Paper II PYQs-2011

1.(a) Show that the set \(G=\left\{f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}\right\}\) of six transformations on the set of Complex numbers defined by \(f_{1}(z)=z\),
\(f_{2}(z)=1-z\),
\(f_{3}(z)=\dfrac{z}{(1-z)}\),
\(f_{4}(z)=\dfrac{1}{z}\),
\(f_{5}(z)=\dfrac{1}{(1-z)}\),
\(f_{6}(z)=\dfrac{(z-1)}{z}\)
is a non-abelian group of order 6 wrt composition of mappings.

[12M]

1.(b) Let \(S=(0, 1)\) and \(f\) be defined by \(f(x)=\dfrac{1}{x}\) where \(0 < x \leq 1\) (in \(R\)). Is \(f\) uniformly continuous on \(S\)? Justify your answer.

[12M]

1.(c) If \(f(z)=u+i v\) is an analytic function of \(z=x+i y\) and \(u-v=\dfrac{e^{y}-\cos x+\sin x}{\cosh y-\cos x}\), find \(f(z)\) subject to the condition, \(f\left(\dfrac{\pi}{2}\right)=\dfrac{3-i}{2}\).

[12M]

1.(d) Solve by Simplex method. Maximize
\(Z=5 x_{1}+x_{2}\),
subject to constraints:
\(3 x_{1}+5 x_{2} \leq 15\)
\(5 x_{1}+2 x_{2} \leq 10\)
\(x_{1}\), \(x_{2} \geq 0\)

[12M]

1.(e)(i) Prove that a group of Prime order is abelian.

[6M]

1.(e)(ii) How many generators are there of the cyclic group \((G, .)\) of order 8?

[6M]

2.(a) Give an example of a group \(G\) in which every proper subgroup is cyclic but the group itself is not cyclic.

[15M]

2.(b) Let \(f_{n}(x)=n x(1-x)^{n}\), \(x \in[0, 1]\). Examine the uniform convergence of \(\left\{f_{n}(x)\right\}\) on \([0, 1]\).

[15M]

2.(c) If the function \(f(z)\) is analytic and one valued in \(\vert z-a \vert < R\), prove that for \(0< r < R\), \(f^{\prime}(a)=\dfrac{1}{\pi r} \int_{0}^{2 \pi} \mathrm{P}(\theta) e^{-i \theta} d \theta\), where \(P(\theta)\) is the real part of \(f\left(a+r e^{i \theta}\right)\).

[15M]

2.(d) Find the shortest distance from the origin \((0,0)\) to the hyperbola \(x^{2}+8 x y+7 y^{2}=225\).

[15M]

3.(a) Let \(F\) be the set of all real valued continuous functions defined on the closed interval \([0,1]\). Prove that \((F,+, .)\) is a Commutative Ring with unity with respect to addition and multiplication of functions defined point wise as below:-
\((f+g) x=f(x)+g(x)\) and \((f g) x=f(x) g(x)\)
\(x \in[0,1]\) where \(f, g \in F\).

[15M]

3.(b) Show that the series for which the sum of first \(n\) terms, \(f_{n}(x)=\dfrac{n x}{1+n^{2} x^{2}}\), \(0 \leq x \leq 1\) cannot be differentiated term-by-term at \(x=0\). What happens at \(x \neq 0\)?

[15M]

3.(c) Evaluate by Contour integration, \(\int_{0}^{1} \dfrac{d x}{\left(x^{2}-x^{3}\right)^{\dfrac{1}{3}}}\).

[15M]

3.(d) Find the Laurent series for the function \(f(z)=\dfrac{1}{1-z^{2}}\) with centre \(z=1\).

[15M]

4.(a) Let \(a\) and \(b\) be elements of a group, with \(a^{2}=e\), \(b^{6}=e\) and \(a b=b^{4} a\). Find the order of \(a b\), and express its inverse in each of the forms \(a^{m} b^{n}\) and \(b^{m} a^{n}\).

[20M]

4.(b) Show that if \(S(x)=\sum_{n=1}^{\infty} \dfrac{1}{n^{3}+n^{4} x^{2}}\), then its derivative \(S^{\prime}(x)=-2 x \sum_{n=1}^{\infty} \dfrac{1}{n^{2}\left(1+n x^{2}\right)^{2}}\), for all \(x\).

[20M]

4.(c) Write down the dual of the following LP problem and hence solve it by graphical method.
Minimize \(Z=6 x_{1}+4 x_{2}\),
subject to constraints:
\(2 x_{1}+x_{2} \geq 1\)
\(3 x_{1}+4 x_{2} \geq 1.5\)
\(x_{1}, x_{2} \geq 0\)

[20M]

5.(a) Solve the PDE \(\left(D^{2}-D^{\prime 2}+D+3 D^{\prime}-2\right) z\)=\(e^{(x-y)}-x^{2} y\).

[12M]

5.(b) Solve the PDE \((x+2 z) \dfrac{\partial z}{\partial x}+(4 z x-y) \dfrac{\partial z}{\partial y}=2 x^{2}+y\).

[12M]

5.(c) Calculate \(\int_{2}^{10} \dfrac{d x}{1+x}\) (up to 3 places of decimal) by dividing the range into 8 equal parts by Simpson’s \(\dfrac{1}{3}\)rd rule.

[12M]

5.(d)(i) Compute \((3205)_{10}\) to the base 8.

[6M]

5.(d)(ii) Let \(A\) be an arbitrary but fixed Boolean algebra with operations \(\wedge\), \(\vee\) and \('\), and the zero and the unit element denoted by 0 and 1 respectively. Let \(x\), \(y\), \(z \ldots\) be elements of \(A\). If \(x, y\in A\) be such that \(x \wedge y=0\) and \(x \vee y=1\), then prove that \(y=x^{\prime}\).

[6M]

5.(e) Let \(a\) be the radius of the base of a right circular cone of height \(h\) and mass \(M\) . Find the moment of inertia of that right circular cone about a line through the vertex perpendicular to the axis.

[12M]

6.(a) Find the surface satisfying \(\dfrac{\partial^{2} z}{\partial x^{2}}=6 x+2\) and touching \(z=x^{3}+y^{3}\) along its section by the plane \(x+y+1=0\).

[20M]

6.(b) Solve \(\dfrac{\partial^{2} u}{\partial x^{2}}+\dfrac{\partial^{2} u}{\partial y^{2}}=0,0 \leq x \leq a, 0 \leq y \leq b\) satisfying the boundary conditions \(u(0, y)=0\), \(u(x, 0)=0\), \(u(x, b)=0\), \(\dfrac{\partial u}{\partial x}(a, y)=T \sin ^{3} \dfrac{\pi y}{a}\).

[20M]

6.(c) Obtain temperature distribution \(y(x, t)\) in a uniform bar of unit length whose one end is kept at \(10^{0}\) and the other end is insulated. Also it is given that \(y(x, 0)=1-x\), \(0<x<1\).

[20M]

7.(a) A solid of revolution is formed by rotating about the \(x-axis\), the area between the \(x-axis\), the line \(x=0\) and \(x=1\) and a curve through the points with the following co-ordinates:
\(\begin{array}{|c|c|c|c|c|c|}\hline x & {0.00} & {0.25} & {0.50} & {0.75} & {1} \\ \hline y & {1} & {0.9896} & {0.9589} & {0.9089} & {0.8415} \\ \hline\end{array}\)

Find the volume of the solid.

[20M]

7.(b) Find the logic circuit that represents the following Boolean function. Find also an equivalent simpler circuit:
\(\quad \begin{array}{|c|c|c|c|}\hline x & {y} & {z} & {f(x, y, z)} \\ \hline 1 & {1} & {1} & {1} \\ \hline 1 & {1} & {0} & {0} \\ \hline 1 & {0} & {1} & {0} \\ \hline 1 & {0} & {1} & {0} \\ \hline 0 & {1} & {1} & {1} \\ \hline 0 & {1} & {0} & {0} \\ \hline 0 & {0} & {1} & {0} \\ \hline 0 & {0} & {0} & {0} \\ \hline\end{array}\)

[20M]

7.(c) Draw a flow chart for Lagrange’s interpolation formula.

[20M]

8.(a) The ends of a heavy rod of length \(2a\) are rigidly attached to two light rings which can respectively slide on the thin smooth fixed horizontal and vertical wires \(O\). The rod starts at an angle \(\alpha\) to the horizon with an angular velocity \(\sqrt{[3 g(1-\sin \alpha) / 2 a]}\) and moves downwards. Show that it will strike the horizontal wire at the end of time \(-2 \sqrt{a /(3 g)} \log \left[\tan \left(\dfrac{\pi}{8}-\dfrac{\alpha}{4}\right) \cot \dfrac{\pi}{8}\right]\).

[30M]

8.(b) An infinite row of the equidistance rectilinear vortices are at distance \(a\) apart. The vortices are of the same numerical strength \(K\) but they are alternately of opposite signs. Find the Complex function that determines the velocity potential and the stream function.

[30M]