STUPIDSID
Short Questions
1(a)
The complex conjugate of i1−i is _______ .
1 M
1(b)
A mapping which preserves only magnitude is known as _______ mapping.
1 M
1(c)
IF z = cos θ + i sin θ then sin nθ = _______ .
1 M
1(d)
The value of ∫cez(z−3)2dz where C: |z| = 2 is _______ .
1 M
1(e)
Is the set |z-1+2i| ≤ 2 domain?
1 M
1(f)
Find the principal value of i(1-i).
1 M
1(g)
Prove limz→1iz3=i3 by definition.
1 M
1(h)
Show that sin(log ii)=-1
1 M
1(i)
Define residue
1 M
1(j)
While evaluating a definite integral by trepezoidal rule, the accuracy can be increased by taking _______ number of sub-intervals
1 M
1(k)
The relationship between E and is _______ .
1 M
1(l)
The order of convergence in Newton - Raphson method is _______ .
1 M
1(m)
Iterative formula for finding the square root of N by Newton - Raphson method is _______ .
1 M
1(n)
Putting n = 1 in the Newton-cote's quadrature formula rule obtained is _______ .
1 M
2(a)
Find all the roots of (1+i)2/3.
3 M
2(b)
Show that f(z)=log z is analytic everywhere except at the origin.
4 M
Solve any one question from Q.2(c) & Q.2(d)
2(c)
Prove that u = x2 - y2 and v=−yx2+y2 are harmonic but u + iv is not regular.
7 M
2(d)
Examine the nature of the function f(z)={x3y(y−ix)x6+y2,z≠00,z=0 in the region including the origin.
7 M
Solve any three question from Q.3(a), Q.3(b), Q.3(c) & Q.3(d), Q.3(e), Q.3(f)
3(a)
Find analytic function f(z) = u + iv if v = ex (x sin y + y cos y).
3 M
3(b)
Evaluate using Cauchy's integral formula ∫c3z2+z+1(z2−1)(z+3)dz where C is the circle |z|=2.
4 M
3(c)
Using contour integration evaluate the real integral ∫2π0cos3θ5−4cosθ.
7 M
3(d)
If f(z)=u+iv is analytic in domain D then prove that (∂2∂x2+∂2∂y2)|Re(f(z)|2=2|f′(z)|2
3 M
3(e)
Determine the linear fractional transformation that maps z1 = 0, z2 = 1, z3=&infin onto w1, w2=-i, w3=1 repectively.
4 M
3(f)
Expand f(z)=1(z+2)(z+4) valid for the region (i) |z| < 2 (ii) 2 < |z| < 4 (iii) |z| > 4
7 M
Solve any three question from Q.4(a), Q.4(b), Q.4(c) & Q.4(d), Q.4(e), Q.4(f)
4(a)
Find the dominant Eigen values of A=[3−5−24] by power method.
3 M
4(b)
Evaluate ∫60dx1+x2 by using (i) Trapezoidal rule (ii) Simpson's 1/3 rule taking h=1.
4 M
4(c)
Prove that the transformation w=z1−z maps the upper half of the z-plane unto the upper half of the w-plane. What is the image of |z| = 1 under this transformation?
7 M
4(d)
Solve the system of equations by Gauss-Seidal Method
10x1 + x2 + x3 = 6
x1 + 10x2 + x3 = 6
x1 + x2 + 10x3 = 6
10x1 + x2 + x3 = 6
x1 + 10x2 + x3 = 6
x1 + x2 + 10x3 = 6
3 M
4(e)
Evaluate the integral ∫10dt1+t by one point, two point and three point Gaussian formula.
4 M
4(f)
The population of the town is given below, estimate the population of the year 1895 and 1930 using suitable interpolation
| Year , x | 1891 | 1901 | 1911 | 1921 | 1931 |
|
Population (in thousand) f(x) |
46 | 66 | 81 | 93 | 101 |
7 M
Solve any three question from Q.5(a), Q.5(b), Q.5(c) & Q.5(d), Q.5(e), Q.5(f)
5(a)
Find up to four decimal places the root of the equation sin x=e-x , using Newtons Rapson's Method starting with x0=0.6.
3 M
5(b)
Find the negative root of x3 - 7x + 3 = 0 by bisection method up to two decimal places.
4 M
5(c)
Apply improved Euler method to solve the initial value problem dydx=log(x+y) with y(1) = 2 taking h = 0.2 for x = 1.2 and x = 1.4 correct up to four decimal places.
7 M
5(d)
Express the function 3x2−12x+11(x−1)(x−2)(x−3) as a sum of partial fraction, using Lagrange's formula.
3 M
5(e)
Using Newton's divided difference formula find a polynomial and also find f(6).
| x | 1 | 2 | 4 | 7 |
| f(x) | 10 | 15 | 67 | 430 |
4 M
5(f)
Apply fourth order Runge ' Kutta Method to find y(0, 2) given dydx=x+y, y(0) = 1 (Taking h = 0.1).
7 M
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