STUPIDSID
1(a)i
Sketch the region \[1<\left | z+i \right |\leq 2\] and check whether the region is domain or not?
3 M
1(a)ii
Find all the roots of the equation \[\log z=\dfrac{i\pi}{2}\] .
2 M
1(a)iii
Evaluate \[\int _c \dfrac{dz}{z^2}\] , C is along a unit circle.
2 M
1(b)i
Obtain the Taylor's series f(z) = sin z in power of \[\left ( z-\dfrac{\pi}{4} \right )\]
3 M
1(b)ii
Find the Laurent's expansion of \[\frac{\sin z}{z^3}\] at z = 0 and classify the singular point z = 0.
2 M
1(b)iii
Is \[f(z)=\sqrt{re}^{\dfrac{i\theta}{2}}\] analytic? ( r>0, -π < \theta < π)
2 M
2(a)
Show that for the function \[f(z)=\left\{\begin{matrix}
\dfrac{z^{-2}}{z}; & z\neq 0\\
0\ ; & z=0
\end{matrix}\right.\] Is not differentiable at z = 0 even though Cauchy Riemann equation
are satisfied at z = 0 .
7 M
Solved any one question from Q.2(b) & Q.2(c)
2(b)i
Discuss the convergence of the series \[\sum \frac{(2n)!(z-3i)^n}{(n!)^2}\]
4 M
2(b)ii
Discuss continuity of \[f(z)=\left\{\begin{matrix}
\dfrac{Re(z^2)}{|z|^2} ;& z\neq 0\\
0; & z=0
\end{matrix}\right.\] at z = 0.
3 M
2(c)i
State De-Moivre's formula .Find all the root of \[(8i)^{\dfrac{1}{3}}\] in the complex plane.
4 M
2(c)ii
Evaluate \[\int _c (x^2-iy^2)dz\] along the parabola y = 2x2 from (1, 2) to (2, 8).
3 M
Solved any one question from Q.3 & Q.4
3(a)
Using Residue theorem, evaluate \[\int ^{2\pi}_0\dfrac{d\theta}{5-3\sin \theta}\]
7 M
3(b)i
Evaluate \[\int _c \dfrac{1+z^2}{1-z^2}dz\] , where c is unit circle centred at
(1) z = -1
(2) z = i
(1) z = -1
(2) z = i
4 M
3(b)ii
Find the image in the w - plane of the circle |z-3|=2 in the z- plane under the inversion mapping \[w=\dfrac{1}{z}\] .
3 M
4(a)i
Show that u(x, y) = x2 - y2 is harmonic in some domain and find the harmonic conjugate v ( x , y ) .
4 M
4(a)ii
Find the Bilinear transformation which maps z = 1 , i , - 1 into ω = 2, i, -2.
3 M
4(b)i
Determine the poles of the function \[f(z)=\dfrac{z^2}{(z-1)^2(z+2)}\] and residue at each pole. Evaluate \[\int _c f(z)dz\] , where c is the circle |z| = 3.
4 M
4(b)ii
Evaluate \[ \int _{c:|z|=2}\dfrac{dz}{z^3(z+4)}\] .
3 M
Solved any one question from Q.5 & Q.6
5(a)
Find the Langrage's interpolation polynomial from the following
Data :
Also find f(2).
| x | 0 | 1 | 4 | 5 |
| f(x) | 1 | 3 | 24 | 39 |
Also find f(2).
7 M
5(b)i
Using partial pivoting solve the system of equation by Gauss
Elimination method :
x+y+z=7; 3x+3y+4z=24; 2x+y+3z=16
x+y+z=7; 3x+3y+4z=24; 2x+y+3z=16
4 M
5(b)ii
Find a root of the equation x3 -4x-9=0 using False-position method correct up to three decimal.
3 M
6(a)i
Using Newton's Backward interpolation formula , evaluate f (300)
from the given table:
| x | 50 | 100 | 150 | 200 | 250 |
| y(x | 618 | 724 | 805 | 906 | 1032 |
4 M
6(a)ii
Find real root of x3 5x + 3 = 0 correct up to three decimal using Newton-Raphson method.
3 M
6(b)i
Solve the system of equation by Gauss Seidel method 10x+y+z = 6; x+10y+z = 6; x+y+10z = 6
4 M
6(b)ii
Using Newton's Divided difference formula find f (3) from the following table
| x | -1 | 2 | 4 | 5 |
| f(x) | -5 | 13 | 255 | 625 |
3 M
7(a)i
Using the power method find the largest Eigen value for the matrix \[A=\begin{bmatrix}
1 & -3 & 2\\
4 & 4 & -1\\
6 & 3 & 5
\end{bmatrix}\]
7 M
7(b)i
Use Runge-Kutta method of second order to find the appropriate value of y (0.2) given that \[\dfrac{dy}{dx}=x-y^2\] ; y(0) and h=0.1
4 M
7(b)ii
Evaluate \[\int ^1_0 \dfrac{dx}{1+x^2}\] using Trapezoidal rule taking \[h=\dfrac{1}{5}\]
3 M
8(a)i
Given \[\dfrac{dy}{dx}=\dfrac{y-x}{y+x}\] with initial condition y = 1 at x = 0 ; find y for x = 1.0 and h = 0.25 by Euler's method
4 M
8(a)ii
Using Gauss Forward interpolation formula , evaluate f (55) form
the given table :
| x | 40 | 50 | 60 | 70 |
| f(x) | 836 | 682 | 436 | 272 |
3 M
8(b)i
Evaluate \[\int ^1_0\dfrac{dt}{1+t}\] by Gaussian formula for n = 2 and n = 3.
4 M
8(b)ii
Prove that E = ehd.
3 M
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