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 $u=(x-1{)}^3-3x{y}^2+3y^2$, 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\ $x_1+2x_2+x_3\le 430$\ $3x_1+2x_3\le 460$\ $x_1+4x_2\le 420$ $x_1,x_2,x_3\le 0$

[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 $f(x,y)=\{\begin{array}{ll}xy{\displaystyle \frac{x^2-y^2}{x^2+y^2}}& \text{where}{x}^2+y^2\ne 0\\ 0& \text{where}{x}^2+y^2=0\end{array}$

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

[10M]

2.(c) Prove that ${\int }_0^{\mathrm{\infty }}cos{x}^{2}dx={\int }_0^{\mathrm{\infty }}sin{x}^{2}dx={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\pi }{2}}}$

[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 ${\int }_0^{2\pi }co{s}^{2n}\theta d\theta$ where n is a positive integer.

[10M]

3.(d) Show that the improper integral ${\int }_0^1{\displaystyle \frac{sin{\displaystyle \frac{1}{x}}}{\sqrt{x}}}dx$ is convergent.

[10M]

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

[10M]

4.(b) Let $f_n(x)={\displaystyle \frac{x}{n+x^2}},x\in [0,1].$ 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 ${\displaystyle \frac{{\mathrm{\partial }}^2\rho }{\mathrm{\partial }{t}^2}}={\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}}\rho (v^2+k)$.

[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,$p={\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }x}},q={\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}}$.

[10M]

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

[10M]

6.(c) For a particle having charge q and moving in an electromagnetic field, the potential energy is $U=q(\varphi -\overrightarrow{v}\cdot \overrightarrow{A})$ where $\varphi$ and $\overrightarrow{A}$ 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 ${\int }_0^{10}{e}^{-x^2}dx$ with 10 subdivisions.

[10M]

7.(a) Solve $(z^2-2yz-y^2)p+(xy+zx)q=xy-zx$, where $p={\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }x}},q={\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }y}}$\ 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 $2^{\mathrm{\prime }}$ 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 ${\displaystyle \frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{x}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^V}{\mathrm{\partial }{y}^2}}=-4\pi (x^2+y^2)$ 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 $(2m{a}^2)/(r_1{r}_2{r}_3)$. where $r_1,r_2,r_3$ are the distances of the point from the sources and the sink.

[10M]