Paper II PYQs-2009

1.(a) Prove that a non-empty subset H of a group G is normal subgroup of G⇔ for all x,y∈H,g∈G,(gx)(gy)−1∈H

[10M]

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

[10M]

1.(c) Show that under the transformation w=z−iz+i real axis in the E plane is mapped into the circle |w|=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⋅m. of O(a),O(b)

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1.(e) Evaluate ∫C2z+1z2+zdz by Cauchy’s integral formula, where C is |z|=12

[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 xa+yb+zc=1

[10M]

2.(b) Determine the analytic function w=u+iv, if u=2sin2xe2y+e−2y−2cos2x

[10M]

2.(c) Find the multiplicative inverse of the element [2513] of the ring, M of all matrices of order two over the integers.

[10M]

3.(a) Evaluate ∬ 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

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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 pth 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.

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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

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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}.

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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.

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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]