Paper II PYQs-2013

1.(a) Evaluate: \(\lim _{x \rightarrow 0}\left(\dfrac{e^{a x}-e^{b x}+\tan x}{x}\right)\)

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1.(b) Prove that if every element of a group \(( G , 0)\) be its own inverse, then it is an abelian group.

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1.(c) Construct an analytic function \(\begin{array}{l} f(z)=u(x, y)+i v(x, y), \text { where } \\ v(x, y)=6 x y-5 x+3 \end{array}\) Express the result as a function of \(z\).

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1.(d)Find the optimal assignment cost from the following cost matrix : | |\(A\)|\(B\)|\(C\)|\(D\)| |—–|—|—|—|—| |\(I\) | 4 | 5 | 4 | 3 | |\(II\) | 3 | 2 | 2 | 6 | |\(III\)| 4 | 5 | 3 | 5 | |\(IV\) | 2 | 4 | 2 | 6 |

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2.(a) Show that any finite integral domain is a field.

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2.(b) Every field is an integral domain. Prove it.

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2.(c) Solve the following Salesman problem : | | \(A\) | \(B\) | \(C\) | \(D\) | |—|——–|——–|——–|——–| |\(A\)|\(\infty\)| 12 | 10 | 15 | |\(B\)| 16 |\(\infty\)| 11 | 13 | |\(C\)| 17 | 18 |\(\infty\)| 20 | |\(D\)| 13 | 11 | 18 |\(\infty\)|

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3.(a) Show that the function \(f ( x )= x ^{2}\) is uniformly continuous in \((0,1)\) but not in \(R\).

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3.(b) Prove that: (i) the intersection of two ideals is an ideal. (ii) a field has no proper ideals.

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4.(a) Find the area of the region between the x-axis and \(y =( x -1)^{3}\) from \(x =0\) to \(x =2\).

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4.(b) Find Laurent series about the indicated singularity. Name the singularity and give the region of convergence. \(\dfrac{z-\sin z}{z^{3}} ; z=0\)

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4.(c) \(x _{1}=4, x _{2}=1, x _{3}=3\) is a feasible solution of the system of equations \(\begin{array}{l} 2 x_{1}-3 x_{2}+x_{3}=8 \\ x_{1}+2 x_{2}+3 x_{3}=15 \end{array}\) Reduce the feasible solution to two different basic feasible solutions.

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5.(a) Use Newton - Raphson method and derive the iteration scheme \(x_{n+1}=\dfrac{1}{2}\left(x_{n}+\dfrac{N}{x_{n}}\right)\) to calculate an approximate value of the square root of a number \(N\). Show that the formula \(\sqrt{ N } \approx \dfrac{ A + B }{4}+\dfrac{ N }{ A + B }\), where \(AB = N ,\) can easily be obtained if the above scheme is applied two times. Assume \(A =1\) as an initial guess value and use the formula twice to calculate the value of \(\sqrt{2}\) [For \(2^{\text {nd }}\) iteration, one may take \(A=\) result of the \(1^{\text {st }}\) iteration].

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5.(b) Eliminate the arbitrary function \(f\) from the given equation \(f\left(x^{2}+y^{2}+z^{2}, x+y+z\right)=0\)

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5.(c) Derive the Hamiltonian and equation of motion for a simple pendulum.

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6.(a) Solve the PDE: \(x u_{x}+y u_{y}+z u_{z}=x y z\)

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6.(b) Convert \((0 \cdot 231)_{5},(104 \cdot 231)_{5}\) and \((247)_{7}\) to base \(10\).

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6.(c) Rewrite the hyperbolic equation \(x^{2} u_{x x}-y^{2} u_{y y}=0(x>0, y>0)\) in canonical form.

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7.(a) Find the values of \(a\) and \(b\) in the \(2-D\) velocity field \(\vec{v}=\left(3 y^{2}-a x^{2}\right) \hat{i}+\) bxy \(\hat{j}\) so that the flow becomes incompressible and irrotational. Find the stream function of the flow.

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7.(b) Write an algorithm to find the inverse of a given non-singular diagonally dominant square matrix using Gauss - Jordan method.

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7.(c) Find the solution of the equation \(\left(\dfrac{\partial u}{\partial x}\right)^{2}+\left(\dfrac{\partial u}{\partial y}\right)^{2}=1\) that passes through the circle \(x^{2}+y^{2}=1, u=1\)

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8.(a) Solve the following heat equation, using the method of separation of variables: \(\dfrac{\partial u }{\partial t}=\dfrac{\partial^{2} u }{\partial{ x }^{2}}, 0< x <1, t>0\) subject to the conditions \(u =0\) at \(x =0\) and \(x =1,\) for \(t >0\) \(u =4 x (1- x ),\) at \(t =0\) for \(0 \leq x \leq 1\)

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8.(b) Use the Classical Fourth-order Runge-Kutta method with \(h=0 \cdot 2\) to calculate a solution at \(x=0 \cdot 4\) for the initial value problem \(\dfrac{ du }{ dx }=4- x ^{2}+ u , u (0)=0\) on the interval \([0,0 \cdot 4]\)

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8.(c) Draw a flow chart for testing whether a given real number is a prime or not.

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