Paper II PYQs-2014

1.(a) If \(G\) is a group in which \((a \cdot (b) ^{4}=a^{4} \cdot b^{4},(a \cdot (b) ^{5}=a^{5} \cdot b^{5}\) and \((a \cdot (b) ^{6}=a^{6} \cdot b^{6},\) for all \(a, b \in G,\) then prove that \(G\) is Abelian.

[10M]

1.(b) Let \(f\) be defined on [0,1] as \(f(x)=\left\{\begin{array}{l} \sqrt{1-x^{2}}, \text { if } x \text { is rational } \\ 1-x, \text { if } x \text { is irrational } \end{array}\right.\) Find the upper and lower Riemann integrals of \(f\) over [0,1] .

[10M]

1.(c) Using Cauchy integral formula, evaluate \(\int_{C} \dfrac{z+2}{(z+1)^{2}(z-2)} d z\) where \(C\) is the circle \(|z-i|=2\)

[10M]

1.(d) Obtain the initial basic feasible solution for the transportation problem by North-West corner rule:

[10M]

1.(e)Find the constants \(a, b, c\) such that the function \(f(z)=2 x^{2}-2 x y-y^{2}+i\left(a x^{2}-b x y+c y^{2}\right)\) is analytic for all \(z(=x+i y)\) and express \(f(z)\) in terms of \(z\).

[10M]

2.(a) Let \(J_{n}\) be the set of integers mod \(n .\) Then prove that \(J_{n}\) is a ring under the operations of addition and multiplication mod \(n .\) Under what conditions on \(n\), \(J_{n}\) is a field? Justify your answer.

[10M]

2.(b) Show that the function \(f(x)=\sin \dfrac{1}{x}\) is continuous but not uniformly continuous on \((0, \pi)\)

[10M]

5.(a) Use Lagrange’s formula to find the form of \(f(x)\) from the following table:
| \(x\) | 0 | 2 | 3 | 6 | |——|—|—|—|—| |\(f(x)\)|648|704|729|792|

[10M]

5.(b) Write a program in BASIC to integrate \(\int_{0}^{1} e^{-2 x} \sin x d x\) by Simpson’s \(\dfrac{1}{3} r d\) rule with 20 subintervals.

[10M]

5.(c) Show that the general solution of the pde \(\dfrac{\partial^{2} z}{\partial x^{2}}=\dfrac{1}{c^{2}} \dfrac{\partial^{2} z}{\partial t^{2}}\) is of the form \(Z(x, y)=F(x+c t)+G(x-c t),\) where \(F\) and \(G\) are arbitrary functions.

[10M]

5.(d) Prove that the vorticity vector \(\vec{\Omega}\) of an incompressible viscous fluid moving in the absence of an external force satisfies the differential equation \(\dfrac{D \vec{\Omega}}{D t}=\vec{\Omega} \cdot \nabla \mid \vec{q}+v \nabla^{2} \vec{\Omega}\) where \(\vec{q}\) is the velocity vector with \(\vec{\Omega}=\nabla \times \vec{q}\)

[10M]

5.(e) Find the condition that \(f(x, y, \lambd(a) =0\) should be a possible system of streamlines for steady irrotational motion in two dimensions, where \(\lambda\) is a variable parameter.

[10M]

6.(a) Verify that the differential equation \(\left(y^{2}+y z\right) d x+\left(x z+z^{2}\right) d y+\left(y^{2}-x y\right) d z=0\) is integrable and find its primitive.

[10M]

6.(b) Show that the moment of inertia of a uniform rectangular mass \(M\) and sides \(2 a\) and \(2 b\) about a diagonal is \(\dfrac{2 M a^{2} b^{2}}{3\left(a^{2}+b^{2}\right)}\).

[10M]

6.(c) The values of \(f(x)\) for different values of \(x\) are given as \(f(1)=4, f(2)=5, f(7)=5\) and \(f(8)=4 .\) Using Lagrange’s interpolation formula, find the value of \(f(6) .\) Also find the value of \(x\) for which \(f(x)\) is optimum.

[10M]

6.(d) TBC Write a BASIC program to sum the series \(S=1+x+x^{2}+\ldots+x^{n},\) for \(n=30,60\) and 90 for the values of \(x=0 \cdot 1(0 \cdot 1) 0 \cdot 3\)

[10M]

7.(a) Solve: \(\left(D-3 D^{\prime}-2\right)^{2} z=2 e^{2 x} \cot (y+3 x)\)

[10M]

7.(b) Solve the following system of equations: \(\begin{aligned} 2 x_{1}+x_{2}+x_{3}-2 x_{4} &=-10 \\ 4 x_{1}+2 x_{3}+x_{4} &=8 \\ 3 x_{1}+2 x_{2}+2 x_{3} &=7 \\ x_{1}+3 x_{2}+2 x_{3}-x_{4} &=-5 \end{aligned}\)

[10M]

7.(c) A uniform rod \(O A\) of length \(2 a\) is free to turn about its end \(O,\) revolves with uniform angular velocity \(\omega\) about a vertical axis \(O Z\) through \(O\) and is inclined at a constant angle \(\alpha\) to \(O Z\). Show that the value of \(\alpha\) is either zero or \(\cos ^{-1}\left(\dfrac{3 g}{4 a \omega^{2}}\right)\)

[10M]

8.(a) Using Runge-Kutta 4-th order method, find \(y\) from \(\dfrac{d y}{d x}=\dfrac{y^{2}-x^{2}}{y^{2}+x^{2}}\) with \(y(0)=1\) at \(x=0 \cdot 2,0 \cdot 4\)

[10M]

8.(b) A plank of mass \(M\) is initially at rest along a straight line of greatest slope of a smooth plane inclined at an angle \(\alpha\) to the horizon and a man of mass \(M^{\prime}\) starting from the upper end walks down the plank so that it does not move. Show that he gets to the other end in time \(\sqrt{\dfrac{2 M^{\prime} a}{\left(M+M^{\prime}\right) g \sin \alpha}}\) where \(a\) is the length of the plank.

[10M]

8.(c) Prove that \(\dfrac{x^{2}}{a^{2}} \tan ^{2} t+\dfrac{y^{2}}{b^{2}} \cot ^{2} t=1\) is a possible form for the bounding surface of a liquid and find the velocity components.

[10M]