Paper II PYQs-2009

1.(a) Prove that a non-empty subset \(H\) of \(a\) group G is normal subgroup of \(G \Leftrightarrow\) for all \(x, y \in H, g \in G,(g x)(g y)^{-1} \in H\)

[10M]

1.(b) Show that the function \(f(x]=\dfrac{1}{x}\) is not uniformly continuous [0,1].

[10M]

1.(c) Show that under the transformation \(w=\dfrac{z-i}{z+i}\) real axis in the \(E\) plane is mapped into the circle \(\vert w\vert =1\) What portion of the Z-plane corresponds to the interior of the circle?

[10M]

1.(d) If \(G\) is a finite Abelian group, then show that \(O(a, b)\) is a divisor of \(1 . c \cdot m\). of \(O(a), O(b)\)

[10M]

1.(e) Evaluate \(\int_{C} \dfrac{2 z+1}{z^{2}+z} d z\) by Cauchy’s integral formula, where \(C\) is \(\vert z\vert =\dfrac{1}{2}\)

[10M]

2.(a) Find the dimensions of the largest rectangular parallelopiped that has three faces in the coordinate planes and one vertex in the plane \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\)

[10M]

2.(b) Determine the analytic function \(w=u+i v,\) if \(u=\dfrac{2 \sin 2 x}{e^{2 y}+e^{-2 y} -2 \cos 2 x}\)

[10M]

2.(c) Find the multiplicative inverse of the element \(\left[\begin{array}{ll} 2 & 5 \\ 1 & 3 \end{array}\right]\) of the ring, \(M\) of all matrices of order two over the integers.

[10M]

3.(a) Evaluate \(\iint x y(x+y) d x d y\) over the area between \(y=x^{2}\) and \(y=x\)

[10M]

3.(b) Evaluate by contour integration \(\int_{0}^{2 \pi} \dfrac{d \theta}{1-2 a \sin \theta+a^{2}}, \quad 0<a<1\)

[10M]

3.(c) Write the dual of the following LPP and hence, solve it by graphical method : Minimize \(\quad Z=6 x_{1}+4 x_{2}\) constraints \(\begin{aligned} 2 x_{1}+x_{2} & \geq 1 \\ 3 x_{1}+4 x_{2} & \geq 1.5 \\ x_{1}, x_{2} & \geq 0 \end{aligned}\)

[10M]

4.(a) Show that \(d(a)< d (a b),\) where \(a, b\) be two non-zero elements of a Euclidean domain \(R\) and \(b\) is not a unit in \(R\)

[10M]

4.(b) Show that a field is an integral domain and a non-zero finite integral domain is a field.

[10M]

4.(c) Solve by simplex method, the following LPP : Maximize \(\quad Z=5 x_{1}+3 x_{2}\) constraints \(\begin{aligned} 3 x_{1}+5 x_{2} & \leq 15 \\ 5 x_{1}+2 x_{2} & \leq 10 \\ x_{1}, x_{2} & \geq 0 \end{aligned}\)

[10M]

5.(a) Find complete and singular integrals of \(\left(p^{2}+q^{2}\right) y=q z\).

[10M]

5.(b) Obtain the iterative scheme for finding \(p\)th root of a function of single variable using Newton-Raphson method. Hence, find \(\sqrt[7]{277} \overline{234}\) correct to four decimal places.

[10M]

5.(c) Convert the following binary numbers to the base indicated: (i) \((10111011001 \cdot 101110)_{2}\) to octal (ii) \((10111011001 \cdot 10111000)_{2}\) to hexadecimal (iii) \((0 \cdot 101)_{2}\) to decimal

[10M]

5.(d) A cannon of mass \(M\), resting on a rough horizontal plane of coefficient of friction \(\mu\) is fired with such a charge that the relative velocity of the ball and cannon at the moment when it leaves the cannon is \(u .\) Show that the cannon will recoil a distance \(\left(\dfrac{m u}{M+m}\right)^{2} \dfrac{1}{2 \mu g}\) along the plane, \(m\) being the mass of the ball.

[10M]

5.(e) If the velocity of an incompressible fluid at the point \((x, y, z)\) is given by \(\left(\dfrac{3 x z}{r^{5}}, \dfrac{3 y z}{r^{5}}, \dfrac{3 z^{2}-r^{2}}{r^{5}}\right)\) where \(r^{2}=x^{2}+y^{2}+z^{2},\) prove that the liquid motion is possible and that the velocity potential is \(\cos \theta / r^{2}\)

[10M]

6.(a) A rod of length \(l\) with insulated sides, is initially at a uniform temperature \(u_{0}\). Its ends are suddenly cooled to \(0^{\circ} C\) and are kept at that temperature. Find the temperature distribution in the rod at any time \(t\).

[10M]

6.(b) Convert the following to the base indicated: (i) \((266 \cdot 375\) ) \(_{10}\) to base 8 (ii) \((341\cdot24)_{5}\) to base 10 (iii) \((43 \cdot 3125)_{10}\) to base 2

[10M]

6.(c) Draw the circuit diagram for \(\bar{F}=A \bar{B} C+\bar{C} B\) using NAND to NAND logic long.

[10M]

6.(d) Using Runge-Kutta method, solve \(y^{\prime \prime}=x y^{'2}-y^{2}\) for \(x=0.2 .\) Initial conditions are at \(x=0, y=1\) and \(y^{\prime}=0 .\) Use four decimal places for computations.

[10M]

7.(a) Prove that the equation of motion of a homageneous inviscid liquid moving under conservative forces may be written as \(\dfrac{\partial \vec{q}}{\partial t}-\vec{q} \times \operatorname{curl} \vec{q}=-\operatorname{grad}\left[\dfrac{p}{\rho}+\dfrac{1}{2} q^{2}+\vec{\Omega}\right]\)

[10M]

7.(b) Find the general solution of \(\begin{aligned} \left\{D^{2}-D D^{\prime}\right.&\left.-2 D^{\prime 2}+2 D+2 D^{\prime}\right\} z \\ &=e^{2 x+3 y}+x y+\sin (2 x+y) \end{aligned}\)

[10M]

7.(c) From the following data \(\begin{array}{cccccc} x & : & 1 & 8 & 27 & 64 \\ y & : & 1 & 2 & 3 & 4 \end{array}\) calculate \(y(20),\) using Lagrangian interpolation decimal points for computations.

[10M]

8.(a) A homogeneous sphere of radius \(a\), rotating with angular velocity about hoizontal diameter, is gently placed on a table whose coefficient of friction is \(\mu\). Show that there will be slipping at the point of contact for a time \(\dfrac{2 \omega a}{7 \mu g}\) and that then the sphere will roll with angular velocity \(\dfrac{2 \omega}{7}\).

[10M]

8.(b) Derive composite \(\dfrac{1}{3} rd\) Simpson’s rule. Hence, evaluate \(\int_{0}^{0.6} e^{-x^{2}} d x\) by taking seven ordinates. Tabulate the integrand for these ordinates to four decimal places.

[10M]

8.(c) Show that for an incompressible steady flow with constant viscosity, the velocity components \(\begin{array}{l} u(y)=y \dfrac{U}{h}+\dfrac{h^{2}}{2 \mu}\left(-\dfrac{d p}{d x}\right) \dfrac{y}{h}\left(1-\dfrac{y}{h}\right) \\ v=0, w=0 \end{array}\) satisfy the equations of motion, when the body force is neglected. \(h, U, \dfrac{d p}{d x}\) are constants and \(p=p(x) .\)

[10M]