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 bab−1=a−1.

[12M]

1.(b) Let
fn(x)={0, if x<1n+1sinπx, if x<1n+1≤x≤1n0, if x>1n
Show that fn(x) converges to a continuous function but not uniformly.

[12M]

1.(c) Show that the function defined by f(z)={x3y5(x+iy)x6+y10,z≠00,z=0 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 ∑∞n=1(ππ+1)nn6 is convergent.

[12M]

2.(a) How many conjugacy classes does the permutation group S5 of permutation 5 numbers have? Write down one element in each class (preferably in terms of cycles).

[15M]

2.(b) Let f(x,y)={(x+y)2x2+y2, if (x,y)≠(0,0)1, if (x,y)=(0,0).
Show that ∂f∂x and ∂f∂y exist at (0,0) though f(x,y) is not continuous at (0,0).

[15M]

2.(c) Use Cauchy integral formula to evaluate ∫ce3z(z+1)4dz, where c is the circle |z=2.

[15M]

2.(d) Find the minimum distance of the line given by the planes 3x+4y+5z=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 ∫10f2(x)dx=1. Prove that ∫10xf(x)f′(x)dx=−12.

[15M]

3.(c) Expand the function f(z)=1(z+1)(z+3) in Laurent series valid for:
i) 1<|z|<3
ii) |z|>3
iii) 0<|z+1|<2
iv) |z|<1

[15M]

3.(d) Evaluate by contour integration I=∫2π0dθ1−2acosθ+a2, a2<1.

[15M]

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

[20M]

4.(b) Give an example of a function f(x), that is not Riemann integrable but |f(x)| 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 P1, P2, P3 and P4 have to be sent of destinations D1, D2 and D3. The cost of sending product Pi to destinations Dj is Cij, where the matrix:
[Cij]=[10015573615011913]
The total requirements of destinations D1, D2 and D3 are given by 45, 45, 95 respectively and the availability of the products P1, P2, P3 and P4 are respectively 25, 35, 55 and 70.

[12M]

5.(a) Solve the partial differential equation (D−2D′)(D−D′)2z=ex+y.

[12M]

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

[12M]

5.(c) Provide a computer algorithm to solve an ordinary differential equation dydx=f(x,y) in the interval [a,b] for n number of discrete points, where the initial value is y(a)=α, 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 px+qy=3z.

[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:

3x+20y−z=−18
20x+y−2z=17
2x−3y+20z=25

[20M]

7.(a) Find dydx at x=0.1 from the following data:

[20M]

7.(b) The edge r=α of a circular plate is kept at temperature f(θ). 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 M1 and for minors by M2 and M3. 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 a3 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 a12 above the centre of the sphere].

[30M]

8.(b) Show that ϕ=xf(r) is a possible form for the velocity potential for an incompressible fluid motion. If the fluid velocity ˙q→0 as r→∞, find the surfaces of constant speed.

[30M]