Paper II PYQs-2016

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∈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,∞)→R given by f(x)=x2sin1x, 0<x<∞, show that there is a differentiable function g:R→R that extends f.

[10M]

1.(c) Two sequences {xn} and {yn} are defined inductively by the following:x1=12, y1=1, xn=√xn−1yn−1, n=2,3,4,…1yn=12(1xn+1yn−1), n=2,3,4,…. Prove that xn−1<xn<yn<yn−1, n=2,3,4,… and deduce that both the sequences converge to the same limit where 12<l<1.

[10M]

1.(d) Is v(x,y)=x3−3xy2+2y a harmonic function? Prove your claim. If yes, find its conjugate harmonic function u(x,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

with constraints

x+2y≥1, 2x+y≤1 x≥0 and y≥0

by graphical method.

[10M]

2.(a) Show that the series ∑∞n=1(−1)n+1n+1 is conditionally convergent. (If you use any theorem, (s) to show it, then you must give a proof of that theorem(s)).

[15M]

2.(b) Let p be prime number and Zp denote the additive group of integers modulo p. Show that that every non-zero element Zp generates Zp.

[15M]

2.(c) Maximize

subject to

2x1+x2+x3≤5 3x2+2x3≤6 x1≥0, x2≥0, x3≥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⊂K⊂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)=x4+y4−2x2+4xy−2y2.

[15M]

3.(c)Let γ:[0,1]→C be the curve γ(t)=e2πit, 0≤t≤1. Find, giving justifications, the value of the contour integral ∫γdz4z2−1.

[15M]

4.(a) Show that every algebraically closed field is infinite.

[15M]

4.(b) Let f:R→R be a continuous function such limx→+∞f(x) and limx→−∞f(x) exist and are finite. Prove that f is uniformly continuous on R.

[15M]

4.(c) Prove that every power series represents an analytic function inside its circle of convergence.

[20M]

5.(a) Find the general equation of surfaces orthogonal to the family of spheres given by x2+y2+z2=cz.

[10M]

5.(b) Does a fluid with velocity q=[z−2xr,2y−3z−2yr,x−3y−2zr] possess vorticity, where q(u,v,w) is the velocity in the cartesian frame →r(x,y,z) and r2=x2+y2+z2? What is the circulation in the circle x2+y2=9, z=0?

[10M]

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 τ, find the Hamiton’s characteristic function S as a function of x,y,z,τ.

[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+zx)p−(x+yz)q=x2−y2.

[10M]

6.(a) Determine the characteristics of the equation z=p2−q2 and find the integral surface which passes though the parabola 4z+x2=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→i, show that the velocity potential ϕ at any point P of the stream is mr−Urcosθ, where OP=r and θ 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 Ur2sin2θ−2mcosθ= constant.

[15M]

6.(c) Let f(x)=e2xcos3x for x∈[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 ∂3z∂x3−2∂3z∂x2∂y∂3z∂x∂y2+2∂3z∂y3=ex+y.

[15M]

7.(b) The space between two concentric spherical shells of radii a, b(a<b) is filled with a liquid of density ρ. 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 ϕ={a3U(1+12b3r−3)x−b3V(1+12a3r−3)y}(b3−a3), where r2=x2+y2+z2, the coordinate being rectangular. Evaluate the velocity at any point of the liquid.

[20M]

7.(c) For an integral ∫1−1f(x)dx, show that the two point Gauss quadrature rule is given by ∫1−1f(x)dx=f(1√3)+f(−1√3). Using this rule, estimate ∫422xexdx.

[15M]

8.(a) Find the temperature u(x,t) in a bar of silver of lengtant cross section of area 1cm2. Let density p=10.6g/cm3, thermal conductivity K=1.04/(cmsec∘C) and specific heat σ=0.056/g∘C the bar is perfectly isolated laterally with ends kept at 0∘C and initial temperature f(x)=sin(0.1πx)∘C note that u(x,t) follows the head equation ut=c2uxx where c2=k/(ρσ).

[20M]

8.(b) A hoop with radius r is rolling, without slipping, down an inclined plane of lengthland with angle of inclination ϕ. 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).(¯A+B+C)⋅(A+¯B+C).(A+B+¯C).

[15M]