Paper II PYQs-2012

1.(a) How many elements of order 2 are there in the group of order 16 generated by \(a\) and \(b\) such that the order of \(a\) is 8, the order of \(b\) is 2 and \(b a b^{-1}=a^{-1}\).

[12M]

1.(b) Let
\(f_{n}(x)=\left\{\begin{array}{cl}{0,} & {\text { if } x<\dfrac{1}{n+1}} \\ {\sin \dfrac{\pi}{x},} & {\text { if } x<\dfrac{1}{n+1} \leq x \leq \dfrac{1}{n}} \\ {0,} & {\text { if } x>\dfrac{1}{n}}\end{array}\right.\)
Show that \(f_{n}(x)\) converges to a continuous function but not uniformly.

[12M]

1.(c) Show that the function defined by \(f(z)=\left\{\begin{array}{c}{\dfrac{x^{3} y^{5}(x+i y)}{x^{6}+y^{10}}, z \neq 0} \\ {0, \quad z=0}\end{array}\right.\) is not analytic at the origin though it satisfies Cauchy Riemann-Equations at the origin.

[12M]

1.(d) For each hour per day that Ashok studies mathematics, it yields him 10 marks and for eaach hour that he studies physics, it yields him 5 marks. He can study at most 14 hours a day and he must get at least 40 marks in each. Determine graphically how many hours a day he should study mathematics and physics each, in order to maximize his marks?

[12M]

1.(e) Show that the series \(\sum_{n=1}^{\infty}\left(\dfrac{\pi}{\pi+1}\right)^{n} n^{6}\) is convergent.

[12M]

2.(a) How many conjugacy classes does the permutation group \(S_{5}\) of permutation 5 numbers have? Write down one element in each class (preferably in terms of cycles).

[15M]

2.(b) Let \(f(x, y)=\left\{\begin{array}{ll}{\dfrac{(x+y)^{2}}{x^{2}+y^{2}},} & {\text { if }(x, y) \neq(0,0)} \\ {1,} & {\text { if }(x, y)=(0,0)}\end{array}\right.\).
Show that \(\dfrac{\partial f}{\partial x}\) and \(\dfrac{\partial f}{\partial y}\) exist at \((0,0)\) though \(f(x, y)\) is not continuous at \((0,0)\).

[15M]

2.(c) Use Cauchy integral formula to evaluate \(\int_{c} \dfrac{e^{3 z}}{(z+1)^{4}} d z\), where \(c\) is the circle \(\vert z=2\).

[15M]

2.(d) Find the minimum distance of the line given by the planes \(3 x+4 y+5 z=7\) and \(x-z=9\) and from the origin, by the method of Lagrange’s multipliers.

[15M]

3.(a) Is the ideal generated by 2 and \(X\) in the polynomial ring \(Z[X]\) of polynomials in a single variable \(X\) with coefficients in the ring of integers \(Z\), a principal ideal? Justify your answer.

[15M]

3.(b) Let \(f(x)\) be differentiable on \([0,1]\) such that \(f(1)=0\) and \(\int_{0}^{1} f^{2}(x) d x=1\). Prove that \(\int_{0}^{1} x f(x) f^{\prime}(x) d x=-\dfrac{1}{2}\).

[15M]

3.(c) Expand the function \(f(z)=\dfrac{1}{(z+1)(z+3)}\) in Laurent series valid for:
i) \(\quad 1 < \vert z \vert < 3\)
ii) \(\vert z \vert>3\)
iii) \(0< \vert z+1 \vert < 2\)
iv) \(\vert z \vert < 1\)

[15M]

3.(d) Evaluate by contour integration \(I=\int_{0}^{2 \pi} \dfrac{d \theta}{1-2a \cos \theta+a^{2}}\), \(a^{2} < 1\).

[15M]

4.(a) Describe the maximal ideals in the ring of Gaussian integers \(Z[i]=\left\{a+ bi \vert a, b \in Z\right\}\).

[20M]

4.(b) Give an example of a function \(f(x)\), that is not Riemann integrable but \(\vert f(x) \vert\) is Riemann integrable. Justify your answer.

[20M]

4.(c) By the method of Vogel, determine an initial basic feasible solution for the following transportation problem. Products \(P_{1}\), \(P_{2}\), \(P_{3}\) and \(P_{4}\) have to be sent of destinations \(D_{1}\), \(D_{2}\) and \(D_{3}\). The cost of sending product \(P_{i}\) to destinations \(D_{j}\) is \(C_{i j}\), where the matrix:
\(\left[C_{i j}\right]=\left[ \begin{array}{cccc}{10} & {0} & {15} & {5} \\ {7} & {3} & {6} & {15} \\ {0} & {11} & {9} & {13}\end{array}\right]\)
The total requirements of destinations \(D_{1}\), \(D_{2}\) and \(D_{3}\) are given by 45, 45, 95 respectively and the availability of the products \(P_{1}\), \(P_{2}\), \(P_{3}\) and \(P_{4}\) are respectively 25, 35, 55 and 70.

[12M]

5.(a) Solve the partial differential equation \(\left(D-2 D^{\prime}\right)\left(D-D^{\prime}\right)^{2} z=e^{x+y}\).

[12M]

5.(b) Use Newton-Raphson method to find the real root of the equation \(3 x=\cos x+1\) correct to four decimal places.

[12M]

5.(c) Provide a computer algorithm to solve an ordinary differential equation \(\dfrac{d y}{d x}=f(x, y)\) in the interval \([a, b]\) for \(n\) number of discrete points, where the initial value is \(y(a)=\alpha\), using Euler’s method.

[12M]

5.(d) Obtain the equations governing the motion of a spherical pendulum.

[12M]

5.(e) A rigid sphere of radius \(a\) is placed in a stream of fluid whose velocity in the undisturbed state is \(V\). Determine the velocity of the fluid at any point of the disturbed stream.

[12M]

6.(a) Solve the partial differential equation \(p x+q y=3 z\).

[20M]

6.(b) A string of length \(l\) is fixed at its ends. The string from the mid-point is pulled up to a height \(k\) and then released from rest. Find the deflection \(y(x, t)\) of the vibrating string.

[20M]

6.(c) Solve the following system of simultaneous equations, using Gauss-Seidel iterative method:

\(3 x+20 y-z=-18\)
\(20 x+y-2 z=17\)
\(2 x-3 y+20 z=25\)

[20M]

7.(a) Find \(\dfrac{d y}{d x}\) at \(x=0.1\) from the following data:

[20M]

7.(b) The edge \(r=\alpha\) of a circular plate is kept at temperature \(f(\theta)\). The plate is insulated so that there is no loss of heat from either surface. Find the temperature distribution in steady state.

[20M]

7.(c) In a certain examination, a candidate has to appear for one major & two major subjects. The rules for declaration of results are marks for major are denoted by \(M_{1}\) and for minors by \(M_{2}\) and \(M_{3}\). If the candidate obtains 75% and above marks in each of the three subjects, the candidate is declared to have passed the examination in first class with distinction. If the candidate obtains 60% and above marks in each of the three subjects, the candidate is declared to have passed the examination in first class. If the candidate obtains 50% or above in major, 40% or above in each of the two minors and an average of 50% or above in all the three subjects put together, the candidate is declared to have passed the examination in second class. All those candidates, who have obtained 50% and above in major and 40% or above in minor, are declared to have passed the examination. If the candidate obtains less than 50% in major or less than 40% in anyone of the two minors, the candidate is declared to have failed in the examinations. Draw a flow chart to declare the results for the above.

[20M]

8.(a) A pendulum consists of a rod of length $2a$ and mass \(m\); to one end of spherical bob of radius \(\dfrac{a}{3}\) and mass \(15m\) is attached. Find the moment of inertia of the pendulum:

(i) About an axis through the other end of the rod and at right angle to the rod.

(ii) About a parallel axis through the centre of mass of the pendulum. [Given: the centre of mass of the pendulum is \(\dfrac{a}{12}\) above the centre of the sphere].

[30M]

8.(b) Show that \(\phi=x f(r)\) is a possible form for the velocity potential for an incompressible fluid motion. If the fluid velocity \(\dot{q} \rightarrow 0\) as \(r \rightarrow \infty\), find the surfaces of constant speed.

[30M]