Paper II PYQs-2017

Section A

1.(a) Let $x_{1} = 2$ and $x_{n + 1} = \sqrt{x_{n} + 20}, n = 1, 2, 3, \dots$ , show that the sequence $x_{1}, x_{2}, x_{3} . .$ is convergent.

[10M]

1.(b) Let $G$ be a group of order $n$ . Show that $G$ is isomorphic to a subgroup of the permutation group $S_{n}$ .

[10M]

1.(c) Find the supremum and infimum of $\frac{x}{\sin x}$ on the interval $(0, \frac{π}{2}]$

[10M]

1.(d) Determine all entire functions $f (z)$ such that 0 is a removable singularity of $f (\frac{1}{z})$ .

[10M]

1.(e) Using graphical method, find the maximum value of

2 x + y

subject to

$4 x + 3 y \leq 12$ $4 x + y \leq 8$ $4 x - y \leq 8$ $x, y \geq 0$

[10M]

2.(a) Let $f (t) = \int_{0}^{t} [x] d x$ where $[x]$ denotes the largest integer less than or equal to $x$ .
i) Determine all the real numbers $t$ at which $f$ is differentiable.
ii) Determine all the real numbers $t$ at which $f$ is continuous but not differentiable.

[15M]

2.(b) Using contour integral method, prove that $\int_{0}^{\infty} \frac{x \sin m x}{a^{2} + x^{2}} d x = \frac{π}{2} e^{- m a}$ .

[15M]

2.(c) Let $F$ be a field and $F [x]$ denote the ring of polynomial over $F$ in a single variable $X$ . For $f (X), g (X) \in F [X]$ with $g (X) \neq 0$ , show that there exist $q (X), r (X) \in F [X]$ such that degree $r (X)$ < degree $g (X)$ and $f (X) = q (X) \cdot g (X) + r (X)$ .

[20M]

3.(a) Show that the groups $Z_{5} \times Z_{7}$ and $Z_{35}$ are isomorphic.

[15M]

3.(b) Let $f = u + i v$ be analytic function on the unit disc $D = {z \in C : | z | < 1}$ . Show that $\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}$ =0= $\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}}$ at all points of $D$ .

[15M]

3.(c) Solve the following linear programming problem by simplex method. Maximize $z = 3 x_{1} + 5 x_{2} + 4 x_{3}$ , subject to:
$2 x_{2} + 3 x_{2} \leq 8$
$2 x_{1} + 5 x_{2} \leq 10$
$3 x_{1} + 2 x_{2} + 4 x_{3} \leq 15 x_{3}$
$x_{1}, x_{2}, x_{3} \geq 0$

[20M]

4.(a) For a function $f : C \to C$ and $n \geq 1$ , let $f^{(n)}$ denotes the $n^{t h}$ derivative of $f$ and $f^{(0)} = f$ . Let $f$ be an entire function such that for some $n \geq 1$ , $f^{(n)} (\frac{1}{k}) = 0$ for all $k = 1$ , 2, 3, show that $f$ is a polynomial.

[15M]

4.(b) Find the initial basic feasible solution of the following transportation problem using Vogel’s approximation methods and find the cost.
$D e s t i n a t i o n s$ $\begin{array}{cccccc} D_{1} & D_{2} & D_{3} & D_{4} & D_{5} \\ O_{1} & 4 & 7 & 0 & 3 & 6 & 14 \\ O_{2} & 1 & 2 & - 3 & 3 & 8 & 9 \\ O_{3} & 3 & - 1 & 4 & 0 & 5 & 17 \\ 8 & 3 & 8 & 13 & 8 \end{array}$

D e m a n d

[15M]

4.(c) Let $\sum_{n = 1}^{\infty} x_{n}$ be a conditionally convergent series of real numbers. Show that there is a rearrangement $\sum_{n = 1}^{\infty} x_{π (n)}$ of the series $\sum_{n = 1}^{\infty} x_{n}$ that converges to 100.

[20M]

Section B

5.(a) Solve $(D^{2} - 2 D D^{'} - D^{2}) z$ = $e^{x + 2 y} + x^{3} + \sin 2 x$ where $D \equiv \frac{\partial}{\partial x}$ , $D^{'} \equiv \frac{\partial}{\partial y}$ , $D^{2} \equiv \frac{\partial^{2}}{\partial x^{2}}$ , $D^{' 2} \equiv \frac{\partial^{2}}{\partial y^{2}}$ .

[10M]

5.(b) Explain the main steps of the Gauss-Jordan method and apply this method to find the inverse of the matrix $[\begin{matrix} 2 & 6 & 6 \\ 2 & 8 & 6 \\ 2 & 6 & 8 \end{matrix}]$ .

[10M]

5.(c) Write the Boolean expression $z (y + z) (x + y + z)$ in the simplest form using Boolean postulate rules. Mention the rules used during simplification. Verify your result by constructing the truth table for the given expression and for its simplest form.

[10M]

5.(d) Let $Γ$ be a closed curve in $x y - p l a n e$ and let $S$ denote the region bounded by the curve $Γ$ . Let $\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}}$ = $f (x, y) \forall (x, y) \in S$ . If $f$ is prescribed at each point $(x, y)$ of $S$ and $w$ is prescribed on the boundary $Γ$ of $S$ then prove that any solution $w = w (x, y)$ , satisfying these conditions, is unique.

[10M]

5.(e) Show that the moment of inertia of an elliptic area of mass $M$ and semi-axis $a$ and $b$ about a semi-diameter of length $r$ is $\frac{1}{4} M \frac{a^{2} b^{2}}{r^{2}}$ . Further, prove that the moment of inertia about a tangent is $\frac{5 M}{4} p^{2}$ , where $p$ is the perpendicular distance from the centre of the ellipse to the tangent.

[10M]

6.(a) Find a complete integral of the partial differential equation $2 (p q + y p + q x) + x^{2} + y^{2} = 0$ .

[15M]

6.(b) For given equidistant values $u_{- 1}$ , $u_{0}$ , $u_{1}$ and $u_{2}$ , a value is interpolated by Lagrange’s formula. Show that it may be written in the form $u_{x} = y u_{0} + \frac{y (y^{2} - 1)}{3!} Δ^{2} u_{- 1} + \frac{x (x^{2} - 1)}{3!} Δ^{2} u_{0}$ , where $x + y = 1$ .

[15M]

6.(c) Two uniform rods $A B$ and $A C$ , each of mass $m$ and length $2 a$ , are smoothly hinged together at $A$ and move on a horizontal plane. At time $t$ , the mass centre of the road is at the point $(ξ, η)$ referred to fixed perpendicular axes $O x$ , $O y$ in the plane, and the rods make angles $θ \pm ϕ$ with $O x$ . Prove that the kinetic energy of the system is $m [ξ^{2} + η^{2} + (\frac{1}{3} + \sin^{2} ϕ) a^{2} θ^{2} + (\frac{1}{3} + \cos^{2} ϕ) a^{2} ϕ^{2}]$ . Also derive Lagrange’s equation of motion for the system if an external force with components $[X, Y]$ along the axes acts at $A$ .

[20M]

7.(a) Reduce the equation $y^{2} \frac{\partial^{2} z}{\partial x^{2}} - 2 x y \frac{\partial^{2} z}{\partial x \partial y} + x^{2} \frac{\partial^{2} z}{\partial y^{2}}$ = $\frac{y^{2}}{\partial x} \frac{\partial z}{\partial x} + \frac{x^{2}}{\partial y} \frac{\partial z}{\partial y}$ to canonical form and hence solve it.

[15M]

7.(b) Derive the formula $\int_{a}^{b} y d x = \frac{3 h}{8} [(y_{0} + y_{n}) + 3 (y_{1} + y_{2} + y_{4} + y_{5} + \dots + y_{n - 1}) + 2 (y_{3} + y_{6} + y_{n - 3})]$ . Is there any restriction on $n$ ? State that condition. What is the error bounded in the case of Simpson’s $\frac{3}{8}$ th rule?

[20M]

7.(c) A stream is rushing from a boiler through a conical pipe, the diameters of the ends of which are $D$ and $d$ . If $V$ and $v$ be the corresponding velocities of the stream and if the motion is assumed to steady and diverging from the vertex of the cone, then prove that $\frac{v}{V} = \frac{D^{2}}{d^{2}} e^{(u^{2} - v^{2}) / 2 K}$ , where $K$ is the pressure divided by the density and is constant.

[15M]

Let $A B$ and $A^{'} B^{'}$ be the ends of the conical pipe such that $A^{'} B^{'} = d$ and $A B = D .$ Let $ρ_{1}$ and $ρ_{2}$ be densities of the stream at $A^{'} B^{'}$ and $A B$ . By principle of conservation of mass, the mass of the stream that enters the end $A B$ and leaves at the end $A^{'} B^{'}$ must be the same. Hence the equation of continuity is $π (d / 2)^{2} v ρ_{1} = π (D / 2)^{2} V ρ_{2}$

so that $\frac{v}{V} = \frac{D^{2}}{d^{2}} \times \frac{ρ_{2}}{ρ_{1}} \dots (1)$

2017-7(c)

By Bernoulli’s theorem (in absence of external forces like gravity), we have
$\int \frac{d p}{ρ} + \frac{1}{2} q^{2} = C \dots (2)$

Given that $p / ρ = k$ so that

d p = k d ρ \dots (3)

$∴$ (2) reduces to
$k \int \frac{d ρ}{ρ} + \frac{1}{2} q^{2} = C$ (using (3))
Integrating above, we get
$k \log ρ + q^{2} / 2 = C$ , $C$ being an arbitrary constant

$\begin{array}{lllll} When & q = v, & ρ = ρ_{1} & and & when & q = V, & ρ = ρ_{2} . \end{array}$ Hence (4) yields
$k \log ρ_{1} + v^{2} / 2 = C$ and $k \log ρ_{2} + V^{2 /} 2 = C$
Subtracting, $k (\log ρ_{2} - \log ρ_{1}) + (V^{2} - v^{2}) / 2 = 0$

$⟹ \log (ρ_{2} / ρ_{1}) = (v^{2} - V^{2}) 2 k$ or $ρ_{2} / ρ_{1} = e^{(v^{2} - V^{2}) 2 k}$
Using (5), (1) reduces to $v / V = (D^{2} / d^{2}) \times e^{(v^{2} - V^{2}) / 2 k}$

Hence, proved.

8.(a) Given the one-dimensional wave equation $\frac{\partial^{2} y}{\partial t^{2}} = c^{2} \frac{\partial^{2} y}{\partial x^{2}}$ ; $t > 0$ , where $c^{2} = \frac{T}{m}$ , $T$ the constant tension in the string and $m$ is the mass per unit length of the string.

(i) Find the appropriate solution of the wave equation.
(ii) Find also the solution under the conditions $y (0, t) = 0$ , $y (l, t) = 0$ for all $t$ and ${[\frac{\partial y}{\partial t}]}_{t = 0} = 0$ , $y (x, 0) = a \sin \frac{π x}{t}$ , $0 < x < l$ , $a > 0$ .

[20M]

8.(b) Write an algorithm in the form of a flow chart for Newton-Raphson method. Describe the cases of failure of this method.

[15M]

8.(c) If the velocity of an incompressible fluid at the point $(x, y, z)$ is given by $(\frac{3 x z}{r^{5}}, \frac{3 y z}{r^{5}}, \frac{3 z^{2} - r^{2}}{r^{5}})$ , $r^{2} = x^{2} + y^{2} + z^{2}$ then prove that the liquid motion is possible and that the velocity potential is $\frac{z}{r^{3}}$ . Further, determine the the streamlines.

[15M]

Here $u = \frac{3 x z}{r^{5}}$ , $v = \frac{3 y z}{r^{5}}$ , $w = \frac{3 z^{2} - r^{2}}{r^{5}} = \frac{3 z^{2}}{r^{5}} - \frac{1}{r^{3}} \dots$ (1) where
$r^{2} = x^{2} + y^{2} + z^{2} \dots$ (2)

From $(2), \partial r / \partial x = x / r, \partial r / \partial y = y / r, \partial r / \partial z = z / r$ From (1), ( 2 ) and (3), we have

\frac{\partial u}{\partial x} = 3 z [\frac{1}{r^{5}} + (- 5 x) r^{- 6} \frac{\partial r}{\partial x}] = \frac{3 z}{r^{5}} - \frac{15 x^{2} z}{r^{7}}

\frac{\partial v}{\partial y} = 3 z [\frac{1}{r^{5}} + (- 5 y) r^{- 6} \frac{\partial r}{\partial y}] = \frac{3 z}{r^{5}} - \frac{15 y^{2} z}{r^{7}}

\frac{\partial w}{\partial z} = \frac{6 z}{r^{5}} - 15 z^{2} r^{- 6} \frac{\partial r}{\partial z} + 3 r^{- 4} \frac{\partial r}{\partial z} = \frac{6 z}{r^{5}} - \frac{15 z^{2}}{r^{6}} \cdot \frac{z}{r} + \frac{3}{r^{4}} \cdot \frac{z}{r} = \frac{9 z}{r^{5}} - \frac{15 z^{3}}{r^{7}}

∴ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = \frac{15 z}{r^{5}} - \frac{15 z}{r^{7}} (x^{2} + y^{2} + z^{2}) = \frac{15 z}{r^{5}} - \frac{15 z}{r^{7}} \times r^{2} = 0

Since the equation of continuity is satisfied by the given values of $u, v$ and $w,$ the motion is possible. Let $ϕ$ be the required velocity potential. Then

\begin{array}{l} d ϕ = \frac{\partial ϕ}{\partial x} d x + \frac{\partial ϕ}{\partial y} d y + \frac{\partial ϕ}{\partial z} d z = - (u d x + v d y + w d z), by definition of ϕ \\ = - [\frac{3 x}{r^{5}} d x + \frac{3 y z}{r^{5}} d y + \frac{3 z^{2} - r^{2}}{r^{5}} d z] = \frac{r^{2} d z - 3 z (x d x + y d y + z d z)}{r^{5}} \\ Thus, d ϕ = \frac{r^{3} d z - 3 r^{2} z d r}{{(r^{3})}^{2}} = d (\frac{z}{r^{3}}), using (2) \end{array}

Integrating,

ϕ = z / r^{3}

[Omitting constant of integration, for it has no significance in $ϕ$ ]

In spherical polar coordinates $(r, θ, ϕ),$ we know that $z = r \cos θ .$ Hence the required potential is given by $ϕ = (r \cos θ) / r^{3} = (\cos θ) / r^{2}$ . We now obtain the streamlines. The equations of streamlines are given by

\frac{d x}{u} = \frac{d y}{v} = \frac{d z}{w} i.e., \frac{d x}{3 x z / r^{5}} = \frac{d y}{3 y z / r^{5}} = \frac{d z}{(3 z^{2} - r^{2}) / r^{5}}

So,

\frac{d x}{3 x z} = \frac{d y}{3 y z} = \frac{d z}{3 z^{2} - r^{2}} \dots (4)

Taking the first two members of (4) and simplifying, we get

d x / x = d y / y

⟹ d x / x - d y / y = 0

Integrating, $\log x - \log y = \log c_{1}$ l.e. $x / y = c_{1}, c_{1}$ being a constant $\dots (5)$

Now, each member in $(4) = \frac{x d x + y d y + z d z}{3 x^{2} z + 3 y^{2} z + 3 z^{3} - r^{2} z} = \frac{x d x + y d y + z d z}{3 z (x^{2} + y^{2} + z^{2}) - r^{2} z}$

= \frac{x d x + y d y + z d z}{3 z (x^{2} + y^{2} + z^{2}) - z (x^{2} + y^{2} + z^{2})} = \frac{x d x + y d y + z d z}{2 z (x^{2} + y^{2} + z^{2})}, by (2) \dots (6)

Taking the first member in (4) and (6), we get

\frac{d x}{3 x z} = \frac{x d x + y d y + z d z}{2 z (x^{2} + y^{2} + z^{2})} or \frac{2}{3} \frac{d x}{x} = \frac{1}{2} \frac{2 x d x + 2 y d y + 2 z d z}{x^{2} + y^{2} + z^{2}}

Integrating, $(2 / 3) \times \log x = (1 / 2) \times \log (x^{2} + y^{2} + z^{2}) + \log c_{2}$ or $x^{2 / 3} = c_{2} {(x^{2} + y^{2} + z^{2})}^{1 / 2}, c_{2}$ being an arbitrary constant. $\dots (7)$ .

The required streamlines are the curves of intersction of (5) and (7).

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