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 \(f: [0,1]\to [0,1]\) is continuous on [0,1]. Prove that there exists a point c in [0,1] such that \(f(c) =c\).

[10M]

1.(c) If \(u=(x-1)^3-3xy^2+3y^2\), 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 \(Z=3x_1+2x_2+5x_3\)\ subjected to the constraints\ \(x_1+2x_2+x_3\leq430\)\ \(3x_1+2x_3\leq460\)\ \(x_1+4x_2\leq420\) \(x_1,x_2,x_3\leq0\)

[10M]

2.(a) Find all the homomorphism from the group \(({\mathbb{Z}},+)\) to \(({\mathbb{Z}}_4 ,+)\)

[10M]

2.(b) Consider the function \(f\) defined by \(f(x,y) = \begin{cases} xy\dfrac{x^2-y^2}{x^2+y^2} & \text{where}\; x^2+y^2\neq 0 \\ 0 & \text{where}\; x^2+y^2= 0 \end{cases}\)

Show that \(f_{xy}\neq f_{yx}\) at (0,0).

[10M]

2.(c) Prove that \(\int_{0}^{\infty} cos x^2 \,dx=\int_{0}^{\infty} sin x^2 \,dx=\dfrac{1}{2}\sqrt{\dfrac{\pi}{2}}\)

[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 \(x^2+y^2+z^2\) 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 \(\int_{0}^{2\pi} cos^{2n} \theta \,d\theta\) where n is a positive integer.

[10M]

3.(d) Show that the improper integral \(\int_{0}^{1} \dfrac{sin\dfrac{1}{x}}{\sqrt x} \,dx\) is convergent.

[10M]

4.(a) Show that \(\iint_{R} x^{m-1}y^{n-1}(1-x-y)^{l-1} \,dxdy=\dfrac{ \Gamma (l) \Gamma (m) \Gamma (n)}{ \Gamma (l+m+n)};\) l,m,n>0 taken over R : the triangle bounded by \(x=0,y=0,x+y=1\).

[10M]

4.(b) Let \(f_n(x)=\dfrac{x}{n+x^2},\; x\in [0,1].\) Show that the sequence \({f_n}\) 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 \(S_1,S_2\;S_3\) and the requirements of four destination \(D_1,D_2,D_3\;\)and \(D_4\) 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 \(\rho\) be the density and \(v\) be the velocity at a distance \(x\) from a fixed point at time \(t\), then \(\dfrac{\partial^2 \rho}{\partial t^2 }=\dfrac{\partial^ 2 }{\partial x^2}{\rho(v^2+k)}\).

[10M]

6.(a) Find the complete integral of the partial differential equation \((p^2+q^2)x=zp\) and deduce the solution which passes through the curve \(x=0,z^2=4y\). Here,\(p=\dfrac{\partial z}{\partial x},\;q=\dfrac{\partial z }{\partial y}\).

[10M]

6.(b) Apply fourth-order Runge-Kutta Method to compute \(y\) at \(x=0.1\;and\; x=0.2\) given that \(\dfrac{dy}{dx}=x+y^2,\;y=1\) at x=0.

[10M]

6.(c) For a particle having charge q and moving in an electromagnetic field, the potential energy is \(U=q(\phi -\vec{v}\cdot \vec{A})\) where \(\phi\) and \(\vec{A}\) 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 \(\int^{10}_{0} e^{-x^2}\, dx\) with 10 subdivisions.

[10M]

7.(a) Solve \((z^2-2yz-y^2)p+(xy+zx)q=xy-zx\), where \(p=\dfrac{\partial z}{\partial x},\;q=\dfrac{\partial z }{\partial y}\)\ 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 \(2^{\prime}\) 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 \(2m\dot u\), 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 \(\dfrac{\partial^2 V}{\partial x^2 }+\dfrac{\partial^ V }{\partial y^2}=-4\pi(x^2+y^2)\) reducing to zero, where \(y=0\).

[10M]

8.(b) The equation \(x^6-x^4-x^3-1=0\) 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 \(\alpha\) 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 \((x^2+y^2)^2=a^2(x^2-y^2+\lambda xy)\), where \(\lambda\) is a variable parameter. Show also that the fluid speed at any point is \((2ma^2)/(r_1 r_2 r_3)\). where \(r_1,r_2,r_3\) are the distances of the point from the sources and the sink.

[10M]