Paper II PYQs-2016
Section A
1.(a) Let \(K\) be a field and \(K[X]\) be the ring of polynomials over \(K\) in a single \(X\) for a polynomial \(f \in K[X]\). Let \((f)\) denote the ideal in \(K[X]\) generated by \(f\). Show that \((f)\) is a maximal ideal in \(K[X]\) if and only iff is an irreducible polynomial over \(K\).
[10M]
1.(b) For the function \(f:(0, \infty) \rightarrow R\) given by \(f(x)=x^{2} \sin \dfrac{1}{x}\), \(0< x < \infty\), show that there is a differentiable function \(g : R \rightarrow R\) that extends \(f\).
[10M]
1.(c) Two sequences \(\left\{x_{n}\right\}\) and \(\left\{y_{n}\right\}\) are defined inductively by the following:\(x_{1}=\dfrac{1}{2}\), \(y_{1}=1\), \(x_{n}=\sqrt{x_{n-1} y_{n-1}}\), \(n=2,3,4, \ldots \dfrac{1}{y_{n}}=\dfrac{1}{2}\left(\dfrac{1}{x_{n}}+\dfrac{1}{y_{n-1}}\right)\), \(n=2,3,4, \ldots\). Prove that \(x_{n-1} < x_{n} < y_{n} < y_{n-1}\), \(n=2,3,4, \ldots\) and deduce that both the sequences converge to the same limit where \(\dfrac{1}{2} < l < 1\).
[10M]
1.(d) Is \(\mathrm{v}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{3}-3 \mathrm{xy}^{2}+2 \mathrm{y}\) a harmonic function? Prove your claim. If yes, find its conjugate harmonic function \(\mathrm{u}(\mathrm{x}, \mathrm{y})\) and hence, obtain the analytic function whose real and imaginary parts are \(u\) and \(v\) respectively.
[10M]
1.(e) Find the maximum value of
\[5x+2y\]with constraints
\(x+2y \geq 1\), \(2x+y \leq 1\) \(x \geq 0\) and \(y \geq 0\)
by graphical method.
[10M]
2.(a) Show that the series \(\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{n+1}\) is conditionally convergent. (If you use any theorem, \((\mathrm{s})\) to show it, then you must give a proof of that theorem(s)).
[15M]
2.(b) Let \(p\) be prime number and \(Z_{p}\) denote the additive group of integers modulo \(p\). Show that that every non-zero element \(Z_{p}\) generates \(Z_{p}\).
[15M]
2.(c) Maximize
\[z=2x_1+3x_2+6x_3\]subject to
\(2x_1+x_2+x_3 \leq 5\) \(3x_2+2x_3 \leq 6\) \(x_1 \geq 0\), \(x_2 \geq 0\), \(x_3 \geq 0\)
Is the optimal solution unique? Justify your answer.
[20M]
3.(a) Let \(K\) be an extension of a field \(F\). Prove that the element of \(K\) which are algebraic over \(F\) form a subfield of \(K\). Further if \(F \subset K \subset L\) Fare fields \(L\) is algebraic over \(K\) and \(K\) is algebraic over \(F\), then prove that \(L\) is algebraic over \(F\).
[20M]
3.(b) Find the relative maximum minimum values of the function \(f(x, y)=x^{4}+y^{4}-2 x^{2}+4 x y-2 y^{2}\).
[15M]
3.(c)Let \(\gamma:[0,1] \rightarrow C\) be the curve \(\gamma(t)=e^{2 \pi i t}\), \(0 \leq t \leq 1\). Find, giving justifications, the value of the contour integral \(\int_{\gamma} \dfrac{d z}{4 z^{2}-1}\).
[15M]
4.(a) Show that every algebraically closed field is infinite.
[15M]
4.(b) Let \(f: R \rightarrow R\) be a continuous function such \(\lim _{x \rightarrow+\infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\) exist and are finite. Prove that \(f\) is uniformly continuous on \(\mathbb{R}\).
[15M]
4.(c) Prove that every power series represents an analytic function inside its circle of convergence.
[20M]
Section B
5.(a) Find the general equation of surfaces orthogonal to the family of spheres given by \(x^{2}+y^{2}+z^{2}=c z\).
[10M]
5.(b) Does a fluid with velocity \(q=\left[z-\dfrac{2 x}{r}, 2 y-3 z-\dfrac{2 y}{r}, x-3 y-\dfrac{2 z}{r}\right]\) possess vorticity, where \(q(u, v, w)\) is the velocity in the cartesian frame \(\vec{r}(x, y, z)\) and \(r^{2}=x^{2}+y^{2}+z^{2}\)? What is the circulation in the circle \(x^{2}+y^{2}=9\), \(z=0\)?
[10M]
Given,
(i) Calculation of vorticity:
velocity \(\vec{q}=\left(z-\frac{2 x}{r}\right) \hat{i}+\left(2 y-3 z-\frac{2 y}{r}\right) \hat{j}+\left(x-3 y-\frac{2 z}{r}\right) \hat{k}\) \(=u \hat{i}+v \hat{j}+\omega \hat{k}\)
Now,
\[\nabla \times \vec{q}=\begin{vmatrix}\hat{i} & \hat{j} & \dot{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ z-\frac{2 x}{r} & 2 y-3 z-\frac{2 y}{r} & x-3 y-\frac{2 z}{r}\end{vmatrix}\]So,
\[\begin{aligned} \nabla \times \vec{q}=& \hat{i}\left(-3+\frac{2 z}{r^{2}} \times \frac{y}{r}+3-\frac{2 y}{r^{2}} \times \frac{z}{r}\right)-\hat{j}\left(1+\frac{2 z}{r^{2}} \times \frac{x}{\sigma}-1-\frac{2 x}{r^{2}} \times \frac{z}{r}\right) \\ &+\hat{k}\left(+\frac{2 y}{r^{2}} \times \frac{x}{r}-\frac{2 x}{r^{2}} \times \frac{y}{\sigma}\right) \\ =& 0 \end{aligned}\]\(\therefore\) fluid does not possess vorticity.
(ii) Calculation of circulation:
Given,
\(u=z-\frac{2x}{r}\), \(v=2y-3z-\frac{2y}{r}\), \(w=x-3y-\frac{2z}{r} \ldots (1)\)
\(C: x^2+y^2=9, z=0\)
Let \(x= 3 \cos \theta\), \(y= 3 \sin \theta\), \(z=0 \ldots (2)\)
\(\implies\) \(dx= -3 \sin \theta d \theta\), \(dy= -3 \cos \theta d \theta\), \(dz=0 \ldots (3)\)
Now, the circulation around the circle \(C\) is given by:
\(\Gamma=\int_{C} \vec{q} \cdot d \vec{r}=\int_{0}^{2 \pi} \vec{q} \cdot \overrightarrow{d r}\)
\(\implies \Gamma=\int_{0}^{2 \pi} u d x+v d y+w d z\)
\(\implies \Gamma = \int_{0}^{2 \pi} (0-2 \cos \theta)(-3 \sin \theta d \theta) + (6 \sin \theta - 2 \sin \theta - 2 )(3 \cos \theta d \theta) + 0\)
\(\implies \Gamma= \int_{0}^{2 \pi} 6 \sin \theta \cos \theta d \theta + 12 \sin \theta \cos \theta d \theta\)
\(\implies \Gamma= \int_{0}^{2 \pi} (2 \sin \theta \cos \theta \d \theta)\)
\(\implies \Gamma = \int_{0}^{2 \pi} \sin 2 \theta d \theta\)
\(\implies \Gamma=0\)
Thus, the circulation is 0.
5.(c) Consider single free particle of mass \(m\), moving in space under no forces. If the particle starts form the origin at \(t=0\) and reaches the position \((x, y, z)\) at time \(\tau\), find the Hamiton’s characteristic function \(S\) as a function of \(x, y, z, \tau\).
[10M]
5.(d) Convert the following decimal numbers to univalent binary and hexadecimal numbers:
(i) 4096,
(ii) 0.4375,
(iii) 2048.0625
[10M]
5.(e) Final the general integral of the partial differential equation \((y+z x) p-(x+y z) q=x^{2}-y^{2}\).
[10M]
6.(a) Determine the characteristics of the equation \(z=p^{2}-q^{2}\) and find the integral surface which passes though the parabola \(4 z+x^{2}=0\).
[15M]
6.(b) A simple source of strength \(m\) is fixed at the origin \(O\) in a uniform stream of incompressible fluid moving with velocity \(U \vec{i}\), show that the velocity potential \(\phi\) at any point \(P\) of the stream is \(\dfrac{m}{r}-U r \cos \theta\), where \(O P=r\) and \(\theta\) is the angle which \(OP\) makes with the direction \(i\). Find the differential equation of the streamlines and show that the lie on the surfaces \(U r^{2} \sin ^{2} \theta-2 m \cos \theta=\) constant.
[15M]
Given, strength of source = \(m\).
\(\implies\) The complex potential at any point \(z\) is given by, \(w=-m \log z\)
\(\begin{aligned} \therefore \frac{d w}{d z} &=\frac{-m}{z}=\frac{-m}{r e^{i \theta}}=\frac{-m}{r} e^{-i \theta} \\ \implies \frac{d w}{d z} &=\frac{-m}{r}(\cos \theta-i \sin \theta)=\frac{-m \cos \theta}{r}+\frac{i m \sin \theta}{r} \end{aligned}\) \(\implies \frac{d w}{d z}=\frac{-m x}{r^{2}}+i \frac{m y}{r^{2}}=-u_{m}+i v_{m}\)
where \(u_{m}\) and \(v_{m}\) are velocity components due to source at \(m\). \(\therefore u_{m}=\frac{m x}{r^{2}}, v_{m}=\frac{m y}{r^{2}}\)
Also, fluid is moving with velocity \(U \hat{i}\). \(\therefore\) Total velocity. \(\vec{q}=\left(U+\frac{m x}{r^{2}}\right) \hat{i}+\left(\frac{m y}{r^{2}}\right) \hat{j}=u \hat{i}+\hat{j} v\) Let \(\phi\) be the velocity potential. \(\therefore u=\frac{-\partial \phi}{\partial x}\)
\[\Rightarrow \frac{-\partial \phi}{\partial x}=U+\frac{m x}{r^{2}}\] \[\Rightarrow d \phi=\left(-U-\frac{m x}{r^{2}}\right) d x\]Integrating both sides, we get \(f=-U x-m \log r+f(y)\) where, \(f(y)\) is arbitrary function of \(y\) Also, \(\frac{-\partial \phi}{\partial y}=v=\frac{m y}{r^{2}} \ldots (i)\)
\[\Rightarrow d \phi=\frac{-m y}{r^{2}} d y\] \[\implies \phi=-m \log r+g(x) \ldots (ii)\]where \(g(x)\) is an arbitrary function of \(x .\) From (i) and (ii) \(\phi=-U x-m \log r\)
\[\Rightarrow \phi=-U r \cos \theta-m \log r\]Now, equation of streamlines is given by:
\[\frac{d x}{u}=\frac{d y}{v}\] \[\Rightarrow \frac{d x}{u+\frac{m x}{r^{2}}}=\frac{d y}{\frac{m y}{r^{2}}}\]\(\Rightarrow \frac{d y}{d x}=\frac{\frac{m y}{r^{2}}}{\left(U+\frac{m x}{r^{2}}\right)}\), which is the differential equation of streamlines.
6.(c) Let \(f(x)=e^{2 x} \cos 3 x\) for \(x \in[0,1]\). Estimate the value of \(f(0.5)\) using Lagrange interpolating polynomial of degree 3 over the nodes \(x=0\), \(x=0.3\), \(x=0.6\) and \(x=1\). Also compute the error bound over the interval \([0,1]\) and the actual error \(E(0.5)\).
[20M]
7.(a) Solve the partial differential equation \(\dfrac{\partial^{3} z}{\partial x^{3}}-2 \dfrac{\partial^{3} z}{\partial x^{2} \partial y} \dfrac{\partial^{3} z}{\partial x \partial y^{2}}+2 \dfrac{\partial^{3} z}{\partial y^{3}}=e^{x+y}\).
[15M]
7.(b) The space between two concentric spherical shells of radii \(a\), \(b(a < b)\) is filled with a liquid of density \(\rho\). If the shells are set in motion, the inner one with velocity \(U\) in the \(x-direction\) and the outer one with velocity \(V\) in the \(y- direction\), then show has the initial motion of the liquid is given by velocity potential \(\phi=\dfrac{\left \{ a^{3} U\left ( 1+\dfrac{1}{2} b^{3} r^{-3}\right ) x-b^{3} V \left ( 1+\dfrac{1}{2} a^{3} r^{-3}\right ) y\right\}}{\left (b^{3}-a^{3}\right )}\), where \(r^{2}=x^{2}+y^{2}+z^{2}\), the coordinate being rectangular. Evaluate the velocity at any point of the liquid.
[20M]
Since the motion is irrotational, consequently, the corresponding velocity potential \(\phi\) exists such that:
\[\nabla^2 \phi =0 \ldots (1)\]The boundary conditions for \(\phi\) are:
\[\left( \frac{- \partial \phi}{\partial r} \right)_{r=a} = U \cos \theta \ldots (2)\] \[\left( \frac{- \partial \phi}{\partial r} \right)_{r=b} = V \sin \theta \ldots (3)\]
The above equations suggest that \(\phi\) must involve terms containing \(\sin \theta\) and \(\cos \theta\). So, we take \(\phi\) as:
\(\phi=\left(A r+\frac{B}{r^{2}}\right) \cos \theta+\left(Cr+\frac{D}{r^{2}}\right) \sin \theta \ldots (3)\) \(\implies \frac{-\partial \phi}{\partial r}=\left(-A+\frac{2 B}{r^{3}}\right) \cos \theta+\left(-C+\frac{2 D}{r^{3}}\right) \sin \theta \ldots (4)\)
Using boundary condiitons for (1) and (2) in (4), we get:
\(\left(-A+\frac{2 B}{a^{3}}\right) \cos \theta+\left(-C+\frac{2 D}{a^{3}}\right) \sin \theta=U \cos \theta \ldots (5)\) \(\left(-A+\frac{2 B}{b^{3}}\right) \cos \theta+\left(-C+\frac{2 D}{b^{3}}\right) \sin \theta=V \sin \theta \ldots (6)\)
Comparing coefficients of \(\cos \theta\) and \(\sin \theta\) in (5) and (6), we get:
\(\frac{-A+2 B}{a^{3}}=U, \frac{-C+2 D}{a^{3}}=0 \ldots (7)\), and
\(\frac{-A+2 B}{b^{3}}=0, \frac{-C+2 D}{b^{3}}=\mathrm{V} \ldots (8)\)
Solving (7) and (8), we get:
\(A=\frac{U a^{3}}{b^{3}-a^{3}}\), \(B=\frac{U a^{3} b^{3}}{2\left(b^{3}-a^{3}\right)}\), \(C=\frac{-U b^{3}}{b^{3}-a^{3}}\), \(D=\frac{-U a^{3} b^{3}}{2\left(b^{3}-a^{3}\right)}\)
Putting these values in (3), we get:
\(\phi=\left(\frac{U a^{3} r}{b^{3}-a^{3}}+\frac{u a^{3} b^{3}}{2\left(b^{3}-a^{3}\right) r^{2}}\right) \cos \theta+\left(\frac{-U b^{3}}{b^{3}-a^{3}} \cdot r-\frac{U a^{3} b^{3}}{2\left(b^{3}-a^{3}\right) r^{2}}\right) \sin \theta\)
\(\implies \phi=\frac{\left\{U a^{3}\left(1+\frac{b^{3}}{2 r^{3}}\right) r \cos \theta-V b^{2}\left(1+\frac{a^{3}}{2 r^{3}}\right) r \sin \theta\right\}}{\left(b^{3}-a^{3}\right)}\)
\(\implies \phi=\frac{\left\{a^{3} U\left(1+\frac{b^{3} r^{-3}}{2}\right) x-b^{3} V\left(1+\frac{a^{3} r^{-3}}{2}\right) y\right\}}{b^{3}-a^{3}}\)
Now, let velocity at any point= \(\vec{q} = u\hat{i}+v\hat{j}\) where, \(u=\frac{-\partial \phi}{\partial x}\) and \(v=\frac{-\partial \phi}{\partial y}\)
Therefore,
\(u=\frac{-\partial \phi}{\partial x}=\frac{-a^{3} U\left(1+\frac{b^{3} r^{-3}}{2}\right)}{b^{3}-a^{3}}-\frac{a^{3} U x\left(\frac{-3}{2} b^{3} r^{-4} \frac{x}{r}\right)}{\left(b^{3}-a^{3}\right)}\) + \(\frac{b^{3} V y\left(-\frac{3}{2} a^{3} r^{-4}, \frac{x}{r}\right)}{b^{3}-a^{3}}\) \(\implies u=\frac{-a^{3} U\left(1+\frac{b^{3} r^{-3}}{2}\right)}{b^{3}-a^{3}}+\frac{\frac{3}{2} a^{3} b^{3} x^{2} U r^{-5}}{\left(b^{3}-a^{3}\right)}-\frac{\frac{3}{2} a^{3} b^{3} x y v r^{-5}}{b^{3}-a^{3}}\)
Also,
\[\implies u=\frac{-a^{3} u\left(1+b^{3} r^{-3}\right)}{\left(b^{3}-a^{3}\right)}+\frac{3 a^{3} b^{3} x^{2} U r^{-5}}{2\left(b^{3}-a^{3}\right)}-\frac{3 a^{3} b^{3} x y V r^{-5}}{2\left(-b^{3}-a^{3}\right)} \ldots (9)\] \[v=-\frac{\partial \phi}{\partial y}=\frac{-a^{3} U x\left(-\frac{3}{2} b^{2} r^{-4} \frac{y}{r}\right)}{b^{3}-a^{3}}+\frac{b^{3} V\left(1+\frac{1}{2} a^{3} r^{-3}\right)}{b^{3}-a^{3}} +\frac{b^{3} V y\left(-\frac{3}{2} a^{3} r^{-4} \frac{y}{r}\right)}{b^{3}-a^{3}}\] \[\implies v=\frac{-3 a^{3} b^{3} x y U r^{-5}}{2\left(b^{3}-a^{3}\right)}+\frac{b^{3} v\left(1+\frac{1}{2} a^{3} r^{-3}\right)}{b^{3}-a^{3}}-\frac{3 a^{3} b^{3} y^{2} V r^{-5}}{2\left(b^{3}-a^{3}\right)} \quad \ldots(10)\]\(\therefore\) velocity is given by:
\(\vec{q}=u \hat{i}+\hat{j}\), where \(u\) and \(v\) are given by (9) and (10).
7.(c) For an integral \(\int_{-1}^{1} f(x) d x\), show that the two point Gauss quadrature rule is given by \(\int_{-1}^{1} f(x) d x=f\left(\dfrac{1}{\sqrt{3}}\right)+f\left(-\dfrac{1}{\sqrt{3}}\right)\). Using this rule, estimate \(\int_{2}^{4} 2 x e^{x} d x\).
[15M]
8.(a) Find the temperature \(u(x, t)\) in a bar of silver of lengtant cross section of area 1\(c m^{2}\). Let density \(p=10.6 g / c m^{3}\), thermal conductivity \(K=1.04 /\left(c m \sec ^{\circ} C\right)\) and specific heat \(\sigma=0.056 / g^{\circ} C\) the bar is perfectly isolated laterally with ends kept at \(0^{\circ} C\) and initial temperature \(f(x)=\sin (0.1 \pi x)^{\circ} C\) note that \(u(x, t)\) follows the head equation \(u_{t}=c^{2} u_{x x}\) where \(c^{2}=k /(\rho \sigma)\).
[20M]
8.(b) A hoop with radius \(r\) is rolling, without slipping, down an inclined plane of lengthland with angle of inclination \(\phi\). Assign appropriate generalized coordinate to the system. Determine the constraints, if any. Write down the Lagrangian equation for the system. Hence or otherwise determine the velocity of the hoop, at the bottom of the inclined plane.
[15M]
8.(c) Let \(A\), \(B\), \(C\) be Boolean variable denote complement \(A\), \(A+B\) of is an expression for \(A\) OR \(B\) and \(B.A\) is an expression for \(A\) AND \(B\). Then simplify the following expression and draw a block diagram of the simplified expression using AND and OR gates.
\(A .(A+B, C) .(\overline{A}+B+C) \cdot(A+\overline{B}+C) .(A+B+\overline{C})\).
[15M]