Paper II PYQs-2018

Section A

1.(a) Let $R$ be an integral domain with unit element. Show that any unit in $R (x)$ is a unit in $R$ .

[10M]

1.(b) Prove the inequality: $\frac{π^{2}}{9} < \int_{\frac{π}{6}}^{\frac{π}{2}} \frac{x}{s i n x} d x$ < $\frac{2 π^{2}}{9}$

[10M]

1.(c) Prove that the function: $u (z, y) = (x - 1)^{3} - 3 x y^{2} + 3 y^{2}$ is harmonic and find its harmonic conjugate and the corresponding analytic function $f (z)$ in terms of $z$ .

[10M]

1.(d) Find the range of $p (> 0)$ for which the series:

$\frac{1}{(1 + a)^{p}} - \frac{1}{(2 + a)^{p}} + \frac{1}{(3 + a)^{p}} - . . . ., a > 0$ , is

(i) absolutely convergent and (ii) conditionally convergent.

[10M]

1.(e) An agricultural firm has 180 tons of nitrogen fertilizer, 250 tons of phosphate and 220 tons of potash.It is able to sell a mixture of these substances in their respective ratio 3:3:4 at a profit of Rs. 1500 per ton and a mixture in the ratio 2:4:2 at a profit of Rs. 1200 per ton. Pose a linear programming problem to show how many tons of these two mixtures should be prepared to obtain the maximum profit.

[10M]

2.(a) Show that the quotiet group of $(R, +)$ modulo $Z$ is isomorphic to the multiplicative group of complex numbers on the unit circle in the complex numbers on the unit circle in the complex plane. Here $R$ is the set of real numbers and $Z$ is the set of integers.

[15M]

2.(b) Solve the following linear programming problem by Big M-method and show that the problem has finite optimal solutions. Also, find the value of the objective function: Minimize $z = 3 x_{1} + 5 x_{2}$ subject to $x_{1} + 2 x_{2} \geq 8$
$3 x_{1} + 2 x_{2} \geq 12$ ,
$5 x_{1} + 6 x_{2} \geq 60$ ,
$x_{1}, x_{2} \geq 0$

[20M]

2.(c) Show that if a function $f$ defined on the open interval $(a, b)$ of $R$ is convex, then $f$ is continuous. Show by example, if the condition of open interval is dropped, then the convex function need not be continuous.

[15M]

3.(a) Find all the proper subgroups of the multiplicative group of the field $(Z_{13}, +_{13}, \times_{13})$ , where $+_{13}$ and $\times_{13}$ represent addition modulo 13 and multiplication modulo 13 respectively.

[20M]

3.(b) Show by applying the residue theorem that $\int_{0}^{\infty} \frac{d x}{(x^{2} + a^{2})^{2}} = \frac{π}{4 a^{3}}$ , $a > 0$ .

[15M]

3.(c) How many basic solutions are there in the following linearly independent set of equations? Find all of them.

$2 x_{1} - x_{2} + 3 x_{3} + x_{4} = 6$ $4 x_{1} - 2 x_{2} - x_{3} + 2 x_{4} = 10$

[15M]`

4.(a) Suppose $R$ be the set of all real numbers and $f : R \to R$ is a function such that the following equations hold for all $x$ , $y \in R$ :

$f (x + y) = f (x) + f (y)$
$f (x y) = f (x) f (y)$

Show that that $\forall c \in R$ , either $f (x) = 0$ or $f (x) = x$ .

[20M]

4.(b) Find the Laurent’s series which represent the function $\frac{1}{(1 + z^{2}) (z + 2)}$ when

(i) $| z | < 1$ (ii) $1 < | z | < 2$ (iii) $| z | > 2$

[15M]

4.(c)

\begin{array}{rcrcrcrcrcrcr} M_{1} & M_{2} & M_{3} & M_{4} & M_{5} \\ O_{1} & 24 & 29 & 18 & 32 & 19 \\ O_{2} & 17 & 26 & 34 & 22 & 21 \\ O_{3} & 27 & 16 & 28 & 17 & 25 \\ O_{4} & 22 & 18 & 28 & 30 & 24 \\ O_{5} & 28 & 16 & 31 & 24 & 27 \end{array}

In a factory there are five operator $O_{1}$ , $O_{2}$ , $O_{3}$ , $O_{4}$ , $O_{5}$ and five machine $M_{1}$ , $M_{2}$ , $M_{3}$ , $M_{4}$ , $M_{5}$ . The operating costs are given when the $O_{i}$ operator operates the $M_{j}$ machine $(i, j = 1, 2, . ., 5)$ . But there is a restriction that $O_{3}$ cannot be allowed to operate the third machine $M_{3}$ and $O_{2}$ cannot be allowed to operate the fifth machine $M_{5}$ . The cost matrix is given above. Find the optional assignment and the optimal assignment cost also.

[15M]

Section B

5.(a) Find the partial differential equation of the family of all tangent planes to the ellipsoid: $x^{2} + 4 y^{2} + 4 z^{2} = 4$ , which are not prependicular the the $x y$ plane.

[10M]

5.(b) Using Newton’s forward difference formula find the lowest degree polynomial $u_{x}$ when it is given that $u_{1} = 1$ , $u_{2} = 9$ , $u_{3} = 25$ , $u_{4} = 55$ , $u_{5} = 105$ .

[10M]

5.(c) For an incompressible fluid flow, two components of velocity $(u, v, w)$ are given by $u = x^{2} + 2 y^{2} + 3 z^{2}$ , $v = x^{2} y - y^{2} z + z x$ . Determine the third component $w$ so that they satisfy the equation of continuity. Also, find the $z - c o m p o n e n t$ of acceleration.

[10M]

Using the continuity equation in cartesian coordinates:

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$
$⟹ 2 x + x^{2} - 2 y z + \frac{\partial w}{\partial z} = 0$
$⟹ \frac{\partial w}{\partial z} = 2 y z - 2 x - x^{2}$
Integrating wrt $z$ , we get:
$w = y z^{2} - 2 x z - x^{2} z + f (x, y)$

$⟹$ The $z$ component of acceleration:
$\begin{aligned} a_{z} & = (q \cdot \nabla) w + \frac{\partial ω}{\partial t} \\ = u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial ω}{\partial z} \end{aligned}$

Therefore,

\begin{aligned} a_{z} & = (x^{2} + 2 y^{2} + 3 z^{2}) (\frac{\partial f}{\partial x} - 2 z - 2 x z) \\ + (x^{2} y - y^{2} z + x z) (\frac{\partial f}{\partial y} + z^{2}) \\ + (y z^{2} - 2 x z - x^{2} z + f (x, y)) (2 y z - 2 z - x^{2}) \end{aligned}

5.(d)

\begin{array}{rcrcrcrcrcrcrcrcrcrcr} T i m e (M i n u t e s) & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\ S p e e d (K m / h) & 10 & 18 & 25 & 29 & 32 & 20 & 11 & 5 & 2 & 8.5 \end{array}

Starting from rest in the beginning, the speed(in Km.h) of a train at different times(in minutes) is given by the above table.

Using Simpson’s $\frac{1}{3}$ rd rule, find the approximate distance travelled (in Km) in 20 minutes fro the beginning.

[10M]

5.(e) Write down the basic algorithm for solving the equation: $x e^{x} - 1 = 0$ by bisection method, correct to 4 decimal places.

[10M]

6.(a) Find the general solution of the partial differential equation

$(y^{3} x - 2 x^{4}) p + (2 y^{4} - x^{3} y) q = 9 z (x^{3} - y^{3})$ ,

where $p = \frac{\partial z}{\partial x}$ , $q = \frac{\partial z}{\partial y}$ , and its integral surface that passes through the curve: $x = t$ , $y = t^{2}$ , $z = 1$ .

[15M]

6.(b) Find the equivalent of numbers given in a specified number system to the system mentioned against them.

(i) $(111011.101)_{2}$ to decimal system
(ii) $(1000111110000.00101100)_{2}$ to hexadecimal system
(iii) $(C 4 F 2)_{16}$ to decimal system
(iv) $(418)_{10}$ to binary system

[15M]

6.(c) Suppose the Lagrangian of a mechanical system is given by $L = \frac{1}{2} m (a {\dot{x}}^{2} + 2 b \dot{x} \dot{y} + c {\dot{y}}^{2}) - \frac{1}{2} k (a x^{2} + 2 b x y + c y^{2})$ ,

where $a$ , $b$ , $c (> 0)$ , $k (> 0)$ are constant and $b^{2} \neq a c$ . Write down the Lagrangian Equations of motion and identify the system.

[20M]

7.(a) Solve the partial differential equation:

(2 D^{2} - 5 D D^{'} + 2 D^{' 2}) z = 5 s i n (2 x + y) + 24 (y - x) + e^{3 x + 4 y}

where $D \equiv \frac{\partial}{\partial x}$ , $D^{'} \equiv \frac{\partial}{\partial y}$ .

[15M]

7.(b) Find the values of the constant $a, b, c$ such that the quadrature formula:

$\int_{o}^{h} f (x) d x = h [a f (o) + b f (\frac{h}{3}) + c f (h)]$ is exact for polynomial of as high degree as possible, and hence find the order of the truncation error.

[15M]

7.(c) The Hamiltonian of a mechanical system is given by,

$H = p_{1} q_{1} - a q_{1}^{2} + b q_{2}^{2} = p_{2} q_{2}$ , where $a$ , $b$ are constants. Solve the Hamiltonian equations and show that $\frac{p_{2} - b q_{2}}{q_{1}} = c o n s t a n t$ .

[20M]

8.(a) Simplify the boolean expression:

$(a + b) \cdot (\bar{b} + c) + b \cdot (\bar{a} + \bar{c})$ by using the laws of boolean algebra. From its truth table write it in minterm normal form.

[15M]

8.(b) For a two-dimensional potential flow, the velocity potential is given by $ϕ = x^{2} y = x y^{2} + \frac{1}{3} (x^{3} - y^{3})$ . Determine the velocity components along the direction $x$ and $y$ . Also, detrmine the stream function $ψ$ and check whether $ϕ$ represents a possible case of flow or not.

[15M]

Let $\vec{q} = u \hat{i} + v \hat{ȷ}$
Then,
$u = - \frac{\partial ϕ}{\partial x}$
$⟹ u = - (2 x y - y^{2} + x^{2})$
$⟹ u = y^{2} - x^{2} - 2 x y$

Also,
$v = - \frac{\partial ϕ}{\partial y}$
$\Rightarrow v = - (x^{2} - 2 x y - y^{2})$
$⟹ v = y^{2} - x^{2} - 2 x y$
We know that $ϕ + i ψ$ is an analytic function and salisfies Cauchy Riemann equations.

So, $\frac{\partial ψ}{\partial y} = u$ and $\frac{\partial ψ}{\partial x} = v$
$\Rightarrow \frac{\partial ψ}{\partial x} = y^{2} - x^{2} + 2 x y$
Integrating wrt $x$ :
$ψ = x y^{2} - \frac{x^{3}}{3} + x^{2} y + f (y)$

Now, $\frac{\partial ψ}{\partial y} = 2 x y + x^{2} + f^{'} (y)$ , and
$\frac{\partial ψ}{\partial y} = - u$

$\Rightarrow x^{2} + 2 x y + f^{'} (y) = x^{2} + 2 x y - y^{2}$
$\Rightarrow f^{'} (y) = - y^{2}$
$\frac{d f}{d y} = - y^{2}$
$\Rightarrow f = \frac{- y^{3}}{3} + c$

So,
$ψ = x y^{2} - \frac{(x^{3} + y^{3})}{3} + x^{2} y + c$
$ψ = 0$ at origin $⟹ c = 0$

So, $ψ = x y^{2} + x^{2} y - \frac{(x^{3} + y^{3})}{3}$

Checking possible flow:
Since $\frac{\partial u}{\partial x} + \frac{\partial y}{\partial y} = - 2 x - 2 y + 2 y + 2 x = 0$
$⟹$ Equation of continuity is satisfied,
$⟹$ It is a possible flow.

8.(c) A thin annulus occupies the region $0 < a \leq r \leq b$ , $0 \leq θ \leq 2 π$ . The faces are insulated. Along the inner edge the temperature is maintained at $0^{\circ}$ , while along the outer egde the temperature is held at $T = K c o s \frac{θ}{2}$ , where $K$ is a constant. Determine the temperature distribution in the annulus.

[20M]

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