Paper II PYQs-2015
Section A
1.(a) If in a group G there is an element \(a\) of order \(360,\) what is the order of \(a^{220}\) ? Show that if \(G\) is a cyclic group of order \(n\) and \(m\) divides \(n ,\) then \(G\) has a subgroup of order \(m\).
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1.(b) Let \(\sum_{ n =1}^{\infty} a _{ n }\) be an absolutely convergent series of real numbers. Suppose \(\sum_{ n =1}^{\infty} a _{2 n }=\dfrac{9}{8}\) and \(\sum_{ n =0}^{\infty} a _{2 n +1}=\dfrac{-3}{8} .\) What is \(\sum_{ n =1}^{\infty} a _{ n } ?\) Justify your answer. (Majority of marks is for the correct justification).
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1.(c) Let \(u ( x , y )=\cos x\sinh y .\) Find the harmonic conjugate \(v ( x , y )\) of \(u\) and express \(u ( x , y )+ i v ( x , y )\) as a function of \(z = x + iy\)
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1.(d) Solve graphically: Maximize \(z=7 x+4 y\) subject to \(2 x+y \leq 2, x+10 y \leq 10\) and \(x \leq 8\) (Draw your own graph without graph paper).
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2.(a) If \(p\) is a prime number and \(e\) a positive integer, what are the elements \('a'\) in the ring \(Z_{ p }e\) of integers modulo \(p ^{ e }\) such that \(a ^{2}= a ?\) Hence (or otherwise) determine the elements in \(Z _{35}\) such that \(a ^{2}= a\).
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2.(b) Let \(X =( a , b ]\). Construct a continuous function \(f : X \rightarrow R (\) set of real numbers) which is unbounded and not uniformly continuous on X. Would your function be uniformly continuous on \([ a +\varepsilon, b ], a +\varepsilon< b ?\) Why?
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2.(c) Evaluate the integral \(\int_{ r } \dfrac{ z ^{2}}{\left( z ^{2}+1\right)( z -1)^{2}} dz\), where \(r\) is the circle \(\vert z \vert=2\).
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3.(a) What is the maximum possible order of a permutation in \(S_{8},\) the group of permutations on the eight numbers \(\{1,2,3, \ldots, 8\}\)? Justify your answer. (Majority of marks will be given for the justification).
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| 3.(b) Let \(f_{n}(x)=\dfrac{x}{1+n x^{2}}\) for all real \(x\). Show that \(f_{n}\) converges uniformly to a function \(f\). What is \(f ?\) Show that for \(x \neq 0, f _{ n }^{\prime}( x ) \rightarrow f ^{\prime}( x )\) but \(f _{ n }^{\prime}(0)\) does not converge to \(f ^{\prime}(0) .\) Show that the maximum value $$\left | f _{ n }( x )\right | \(can take is\)\dfrac{1}{2 \sqrt{ n }}$$ |
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3.(c) A manufacturer wants to maximise his daily output of bulbs which are made by two processes \(P_{1}\) and \(P_{2}\). If \(x_{1}\) is the output by process \(P_{1}\) and \(x _{2}\) is the output by process \(P _{2}\), then the total labour hours is given by \(2 x_{1}+3 x_{2}\) and this cannot exceed \(130,\) the total machine time is given by \(3 x _{1}+8 x _{2}\) which cannot exceed 300 and the total raw material is given by \(4 x _{1}+2 x _{2}\) and this cannot exceed \(140 .\) What should \(x _{1}\) and \(x _{2}\) be so that the total output \(x_{1}+x_{2}\) is maximum \(?\) Solve by the simplex method only.
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4.(a) Compute the double integral which will give the area of the region between the \(y\) -axis, the circle \((x-2)^{2}+(y-4)^{2}=z^{2}\) and the parabola \(2 y=x^{2}\). Compute the integral and find the area.
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4.(b) Show that \(\int_{-\infty}^{\infty} \dfrac{ x ^{2}}{1+ x ^{4}} d x =\dfrac{\pi}{\sqrt{2}}\) by using contour integration and the residue theorem.
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4.(c) Solve the following transportation problem:
| \(D_1\) | \(D_2\) | \(D_3\) | Supply | |
|---|---|---|---|---|
| \(O_1\) | 5 | 3 | 6 | 20 |
| \(O_2\) | 4 | 7 | 9 | 40 |
| Demand | 15 | 22 | 23 | 60 |
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Section B
5.(a) Store the value of -1 in hexadecimal in a 32-bit computer.
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5.(b) Show that \(\sum_{ k =1}^{ n } l_{ k }( x )=1,\) where \(l_{ k }( x ), k =1\) to \(n ,\) are Lagrange’s fundamental polynomials.
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5.(c) Derive the Hamiltonian and equation of motion for a simple pendulum.
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5.(d) Find the solution of the equation \(u_{x x}-3 u_{x y}+u_{y y}=\sin (x-2 y)\).
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6.(a) Solve the following system of linear equations correct to two places by Gauss-Seidel method: \(x+4 y+z=-1,3 x-y+z=6, x+y+2 z=4\)
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6.(b) Solve the differential equation \(u_{x}^{2}-u_{y}^{2}\) by variable separation method.
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6.(c) In a steady fluid flow, the velocity components are \(u =2 k x , v =2 ky\) and \(w =-4 kz\). Find the equation of a streamline passing through (1,0,1)
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7.(a) Solve the heat equation \(\dfrac{\partial u }{\partial t }=\dfrac{\partial^{2} u }{\partial x ^{2}}, 0< x <1, t >0\) subject to the conditions \(u (0, t )= u (1, t )=0\) for \(t >0\) and \(u(x, 0)=\sin \pi x, 0< x <1\)
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7.(b) Find the moment of inertia of a uniform mass \(M\) of a square shape with each side a about its one of the diagonals.
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7.(c) Use the classical fourth order Runge-Kutta methods to find solutions at \(x =0 \cdot 1\) and \(x =0 \cdot 2\) of the differential equation \(\dfrac{ dy }{ dx }= x + y , y (0)=1\) with step size \(h =0 \cdot 1\).
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8.(a) Write a BASIC program to compute the product of two matrices.
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8.(b) Suppose \(\overrightarrow{ v }=( x -4 y ) \hat{ i }+(4 x - y ) \hat{ j }\) represents a velocity field of an incompressible and irrotational flow. Find the stream function of the flow.
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8.(c) Solve the wave equation \(\dfrac{\partial^{2} u }{\partial t^{2}}=c^{2} \dfrac{\partial^{2} u}{\partial x^{2}}\) for a string of length \(l\) fixed at both ends. The string is given initially a triangular deflection
\(u ( x , 0)=\left\{\begin{array}{ccc}\dfrac{2}{l} x, & \text { if } 0< x <\dfrac{l}{2} \\ \dfrac{2}{l}(l- x ), & \text { if } & \dfrac{l}{2} \leq x < l\end{array}\right.\)
with initial velocity \(u_t(x,0)=0\).
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