Paper II PYQs-2013
Section A
1.(a) Evaluate: limx→0(eax−ebx+tanxx)
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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 f(z)=u(x,y)+iv(x,y), where v(x,y)=6xy−5x+3 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|∞| 12 | 10 | 15 | |B| 16 |∞| 11 | 13 | |C| 17 | 18 |∞| 20 | |D| 13 | 11 | 18 |∞|
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3.(a) Show that the function f(x)=x2 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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| 3.(c) Evaluate ∮ce2z(z+1)4dz where c is the circle $$ | z | =3$$ |
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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. z−sinzz3;z=0
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4.(c) x1=4,x2=1,x3=3 is a feasible solution of the system of equations 2x1−3x2+x3=8x1+2x2+3x3=15 Reduce the feasible solution to two different basic feasible solutions.
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Section B
5.(a) Use Newton - Raphson method and derive the iteration scheme xn+1=12(xn+Nxn) to calculate an approximate value of the square root of a number N. Show that the formula √N≈A+B4+NA+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 √2 [For 2nd iteration, one may take A= result of the 1st iteration].
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5.(b) Eliminate the arbitrary function f from the given equation f(x2+y2+z2,x+y+z)=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: xux+yuy+zuz=xyz
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6.(b) Convert (0⋅231)5,(104⋅231)5 and (247)7 to base 10.
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6.(c) Rewrite the hyperbolic equation x2uxx−y2uyy=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 →v=(3y2−ax2)ˆi+ bxy ˆj 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 (∂u∂x)2+(∂u∂y)2=1 that passes through the circle x2+y2=1,u=1
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8.(a) Solve the following heat equation, using the method of separation of variables: ∂u∂t=∂2u∂x2,0<x<1,t>0 subject to the conditions u=0 at x=0 and x=1, for t>0 u=4x(1−x), at t=0 for 0≤x≤1
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8.(b) Use the Classical Fourth-order Runge-Kutta method with h=0⋅2 to calculate a solution at x=0⋅4 for the initial value problem dudx=4−x2+u,u(0)=0 on the interval [0,0⋅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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