Paper II PYQs-2018

1.(a) Prove that a non-commutative group of order 2n,where n is an odd prime must have a subgroup of order n.

[10M]

1.(b) A function f:[0,1]→[0,1] is continuous on [0,1]. Prove that there exists a point c in [0,1] such that f(c)=c.

[10M]

1.(c) If u=(x−1)3−3xy2+3y2, determine v so that u+iv is a regular functionof x+iy.

[10M]

1.(d) Solve by simplex method the following Linear Programming Problem:\ Maximize Z=3x1+2x2+5x3\ subjected to the constraints\ x1+2x2+x3≤430\ 3x1+2x3≤460\ x1+4x2≤420 x1,x2,x3≤0

[10M]

2.(a) Find all the homomorphism from the group (Z,+) to (Z4,+)

[10M]

2.(b) Consider the function f defined by f(x,y)={xyx2−y2x2+y2wherex2+y2≠00wherex2+y2=0

Show that fxy≠fyx at (0,0).

[10M]

2.(c) Prove that ∫∞0cosx2dx=∫∞0sinx2dx=12√π2

[10M]

2.(d) Let R be a commutative ring with unity. Prove that an ideal P of R is prime if and only if the quotient ring R/P is an integgral domain.

[10M]

3.(a) Find the minimum value of x2+y2+z2 subject to the condition ax+by+cz=p.

[10M]

3.(b) Show by an example that in a finite commutative ring, every maximal ideal need not be prime.

[10M]

3.(c) Evalute the integral ∫2π0cos2nθdθ where n is a positive integer.

[10M]

3.(d) Show that the improper integral ∫10sin1x√xdx is convergent.

[10M]

4.(a) Show that ∬Rxm−1yn−1(1−x−y)l−1dxdy=Γ(l)Γ(m)Γ(n)Γ(l+m+n); l,m,n>0 taken over R : the triangle bounded by x=0,y=0,x+y=1.

[10M]

4.(b) Let fn(x)=xn+x2,x∈[0,1]. Show that the sequence fn is uniformly convergent on [0,1].

[10M]

4.(c) Let H be a cyclic subgroup of a group G. If H be a normal subgroup of G, prove that every subgroup of H is a normal subgroup of G.

[10M]

4.(d) The capacities of three production facilities S1,S2S3 and the requirements of four destination D1,D2,D3and D4 and transportation costs inrupees are given in the following table:

Find the minimum transportation cost using Vogel’s Approximation Method (VAM).

[10M]

5.(a) Find the partia; differential equation of all planes which are at a constant distance a from the origin.

[10M]

5.(b) A solid of revolution is formed by rotating abput the x-axis,the area between the x-axis, the line x=0 and a curve through the points with the following coordinates:

Estimate the volume of the solid formed using Weddle’s rule.

[10M]

5.(c) Write a program in BASIC to multiply two matrices (checking for consistency for multiplication is required).

[10M]

5.(d) Air,obeying Boyle’s law ,is in motion in a uniform tube of a small section. Prove that if ρ be the density and v be the velocity at a distance x from a fixed point at time t, then ∂2ρ∂t2=∂2∂x2ρ(v2+k).

[10M]

6.(a) Find the complete integral of the partial differential equation (p2+q2)x=zp and deduce the solution which passes through the curve x=0,z2=4y. Here,p=∂z∂x,q=∂z∂y.

[10M]

6.(b) Apply fourth-order Runge-Kutta Method to compute y at x=0.1andx=0.2 given that dydx=x+y2,y=1 at x=0.

[10M]

6.(c) For a particle having charge q and moving in an electromagnetic field, the potential energy is U=q(ϕ−→v⋅→A) where ϕ and →A are, respectively, known as the scalar and vector potentials. Derive expression for Hamiltonian for the particle in the electromagnetic field.

[10M]

6.(d) Write a program in BASIC to implement trapezoidal rule to compute ∫100e−x2dx with 10 subdivisions.

[10M]

7.(a) Solve (z2−2yz−y2)p+(xy+zx)q=xy−zx, where p=∂z∂x,q=∂z∂y\ If the solution of the above equation represents a sphere,what will be the coordinates of its center?

[10M]

7.(b) The velocity v(km/min) of a moped is given at fixed interval of time(min) as below:

Estimate the distance covered during the time (use Simpson’s one-third rule).

[10M]

7.(c) Assuming a 16-bit computer representation of signed integers, represent -44 in 2′ s complement represcntation.

[10M]

7.(d) In the case of two-dimensional motion of a liquid streaming past a fixed circular disc, the velocity at infinity is u in a fixed distance, where u is a variable. Show that the maximum value of the velocity at any point of the fluid is 2u. Prove that the force necessary to hold the disc is 2m˙u, where m is the mass of the liquid displaced by the disc.

[10M]

8.(a) Find a real function V of x and y,satisfying ∂2V∂x2+∂V∂y2=−4π(x2+y2) reducing to zero, where y=0.

[10M]

8.(b) The equation x6−x4−x3−1=0 has one root between 1.4 and1.5. Find the root to four places of decimal by regula-falsi method.

[10M]

8.(c) A particle of mass m is constrained to move on the inner surface of a cone of semi-angle α under the action of gravity. Write the equation of constraint and mention the generalized coordinates. Write down the equation of motion.

[10M]

8.(d)Two sources each of strength m,are placed at the points (−a,0),(a,0) and a sink of strength 2m at the origin. Show that the streamlines 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 r1,r2,r3 are the distances of the point from the sources and the sink.

[10M]