Paper II PYQs-2017

1.(a) Prove that every group of order four is Abelian.

[10M]

1.(b) A function \(f: {\mathbb{R}}\to {\mathbb{R}}\) is defined as below: \(f(x) = \begin{cases} x & \text{,if}\; x\; is\; rational \\ 1-x & \text{,if}\; x\; is\; irrational \end{cases}\)
Prove that \(f\) is continuous at \(x=\dfrac{1}{2}\) but distinuous at all the other points in \({\mathbb{R}}\).

[10M]

1.(c) If \(f(z)=u(x,y)+iv(x,y)\) is an analytic function of \(z=x+iy\) and \(u+2v=x^3-2y^3+3xy(2x-y)\) then find \(f(z)\) in terms of z.

[10M]

1.(d) Solve by simple method the folllowing LPP:
Minimize \(Z=x_1-3x_2+2x_3\) subject to the constraints: \(3x_1-x_2+2x_3\leq 7\)
\(-2x_1+4x_2\leq 12\)
\(-4x_1+3x_2+8x_3\leq 0\), and \(x_1,x_2,x_3\geq 0\)

[10M]

2.(a) Let \(G\) be the set of all real numbers except -1 and define \(a*b=a+b+ab \;\forall a,b\in G\). Examine if \(G\) is an Abelian group under \(*\).

[10M]

2.(b) Let \(H\) and \(K\) are two finite normal subgroups of co-prime order of a group \(G\). Prove that \(hk=kh \;\forall h\in H\;and\; k\in K\).

[10M]

2.(c) Let \(A\) be an ideal of commutative ring \(R\) and \(B\)={\(x\in R\;:x^n\in A\) for some positive integer \(n\)}. Is \(B\) an ideal \(R\)? Justify your answer.

[10M]

2.(d) Prove that the ring \({\mathbb{Z}}[i]=\{a+ib\;:a,b\in {\mathbb{Z}}\), \(i=\sqrt{-1}\}\) of Gaussian integers is a Euclidean domain.

[10M]

3.(a) Evaluate \(f_{xy}(0,0)\) and \(f_{yx}(0,0)\) given that\ \(f(x,y) = \begin{cases} x^2 tan^{-1}\dfrac{y}{x}-y^2\dfrac{x}{y} & \text{,if}\; xy\neq 0 \\ 0 & \text{,if}\; otherwise \end{cases}\)\

[10M]

3.(b) Find the maximum and minimum values of \(x^2+y^2+z^2\) subject to the condition \(\dfrac{x^2}{4}+\dfrac{y^2}{5}+\dfrac{z^2}{25}=1\).

[10M]

3.(c) Prove that \(\int^{\infty}_0 \dfrac{sin x}{x}\, dx\) is convergent but not absolutely convergent.

[10M]

3.(d) Find the volume of the region common to the cylinder \(x^2+y^2=a^2\) and \(x^2+z^2=a^2\).

[10M]

4.(a) Prove by the method of contour integration that \(\int^{\pi}_0 \dfrac{1+2cos\theta}{5+4cos\theta}\, \theta=0\)

[10M]

4.(b) Find the sum of residues of \(f(z)=\dfrac{sinz}{cosz}\) at its poles inside the circle \(\vert z\ vert=2\).

[10M]

4.((c) Evaluate \(\int^{\infty}_{x=0} \int^{\infty}_{y=0} xe^{-x^2/y}\, dydx\).

[10M]

4.(d) A computer centre has four expert programmers. The centre needs four application programs to be developed. The head of the centre after studying carefully the programs to be developed, estimates the computer times in hours required by the experts to the application programs as follows:

Here, \(P_i\) represents Programmers \(i=1,2,3,4\). Assign the programs to the programmers in such a way that total computer time is least.

[10M]

5.(a) Form the partial differential equation by eliminating arbitary function \(\varphi\) and \(\psi\) from the relation \(z=\varphi(x^2-y)+\psi(x^2+y)\).

[10M]

5.(b) Write a BASIC program to compute the multiplicative inverse of a non-singular square matrix.

[10M]

5.(c) A uniform rectangular parallelopiped of mass \(M\) has edges of length \(2a,2b,2c\). Find the moment of interia of this rectangular parallelopipedabout the line through its centre parallel to the edge of length \(2a\).

[10M]

5.(d) Evaluate \(\int^1_0 e^{-x^2}\, dx\) using the composite trapezoidal rule with four decimal precision, i.e., with the absolute value of the error not exceeding \(5\times 10^{-5}\).

[10M]

6.(a) Solve the partial differential equation: \((x-y)\dfrac{\partial z}{\partial x }+(x+y)\dfrac{\partial z}{\partial y }=2xz\).

[10M]

6.(b) Find the surface which is orthogonal to the family of surface \(z(x+y)=c(3z+1)\) and which passes through the circle \(x^2+y^2=1, z=1\).

[10M]

6.(c) Find complete integral of \(xp-yq=xqf(z-px-qy)\) where \(p=\dfrac{\partial z}{\partial x },\;q=\dfrac{\partial z}{\partial y }\).

[10M]

6.(d) A tightly stretched string with fixed end points \(x=0\; and \; x=l\) is initially in a position given by \(y=y_osin^3(\dfrac{\pi x}{l})\). It is released from rest from this position, find the the displacement \(y(x,t)\).

[10M]

7.(a) Find the roots of the equation \(x^3+x^2+3x+4=0\) correct up to five places of decimal using Newton-Raphhson method.

[10M]

7.(b) A river is 80 metre wide, the depth \(y\), in metre of the river at a distance \(x\) from one bank is given by the following table:

Find the area of cross-section of the river using Simpson’s \(\dfrac{1}{3}\)rd rule.

[10M]

7.(c) Find \(y\) for \(x=0.2\) taking \(h=0.1\) by modified Euler’s method and compute the error, given that: \(\dfrac{dy}{dx}=x+y\); \(y(0)=1\).

[10M]

7.(d)Assuming a 32 bit computer representation of signed integers using 2’s complement representation, add the two numbers -1 and -1024 and give the answer in 2’s complement representation.

[10M]

8.(a) Consider a mass \(m\) on the end of a spring of natural length \(l\) and spring constant \(k\). Let \(y\) be the vertical coordinate of the mass as measured from the top of the spring. Assume that the mass can only move up and down in the vertical direction. Show that \(L=\dfrac{1}{2} my^{'^2}-\dfrac{1}{2}k(y-l)^2+mgy\).

Also determine and solve the corresponding Euler-Lagrange equation of motion.

[10M]

8.(b) Find the streamlines and pathlines of the two dimensional velocity field: \(u=\dfrac{x}{1+t},\; v=y,\;w=0\).

[10M]

8.(c) The velocity vector in the flow field is given by \(\vec{q}=(az-by)\hat{i}+(bx-cz)\hat{j}+(cy-ax)\hat{k}\), where \(a\), \(b\), \(c\) are non-zero constants. Determine the equations of vortex lines.

[10M]

8.(d) Solve Laplace’s Equation \(\dfrac{\partial^2 u}{\partial x^2 }+\dfrac{\partial^2 u}{\partial y^2 }=0\) subject to the conditions \(u(0,y)=u(l,y)=u(x,0)=0 \;and\; u(x,(a) =sin(\dfrac{n\pi x}{l})\).

[10M]