Paper II PYQs-2010

1.(a) Let \(G=\left\{\left[\begin{array}{ll} a & a \\ a & a \end{array}\right] \mid a \in \cdot R , a \neq 0\right\}\) Show that \(G\) is a group under matrix multiplication.

[10M]

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

[10M]

1.(c) If \(f: R \rightarrow R\) is such that \(f(x+y)=f(x) f(y)\) for all \(x, y\) in \(R\) and \(f(x) \neq 0\) for any \(x\) in \(R ,\) show that \(f^{\prime}(x)=f(x)\) for all \(x\) in \(R\) given that \(f^{\prime}(0)=f(0)\) and the function is differentiable for all \(x\) in \(R\).

[10M]

1.(d) Determine the analytic function \(f(z)=u+i v\) if \(v=e^{x}(x \sin y+y \cos y)\)

[10M]

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 :

[10M]

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.

[10M]

2.(b) Prove or disprove that \(( R ,+)\) and \(\left( R ^{+}, \cdot\right)\) are isomorphic groups where \(R ^{+}\) denotes the set of all positive real numbers.

[10M]

2.(c) Using the method of contour integration, evaluate \(\int_{-\infty}^{\infty} \dfrac{x^{2} d x}{\left(x^{2}+1\right)^{2}\left(x^{2}+2 x+2\right)}\)

[10M]

3.(a) Show that zero and unity are only idempotents of \(Z_{n}\) if \(n=p^{r},\) where \(p\) is a prime.

[10M]

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

[10M]

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 \(\begin{array}{r} x_{1}+2 x_{2}+7 x_{3} \leq 8 \\ x_{1}+x_{2}-2 x_{3} \leq 6 \\ x_{1}, x_{2}, x_{3} \geq 0 \end{array}\)

[10M]

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.

[10M]

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

[10M]

4.(c) \(A B C\) Electricals manufactures and sells two models of lamps, \(L_{1}\) and \(L_{2},\) the profit per unit being Rs\(50\) and Rs\(30\) 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.

[10M]

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

[10M]

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

[10M]

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.

[10M]

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.

[10M]

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: \(\sin (0 \cdot 1)=0 \cdot 09983, \sin (0 \cdot 2)=0 \cdot 19867\)

[10M]

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

[10M]

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) \(u(x, 0)=\sin 2 x, \quad 0< x< \pi\)

[10M]

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

[10M]

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 \((5,147)\).

[10M]

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 \(m_{1}: m_{2}=2: 1\)

[10M]

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.

[10M]

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.

[10M]