Paper II PYQs-2009

1.(a) Prove that a non-empty subset H of a group G is normal subgroup of G⇔ for all x,y∈H,g∈G,(gx)(gy)−1∈H

[10M]

1.(b) Show that the function f(x]=1x is not uniformly continuous [0,1].

[10M]

1.(c) Show that under the transformation w=z−iz+i real axis in the E plane is mapped into the circle |w|=1 What portion of the Z-plane corresponds to the interior of the circle?

[10M]

1.(d) If G is a finite Abelian group, then show that O(a,b) is a divisor of 1.c⋅m. of O(a),O(b)

[10M]

1.(e) Evaluate ∫C2z+1z2+zdz by Cauchy’s integral formula, where C is |z|=12

[10M]

2.(a) Find the dimensions of the largest rectangular parallelopiped that has three faces in the coordinate planes and one vertex in the plane xa+yb+zc=1

[10M]

2.(b) Determine the analytic function w=u+iv, if u=2sin2xe2y+e−2y−2cos2x

[10M]

2.(c) Find the multiplicative inverse of the element [2513] of the ring, M of all matrices of order two over the integers.

[10M]

3.(a) Evaluate ∬xy(x+y)dxdy over the area between y=x2 and y=x

[10M]

3.(b) Evaluate by contour integration ∫2π0dθ1−2asinθ+a2,0<a<1

[10M]

3.(c) Write the dual of the following LPP and hence, solve it by graphical method : Minimize Z=6x1+4x2 constraints 2x1+x2≥13x1+4x2≥1.5x1,x2≥0

[10M]

4.(a) Show that d(a)<d(ab), where a,b be two non-zero elements of a Euclidean domain R and b is not a unit in R

[10M]

4.(b) Show that a field is an integral domain and a non-zero finite integral domain is a field.

[10M]

4.(c) Solve by simplex method, the following LPP : Maximize Z=5x1+3x2 constraints 3x1+5x2≤155x1+2x2≤10x1,x2≥0

[10M]

5.(a) Find complete and singular integrals of (p2+q2)y=qz.

[10M]

5.(b) Obtain the iterative scheme for finding pth root of a function of single variable using Newton-Raphson method. Hence, find 7√277¯234 correct to four decimal places.

[10M]

5.(c) Convert the following binary numbers to the base indicated: (i) (10111011001⋅101110)2 to octal (ii) (10111011001⋅10111000)2 to hexadecimal (iii) (0⋅101)2 to decimal

[10M]

5.(d) A cannon of mass M, resting on a rough horizontal plane of coefficient of friction μ is fired with such a charge that the relative velocity of the ball and cannon at the moment when it leaves the cannon is u. Show that the cannon will recoil a distance (muM+m)212μg along the plane, m being the mass of the ball.

[10M]

5.(e) If the velocity of an incompressible fluid at the point (x,y,z) is given by (3xzr5,3yzr5,3z2−r2r5) where r2=x2+y2+z2, prove that the liquid motion is possible and that the velocity potential is cosθ/r2

[10M]

6.(a) A rod of length l with insulated sides, is initially at a uniform temperature u0. Its ends are suddenly cooled to 0∘C and are kept at that temperature. Find the temperature distribution in the rod at any time t.

[10M]

6.(b) Convert the following to the base indicated: (i) (266⋅375 ) 10 to base 8 (ii) (341⋅24)5 to base 10 (iii) (43⋅3125)10 to base 2

[10M]

6.(c) Draw the circuit diagram for ˉF=AˉBC+ˉCB using NAND to NAND logic long.

[10M]

6.(d) Using Runge-Kutta method, solve y′′=xy′2−y2 for x=0.2. Initial conditions are at x=0,y=1 and y′=0. Use four decimal places for computations.

[10M]

7.(a) Prove that the equation of motion of a homageneous inviscid liquid moving under conservative forces may be written as ∂→q∂t−→q×curl→q=−grad[pρ+12q2+→Ω]

[10M]

7.(b) Find the general solution of {D2−DD′−2D′2+2D+2D′}z=e2x+3y+xy+sin(2x+y)

[10M]

7.(c) From the following data x:182764y:1234 calculate y(20), using Lagrangian interpolation decimal points for computations.

[10M]

8.(a) A homogeneous sphere of radius a, rotating with angular velocity about hoizontal diameter, is gently placed on a table whose coefficient of friction is μ. Show that there will be slipping at the point of contact for a time 2ωa7μg and that then the sphere will roll with angular velocity 2ω7.

[10M]

8.(b) Derive composite 13rd Simpson’s rule. Hence, evaluate ∫0.60e−x2dx by taking seven ordinates. Tabulate the integrand for these ordinates to four decimal places.

[10M]

8.(c) Show that for an incompressible steady flow with constant viscosity, the velocity components u(y)=yUh+h22μ(−dpdx)yh(1−yh)v=0,w=0 satisfy the equations of motion, when the body force is neglected. h,U,dpdx are constants and p=p(x).

[10M]