Paper II PYQs-2010

1.(a) Let G={[aaaa]∣a∈⋅R,a≠0} Show that G is a group under matrix multiplication.

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1.(b) Let F be a field of order 32. Show that the only subfields of F are F itself and {0,1}

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1.(c) If f:R→R is such that f(x+y)=f(x)f(y) for all x,y in R and f(x)≠0 for any x in R, show that f′(x)=f(x) for all x in R given that f′(0)=f(0) and the function is differentiable for all x in R.

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1.(d) Determine the analytic function f(z)=u+iv if v=ex(xsiny+ycosy)

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1.(e) A captain of a cricket team has to allot four middle-order batting positions(IV,V,VI.VII) to four batsmen(A,B,C,D). The average number of runs scored by each batsman at these positions are as follows. Assign each batsman his batting position for maximum performance :

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2.(a) A rectangular box open at the top is to have a surface area of 12 square units. Find the dimensions of the box so that the volume is maximum.

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2.(b) Prove or disprove that (R,+) and (R+,⋅) are isomorphic groups where R+ denotes the set of all positive real numbers.

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2.(c) Using the method of contour integration, evaluate ∫∞−∞x2dx(x2+1)2(x2+2x+2)

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3.(a) Show that zero and unity are only idempotents of Zn if n=pr, where p is a prime.

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3.(b) Evaluate ∬ where R is the region inside the unit square in which x+y \geq \dfrac{1}{2}

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3.(c) Solve the following linear programming problem by the simplex method: Maximize \quad Z=3 x_{1}+4 x_{2}+x_{3} subject to \begin{array}{r} x_{1}+2 x_{2}+7 x_{3} \leq 8 \\ x_{1}+x_{2}-2 x_{3} \leq 6 \\ x_{1}, x_{2}, x_{3} \geq 0 \end{array}

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4.(a) Let R be a Euclidean domain with Euclidean valuation d. Let n be an integer such that d(1)+n \geq 0 . Show that the function d_{n}: R-\{0\} \rightarrow S, where S is the set of all negative integers defined by d_{n}(a)=d(a)+n for all a \in R-\{0\} is a Euclidean valuation.

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4.(b) Obtain Laurent’s series expansion of the function f(z)=\dfrac{1}{(z+1)(z+3)} in the region 0<|z+1|<2

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4.(c) A B C Electricals manufactures and sells two models of lamps, L_{1} and L_{2}, the profit per unit being Rs50 and Rs30 respectively. The process involves two workers W_{1} and W_{2}, who are available for 40 hours and 30 hours per week, respectively. W_{1} assembles each unit of L_{1} in 30 minutes and that of L_{2} in 40 minutes. W_{2} paints each unit of L_{1} in 30 minutes and that of L_{2} in 15 minutes. Assuming that all lamps made can be sold, determine the weekly production figures that maximize the profit.

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5.(a) Find the general solution of x\left(y^{2}+z\right) p+y\left(x^{2}+z\right) q=z\left(x^{2}-y^{2}\right)

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5.(b) Solve x \log _{10} x=1 \cdot 2 by regula falsi method.

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5.(c) Convert the following: (i) (736 \cdot 4)_{8} to decimal number (ii) (41.6875)_{10} to binary number (iii) (101101)_{2} to decimal number (iv) ( AF 63)_{16} to decimal number (v) (101111011111)_{2} to hexadecimal number

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5.(d) Show that the sum of the moments of inertia of an elliptic area about any two tangents at right angles is always the same.

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5.(e) A two-dimensional flow field is given by \psi=x y. Show that (i) the flow is irrotational; (ii) \psi and \phi satisfy Laplace equation. Symbols \psi and \phi convey the usual meaning.

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6.(a) Using Lagrange interpolation, obtain an approximate value of \sin (0 \cdot 15) and a bound on the truncation error for the given data: \sin (0 \cdot 1)=0 \cdot 09983, \sin (0 \cdot 2)=0 \cdot 19867

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6.(b) Draw a flow chart for finding the roots of the quadratic equation a x^{2}+b x+c=0.

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6.(c) Solve \dfrac{\partial u}{\partial t}=4 \dfrac{\partial^{2} u}{\partial x^{2}} given the conditions (i) u(0, t)=u(\pi, t)=0, n t>0 (ii) u(x, 0)=\sin 2 x, \quad 0< x< \pi

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7.(a) Find the gereral solution of \left(D-D^{\prime}-1\right)\left(D-D^{\prime}-2\right) z=e^{2 x-y}+\sin (3 x+2 y)

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7.(b) Show that \phi=(x-t)(y-t) represents the velocity potential of an incompressible two-dimensional fluid. Further show that the streamlines at time t are the curves (x-t)^{2}-(y-t)^{2}=\text { constant }

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7.(c) Find the interpolating polynomial for (0,2),(1,3),(2,12) and (5,147).

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8.(a) A mass m_{1}, hanging at the end of a string, draws a mass m_{2} along the surface of a smooth table. If the mass on the table be doubled, the tension of the string is increased by one-half. Show that m_{1}: m_{2}=2: 1

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8.(b) Solve the initial value problem \dfrac{d y}{d x}=\dfrac{y-x}{y+x}, \quad y(0)=1 for x=0.1 by Euler’s method.

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8.(c) Show that the vorticity vector \vec{\Omega} of an incompressible viscous fluid moving under no external forces satisfies the differential equation \dfrac{D \vec{\Omega}}{D t}=\vec{\Omega} \cdot \nabla \vec{q}+v \nabla^{2} \vec{\Omega} where v is the kinematic viscosity.

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