Paper II PYQs-2012

Section A

1.(a) Show that the function $f : R \to R$ defined by $f (x) = {\begin{aligned} 1, & x is irrational \\ - 1, & x is rational \end{aligned}$ is discontinuous at every point in’ $R$ .

[10M]

1.(b) Show that every field is without zero divisor.

[10M]

1.(c) Evaluate the integral $\int_{2 - i}^{4 + i} (x + y^{2} - i x y) d z$ along the line segment AB joining the points $A (2, - 1)$ and $B (4, 1)$

[10M]

1.(d) Show that the functions: $\begin{array}{l} u = x^{2} + y^{2} + z^{2} \\ v = x + y + z \\ w = y z + z x + x y \end{array}$ are not independent of one another.

[10M]

2.(a) Show that in a symmetric group $S_{3},$ there are four elements $σ$ satisfying $σ^{2} =$ Identity and three elements satisfying $σ^{3} =$ Identity.

[10M]

2.(b) If $u = x^{2} \tan^{- 1} (\frac{y}{x}) - y^{2} \tan^{- 1} (\frac{x}{y})$ show that $x^{2} \frac{\partial^{2} u}{\partial x^{2}} + 2 x y \frac{\partial^{2} u}{\partial x \partial y} + y^{2} \frac{\partial^{2} u}{\partial y^{2}} = 2 u$

[10M]

2.(c) Solve the following problem by Simplex Method. How does the optimal table indicate that the optimal solution obtained is not unique ? Maximize $z = 8 x_{1} + 7 x_{2} - 2 x_{3}$ subject to the constraints $\begin{array}{l} x_{1} + 2 x_{2} + 2 x_{3} \leq 12 \\ 2 x_{1} + x_{2} - 2 x_{3} \leq 12 \\ x_{1}, x_{2}, x_{3} \geq 0 \end{array}$

[10M]

3.(a) Find the volume of the solid bounded above by the parabolic cylinder $z = 4 - y^{2}$ and bounded below by the elliptic paraboloid $z = x^{2} + 3 y^{2}$

[10M]

3.(b) Show that the function $u (x, y) = e^{- x} (x \cos y + y \sin y)$ is harmonic. Find its conjugate harmonic function $v (x, y)$ and the corresponding analytic function $f (z)$

[10M]

3.(c) If $R$ is an integral domain, show that the polynomial ring $R [x]$ is also an integral domain.

[10M]

4.(a) Using the Residue Theorem, evaluate the integral $\int_{C} \frac{e^{2} - 1}{z (z - 1) (z + i)^{2}} d z$ where $C$ is the circle $| z | = 2$

[10M]

4.(b) Examine the series $\sum_{n = 1}^{\infty} u_{n} (x) = \sum_{n = 1}^{\infty} [\frac{n x}{1 + n^{2} x^{2}} - \frac{(n - 1) x}{1 + (n - 1)^{2} x^{2}}]$ for uniform conyergence. Also, with the help of this example, show that the condition of uniform convergence of $\sum_{n = 1}^{\infty} u_{n} (x)$ is sufficient but not necessary for the sum $S (x)$ of the series to be continuous.

[10M]

4.(c) Find the initial basic feasible solution of the following minimum cost transportation problem by Least cost (Matrix Minima) Method and using it find the optimal transportation cost :- | | $D_{1}$ | $D_{2}$ | $D_{3}$ | $D_{4}$ | $S u p p l y$ | |————-|—–|—–|—–|—–|——–| | $S_{1}$ | 5 | 11 | 12 | 13 | 10 | | $S_{2}$ | 8 | 12 | 7 | 8 | 30 | | $S_{3}$ | 12 | 7 | 15 | 6 | 35 | | $R e q u i r e m e n t$ | 15 | 15 | 20 | 25 | | Here, $S_{i}$ and $D_{i}$ are sources and destinations $i = 1, 2, 3$

[10M]

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Section B

5.(a) Using Lagrange’s interpolation formula, show that $32 f (1) = - 3 f (- 4) + 10 f (- 2) + 30 f (2) - 5 f (4)$ .

[10M]

5.(b) Solve $(D^{3} D^{' 2} + D^{2} D^{' 3}) z = 0$ where $D$ stands for $\frac{\partial}{\partial x}$ and $D^{'}$ stands for $\frac{\partial}{\partial y}$

[10M]

5.(c) Write a computer program to implement trapezoidal rule to evaluate $\int_{0}^{10} (1 - e^{- \frac{x}{2}}) d x$

[10M]

5.(d) Prove that the vorticity vector $\vec{Ω}$ of an incompressible viscous fluid moving in the absence of an external force satisfies the differential equation $\frac{D \vec{Ω}}{D t} = (\vec{Ω} \cdot \nabla) \vec{q} + v \nabla^{2} \vec{Ω}$ where $v$ is kinematic viscosity.

[10M]

6.(a) Using Method of Separation of Variables, solve Laplace Equation in three dimensions.

[10M]

6.(b) Derive the differential equation of motion for a spherical pendulum.

[10M]

6.(c) A river is 80 meters wide. The depth $d$ (in meters) of the river at a distance $x$ from one bank of the river is given by the following table : | $x$ | 0 | 10| 20| 30| 40| 50| 60| 70| 80| |—|—|—|—|—|—|—|—|—|—| | $d$ | 0 | 4 | 7 | 9 | 12| 15| 14| 8 | 3 | Find approximately the area of cross-section of the river.

[10M]

7.(a) Show that $u = \frac{A (x^{2} - y^{2})}{{(x^{2} + y^{2})}^{2}}, v = \frac{2 A x y}{{(x^{2} + y^{2})}^{2}}, w = 0$ are components of a possible velocity vector for inviscid incompressible fluid flow. Determine the pressure associated with this velocity field.

[10M]

7.(b) Solve the following system of equations using Gauss-Seidel Method: $\begin{array}{l} 28 x + 4 y - z = 32 \\ 2 x + 17 y + 4 z = 35 \\ x + 3 y + 10 z = 24 \end{array}$ correct to three decimal places.

[10M]

7.(c) Draw a flow chart for interpolation using Newton’s forward difference formula.

[10M]

8.(a) Solve $(x^{2} - y z) p + (y^{2} - z x) q = z^{2} - x y$ using Lagrange’s Method.

[10M]

8.(b) A weightless rod $A B C$ of length $2 a$ is movable about the end $A$ which is fixed and carries two particles of mass $m$ each one attached to the mid-point $B$ of the rod and the other attached to the end $C$ of the rod. If the rod is held in the horizontal position and released from rest and allowed to move, show that the angular velocity of the rod when it is vertical is $\sqrt{\frac{6 g}{5 a}}$

[10M]

8.(c) Using Euler’s Modified Method, obtain the solution of $\frac{d y}{d x} = x + | \sqrt{y} |, y (0) = 1$ for the range $0 \leq x \leq 0.6$ and step size $0.2 .$

[10M]

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