm in the form of a flow chart for Newton-Raphson method. Describe the cases of failure of this method.
[15M]
8.(c) If the velocity of an incompressible fluid at the point \((x, y, z)\) is given by \(\left(\dfrac{3 x z}{r^{5}}, \dfrac{3 y z}{r^{5}}, \dfrac{3z^2 - r^2}{r^{5}}\right)\), \(r^{2}=x^{2}+y^{2}+z^{2}\) then prove that the liquid motion is possible and that the velocity potential is \(\dfrac{z}{r^{3}}\). Further, determine the the streamlines.
[15M]
Here \(u=\frac{3 x z}{r^{5}}\), \(v=\frac{3 y z}{r^{5}}\), \(w=\frac{3 z^{2}-r^{2}}{r^{5}}=\frac{3 z^{2}}{r^{5}}-\frac{1}{r^{3}} \ldots\) (1)
where
\(r^{2}=x^{2}+y^{2}+z^{2} \ldots\) (2)
From \((2), \quad \partial r / \partial x=x / r, \quad \partial r / \partial y=y / r, \quad \quad \partial r / \partial z=z / r\) From (1), ( 2 ) and (3), we have
\[\frac{\partial u}{\partial x}=3 z\left[\frac{1}{r^{5}}+(-5 x) r^{-6} \frac{\partial r}{\partial x}\right]=\frac{3 z}{r^{5}}-\frac{15 x^{2} z}{r^{7}}\] \[\frac{\partial v}{\partial y}=3 z\left[\frac{1}{r^{5}}+(-5 y) r^{-6} \frac{\partial r}{\partial y}\right]=\frac{3 z}{r^{5}}-\frac{15 y^{2} z}{r^{7}}\] \[\frac{\partial w}{\partial z}=\frac{6 z}{r^{5}}-15 z^{2} r^{-6} \frac{\partial r}{\partial z}+3 r^{-4} \frac{\partial r}{\partial z}=\frac{6 z}{r^{5}}-\frac{15 z^{2}}{r^{6}} \cdot \frac{z}{r}+\frac{3}{r^{4}} \cdot \frac{z}{r}=\frac{9 z}{r^{5}}-\frac{15 z^{3}}{r^{7}}\] \[\therefore \quad \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=\frac{15 z}{r^{5}}-\frac{15 z}{r^{7}}\left(x^{2}+y^{2}+z^{2}\right)=\frac{15 z}{r^{5}}-\frac{15 z}{r^{7}} \times r^{2}=0\]Since the equation of continuity is satisfied by the given values of \(u, v\) and \(w,\) the motion is possible. Let \(\phi\) be the required velocity potential. Then
\[\begin{array}{l} d \phi=\frac{\partial \phi}{\partial x} d x+\frac{\partial \phi}{\partial y} d y+\frac{\partial \phi}{\partial z} d z=-(u d x+v d y+w d z), \text { by definition of } \phi \\ =-\left[\frac{3 x}{r^{5}} d x+\frac{3 y z}{r^{5}} d y+\frac{3 z^{2}-r^{2}}{r^{5}} d z\right]=\frac{r^{2} d z-3 z(x d x+y d y+z d z)}{r^{5}} \\ \text { Thus, } \quad d \phi=\frac{r^{3} d z-3 r^{2} z d r}{\left(r^{3}\right)^{2}}=d\left(\frac{z}{r^{3}}\right), \text { using }(2) \end{array}\]Integrating,
\[\phi=z / r^{3}\][Omitting constant of integration, for it has no significance in \(\phi\)]
In spherical polar coordinates \((r, \theta, \phi),\) we know that \(z=r \cos \theta .\) Hence the required potential is given by \(\phi=(r \cos \theta) / r^{3}=(\cos \theta) / r^{2}\). We now obtain the streamlines. The equations of streamlines are given by
\[\frac{d x}{u}=\frac{d y}{v}=\frac{d z}{w} \quad \text { i.e., } \quad \frac{d x}{3 x z / r^{5}}=\frac{d y}{3 y z / r^{5}}=\frac{d z}{\left(3 z^{2}-r^{2}\right) / r^{5}}\]So,
\[\frac{d x}{3 x z}=\frac{d y}{3 y z}=\frac{d z}{3 z^{2}-r^{2}} \ldots (4)\]Taking the first two members of (4) and simplifying, we get
\[d x / x=d y / y\] \[\implies d x / x-d y / y=0\]Integrating, \(\log x-\log y=\log c_{1} \quad\) l.e. \(\quad x / y=c_{1}, c_{1}\) being a constant \(\ldots (5)\)
Now, each member in \((4)=\frac{x d x+y d y+z d z}{3 x^{2} z+3 y^{2} z+3 z^{3}-r^{2} z}=\frac{x d x+y d y+z d z}{3 z\left(x^{2}+y^{2}+z^{2}\right)-r^{2} z}\)
\[=\frac{x d x+y d y+z d z}{3 z\left(x^{2}+y^{2}+z^{2}\right)-z\left(x^{2}+y^{2}+z^{2}\right)}=\frac{x d x+y d y+z d z}{2 z\left(x^{2}+y^{2}+z^{2}\right)}, \text { by }(2) \ldots (6)\]Taking the first member in (4) and (6), we get
\[\frac{d x}{3 x z}=\frac{x d x+y d y+z d z}{2 z\left(x^{2}+y^{2}+z^{2}\right)} \quad \text { or } \quad \frac{2}{3} \frac{d x}{x}=\frac{1}{2} \frac{2 x d x+2 y d y+2 z d z}{x^{2}+y^{2}+z^{2}}\]Integrating, \((2 / 3) \times \log x=(1 / 2) \times \log \left(x^{2}+y^{2}+z^{2}\right)+\log c_{2}\) or \(x^{2 / 3}=c_{2}\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}, c_{2}\) being an arbitrary constant. \(\ldots (7)\).
The required streamlines are the curves of intersction of (5) and (7).