Paper II PYQs-2010
Section A
1.(a) Let \(G=R-\{-1\}\) be the set of all real numbers omitting -1. Define the binary relation \(*\) on \(G\) by \(a * b=a+b+a b\). Show \((G, *)\) is a group and it is abelian.
[12M]
1.(b) Show that a cyclic group of order 6 is isomorphic to the product of a cyclic group of order 2 and a cyclic group of order 3. Can you generalize this? Justify.
[12M]
1.(c) Discuss the convergence of the sequence \(\left\{x_{n}\right\}\) where \(X_{n}=\dfrac{\sin \left(\dfrac{n \pi}{2}\right)}{8}\).
[12M]
1.(d) Define \(\left\{x_{n}\right\}\) by \(x_{1}=5\) and \(x_{n+1}=\sqrt{4+x_{n}}\) for \(n>1\). Show that the sequence converges to \(\left(\dfrac{1+\sqrt{17}}{2}\right)\).
[12M]
1.(e) Show that \(u(x, y)=2 x-x^{3}+3 x y^{2}\) is a harmonic function. Find a harmonic conjugate of \(u(x, y)\). Hence, find the analytic function \(f\) for which \(u(x, y)\) is the real part.
[12M]
1.(f) Construct the dual of the primal problem.
Maximize \(Z=2 x_{1}+x_{2}+x_{3}\), subject to the constraints:
\(x_{1}+x_{2}+x_{3} \geq 6\)
\(3 x_{1}-2 x_{2}+3 x_{3}=3\)
\(-4 x_{1}+3 x_{2}-6 x_{3}=3\)
\(x_{1}, x_{2}, x_{3} \geq 0\)
[12M]
2.(a) Let \(\left(R^{*}, .\right)\) be the multiplicative group of non-zero reals and \((G L(n, R), X)\) be the multiplicative group of \(n \times n\) non-singular real matrices. Show that the quotient group \(\dfrac{G L(n, R)}{S L(n, R)}\) and \(\left(R^{*}, .\right)\) are isomorphic where \(S L(n, R)=\{A \in G L(n, R) / \operatorname{det} A=1\}\). What is the center of \(G L(n, R)\)?
[15M]
2.(b) Let \(C=\{f : I=[0,1] \rightarrow R / f\) is continuous \(\}\). Show \(C\) is a commutative ring with 1 under point wise addition and multiplication. Determine whether \(C\)s is an integral domain. Explain.
[15M]
2.(c) Define the function
\(f(x)=\left \{ {\begin{array}{ll}x^{2} \sin \dfrac{1}{x}, & \text { if } x \neq 0 \\ {0,} & {\text { if } x=0}\end{array}} \right .\).
Find \(f^{\prime}(x)\). Is \(f^{\prime}(x)\) continuous at \(x=0\)? Justify your answer.
[15M]
2.(d) Consider the series \(\sum_{n=0}^{\infty} \dfrac{x^{2}}{\left(1+x^{2}\right)^{2}}\). Find the values of \(x\) for which it is convergent and also the sum function. Is the convergence uniform? Justify your answer.
[15M]
3.(a) Consider the polynomial ring \(Q[x]\). Show \(p(x)=x^{3}-2\) is irreducible over \(Q\). Let \(I\) be the ideal \(Q[x]\) in generated by \(p(x)\). Then show that \(\dfrac{Q[x]}{I}\) is a field and that each element of it is of the form \(a_{0}+a_{1} t+a_{2} t^{2}\) with \(\alpha_{0}, a_{1}, a_{2}\) in \(Q\) and \(t=x+I\).
[15M]
3.(b) Show that the quotient ring \(\dfrac{Z[i]}{1+3 i}\) is isomorphic to the ring \(\dfrac{Z}{10 Z}\) where \(Z[i]\) denotes the ring of Gaussian integers.
[15M]
3.(c) Let \(f_{n}(x)=x^{n}\) on \(-1 < x \leq 1\) for \(n=1,2 \ldots \ldots\). Find the limit function. Is the convergence uniform? Justify your answer.
[15M]
3.(d) Consider the series \(\sum_{n=0}^{\infty} \dfrac{x^2}{ (1+x^2)^n }\).
Find the values of \(x\) for which it is convergent and also the sum function. Is the convergence uniform? Justify your answer.
[15M]
4.(a)(i) Evaluate the line integral \(\oint f(z) d z\) where \(f(z)=z^{2}\), \(c\) is the boundary of the triangle with vertices \(A(0,0)\), \(B(1,0)\), \(C(1,2)\) in that order.
[7.5M]
4.(a)(ii) Find the image of the finite vertical strip \(R\): \(x=5\) to \(x=5\), \(-\pi \leq \gamma \leq \pi\) of \(z-plane\) exponential function.
[7.5M]
4.(b) Find the Laurent series of the function \(f(z)=\exp \left[\dfrac{\lambda}{2}\left(z-\dfrac{1}{z}\right)\right]\) as \(\sum_{n=-\infty}^{\infty} C_{n} z^{n}\) for 0, \(\vert z \vert<\infty\) \(n=0\), \(\pm 1\), \(\pm 2\), \(\ldots\) with \(\lambda\) a given complex number and taking the unit circle \(C\) given by \(z=e^{i \phi}(-\pi \leq \phi \leq \pi)\) as contour in this region.
[15M]
4.(c) Determine an optimal transportation programme so that the transportation cost of 340 tons of a certain type of material from three factories to five warehouses \(W_{1}\), \(W_{2}\), \(W_{3}\), \(W_{4}\), \(W_{5}\) is minimized. The five warehouses must receive 40 tons, 50 tons, 70 tons, 70 tons and 90 tons respectively. The availability of the material at \(F_{1}\), \(F_{2}\), \(F_{3}\) is 100 tons, 120 tons, 120 tons respectively. The transportation costs per ton factories to warehouses are given in the table below:
\(\begin{array}{|c|c|c|c|c|}\hline W_{1} & {W_{2}} & {W_{3}} & {W_{4}} & {W_{5}} \\ \hline F_{1} & {4} & {1} & {2} & {6} & {9} \\ {F_{2}} & {6} & {4} & {3} & {5} & {7} \\ {F_{3}} & {5} & {2} & {6} & {4} & {8} \\ \hline\end{array}\)
Use Vogel’s approximation method to obtain the initial basic feasible solution.
[30M]
Section B
5.(a) Solve the PDE \(\left(D^{2}-D^{\prime}\right)\left(D-2 D^{\prime}\right) Z=e^{2 x+y}+x y\).
[12M]
5.(b) Find the surface satisfying the PDE \(\left(D^{2}-2 D D^{\prime}+D^{\prime 2}\right) Z=0\) and the conditions that \(b Z=y^{2}\) when \(x=0\) and \(a Z=x^{2}\) when \(y=0\).
[12M]
5.(c) Find the positive root of the equation \(10 x e^{-x^{2}}-1=0\) correct up to 6 decimal places by using Newton-Raphson method. Carry out computations only for three iterations.
[12M]
5.(d)(i) Suppose a computer spends 60 per cent of its time handling a particular type of computation when running a given program and its manufacturers make a change that improves its performance on that type of computation by a factor of 10. If the program takes 100 sec to execute, what will its execution time be after the change?
[6M]
5.(d)(ii) If \(A \oplus B=A B^{\prime}+A^{\prime} B\), find the value of \(x \oplus y \oplus z\).
[6M]
5.(e) A uniform lamina is bounded by a parabolic are of latus rectum 4$a$ and a double ordinate at a distance $b$ from the vertex. If \(b =\dfrac{a}{3}(7+4 \sqrt{7})\), show that two of the principal axis at the end of a latus rectum are the tangent and normal there.
[12M]
5.(f) In an incompressible fluid the vorticity at every point is constant in magnitude and direction, show that the components of velocity \(u\), \(v\), \(w\) are solutions of Laplace’s equation.
[12M]
6.(a) Solve the following partial differential equation, \(z p+y q=x\), \(x_{0}(s)=s\), \(y_{0}(s)=1\), \(z_{0}(s)=2 s\), by the method of characteristics.
[20M]
6.(b) Reduce the following \(2^{nd}\) order partial differential equation into canonical form and find find its general solution.
\(x u_{x x}+2 x^{2} u_{x y}-u_{x}=0\).
[20M]
6.(c) Solve the following heat equation
\(\begin{array}{ll}{u_{t}-u_{x x}=0,} & {0 < x< 2, t > 0} \\ {u(0, t)=u(2, t)=0} & {t > 0} \\ {u(x, 0)=x(2-x),} & {0 \leq x \leq 2}\end{array}\).
[20M]
7.(a) Given the system of equations:
\(2x+3y=1\)
\(2x+4y+z=2\)
\(2x+6z+Aw=4\)
\(4z+Bw=C\)
State the solvability and uniqueness conditions for the system. Give the solution when it exists.
[20M]
7.(b) Find the value of the integral \(\int_{1}^{\infty} \log _{10} x d x\) by using Simpson’s \(\dfrac{1}{3}\)rd rule, correct up to 4 decimal places. Take 8 subintervals in your computation.
[20M]
7.(c)(i) Find the hexadecimal equivalent of the decimal number \((587632)_{10}\).
[5M]
7.(c)(ii) For the given set of data points \((x_1, f(x_1))\), \((x_2, f(x_2))\), \(\cdots\) \((x_n, f(x_n))\), write an algorithm to find the value of \(f(x)\) by using Lagrange’s interpolation formula.
[10M]
7.(c)(iii) Using Boolean algebra, simplify the following expressions:
(a) \(a+a^{\prime} b+a^{\prime} b^{\prime} c+a^{\prime} b^{\prime} c^{\prime} d+\ldots\)
(b) \(x^{\prime} y^{\prime} z+y z+x z\), where \(x'\) represents the complement of \(x\).
[5M]
8.(a) A sphere of radius \(a\) land mass \(m\) rolles down a rough plane inclined at an angle \(\alpha\) to the horizontal. If \(x\) be the distance of the point of contact of the sphere from a fixed point on the plane, find the acceleration by using Hamilton’s equation.
[30M]
8.(b) When a pair of equal and opposite rectilinear vortices are situated in a long circle cylinder at equal distance from its axis, show that the path of each vortex is given by the equation \(\left(r^{2} \sin ^{2} \theta-b^{2}\right)\left(r^{2}-a^{2}\right)^{2}=4 a^{2} b^{2} r^{2} \sin ^{2} \theta\), \(\theta\) being measured from the line through the centre perpendicular to the joint of the vortices.
[30M]