Paper II PYQs-2009

1.(a) If R is the set of real numbers and Rt is the set of positive real numbers, show that R under addition (R,+) and R+ under multiplication (R+,.) are isomorphic. Similarly if Q is set of rational numbers and Q+ is the set of positive rational numbers, are (Q,+) and (Q,.) isomorphic? Justify your answer.

[12M]

1.(b) Determine the number of homomorphisms from the additive group Z15 to the additive group Z10. ( Zn is the cyclic group of order n).

[4+8=12M]

1.(c) State Rolle’s theorem. Use it to prove that between two roots of excosx=1, there will be a root of exsinx=1.

[12M]

1.(d) Let f(x)={|x|2+1, if x<1x2+1, if 1≤x<2−|x|2+1, if 2≤x.

What are the points of discontinuity of f, if any? What are the points where f is not differentiable, if any? Justify your answers.

[12M]

1.(e) Let f(z)=a0+a1……+an−1zn−1b0+b1z+………+bnzn, bn≠0. Assume that the zeros of the denominator are simple. Show that the sum of the residues of f(z) at its poles is equal to an−1bn.

[12M]

1.(f) A paint factory produces both interior and exterior paint from materials $M_{1}$ and $M_{2}$ . The basic data is as follows:
Exterior PaintInterior PaintMax. Daily Availability Raw Material M16424 Raw Material M2122 Profit per ton (Rs. 1000) 54

A market survey indicates that the daily demand interior paint cannot exceed that of exterior paint by more than 1 ton. The maximum daily demand of interior paint is 2 tons. The factory wants to determine the optimum product mix of interior and exterior paint that maximizes daily profits. Formulate the LP problem for this situation.

[12M]

2.(a) How many proper, non-zero ideals does the ring Z12 have? Justify your answer. How many ideals does the ring Z12⊕Z12 have? Why?

[2+3+4+6=15M]

2.(b) Show that the alternating group of four letters A4 has no subgroup of order 6.

[15M]

2.(c) Show that the series (13)2+(1.43.6)2+….+(1.4.7…..(3n−2)3.6.9………..3n)2 till infinity, converges.

[15M]

2.(d) Show that if f:[a,b]→R is a continuous function, then f([a,b])=[c,d] for some real numbers c and d, c≤d.

[15M]

3.(a) Show that Z[X] is a unique factorization domain that is not a principal ideal domain (Z is the ring of integers). Is it possible to give an example of principal ideal domain that is not a unique factorization domain? (Z[X] is the ring of polynomials in the variable X with integer.

[15M]

3.(b) How many elements does the quotient ring Z5[X]X2+1 have? Is it an integral domain? Justify your answers.

[15M]

3.(c) Show that:

\Ltx→1∑∞n=1n2x2n4+x4= ∑∞n=1n2n4+1

Justify all steps of your answer by quoting the theorems you are using.

[15M]

3.(d) Show that a bounded infinite subset of R must have a limit point.

[15M]

4.(a) If α, β, γ are real numbers such that α2>β2+γ2 show that: ∫2π0dθα+βcosθ+γsinθ=2π√α2−β2−γ2.

[30M]

4.(b) Maximize: Z=3x1+5x2+4x3 subject to:

2x1+3x2≤8,
3x1+2x2+4x3≤15,
2x2+5x3≤10,
xi≥0

[30M]

5.(a) Show that the differential equation of all cones which have their vertex at the origin is px+qy=z. Verify that this equation is satisfied by the surface yz+zx+xy=0.

[12M]

5.(b)(i) Form the partial differential equation by elimination the arbitrary function f given by: f(x2+y2,z−xy)=0.

[6M]

5.(b)(ii) Find the integral surface of: x2p+y2p+z2=0 which passes through the curve: xy=x+y, z=1.

[6M]

5.(c)(i) The equation x2+ax+b=0 has two real roots α and β. Show that the iterative method given by: xk+1=−(axk+b)xk, k=0,1,2… is convergent near x=α, if |α|>|β|.

[6M]

5.(c)(ii) Find the values of two valued Boolean variables A, B, C, D by solving the following simultaneous equations:
¯A+AB=0AB+ACAB+A¯C+CD=¯CD
where ¯x represents the complement of x.

[6M]

5.(d)(i) Realize the following expressions by using NAND gates only:
g=(¯a+¯b+c)¯d(¯a+e)f, where ¯x represents the complement of x.

[6M]

5.(d)(ii) Find the decimal equivalent of (357.32)8.

[6M]

5.(e) The flat surface of a hemisphere of radius r is cemented to one flat surface of a cylinder of the same radius and of the same material. If the length of the cylinder be l and the total mass be m, show that the moment of inertia of the combination about the axis of the cylinder is given by: mr2(l2+415r)(l+2r3).

[12M]

5.(f) Two sources, each of strength m are placed at the point (−a,0), (a,0) and a sink of strength 2m is at the origin. Show that the stream lines are the curves: (x2+y2)2=a2(x2−y2+λxy) where λ is a variable parameter. Show also that the fluid speed at any point is (2ma2)/(r1r2r3), where r1r2 and r3 are the distance of the points from the source and the sink.

[12M]

6.(a) Find the characteristics of: y2r−x2t=0 where r and t have their usual meanings.

[15M]

6.(b) Solve: (D2−DD′−2D′2)z=(2x2+xy−y2)sinxy−cosxy where D and D′ represent ∂∂x and ∂∂y and ∂∂y.

[15M]

6.(c) A tightly stretched string has its ends fixed at x=0 and x=1. At time t=0, the string is given a shape defined by f(x)=μx(l−x), where μ is a constant, and then released. Find the displacement of any point x of the string at time t>0.

[30M]

7.(a) Develop an algorithm for Regula-Falsi method to find a root of f(x)=0 starting with two initial iterates x0 and x1 to the root such that sign (f(x0))≠sign(f(x1)). Take n as the maximum number of iterations allowed and epsilon be the prescribed error.

[30M]

7.(b) Using Lagrange interpolation formula, calculate the value of f(3) from the following table of values of x and f(x):
x012456f(x)114155619

[15M]

7.(c) Find the value of y(1.2) using Runge-Kutta fourth order method with step size h=0.2 from the initial value problem: y′=xy, y(1)=2.

[15M]

8.(a) A perfectly rough sphere of mass m and radius b, rests on the lowest point of a fixed spherical cavity a radius a. To the highest point of the movable sphere is attached a particle of mass m′ and the system is disturbed. Show that the oscillations are the same as of a simple pendulum of length (a−b)4m′+75mm+m′(2−ab).

[30M]

8.(b) An infinite mass of fluid is acted on by a force μr3/2 per unit mass directed to the origin. If initially the fluid is at rest and there is a cavity in the form of the sphere r=C in it, show the cavity will be filled up after an interval of time (25μ)12⋅C54.

[30M]