STUPIDSID
Do as Direct:
1 (a)
Solve the equation z2+(2i-3)z+5-i=0.
2 M
1 (b)
Discuss the differentiability of f(z)=x2+iy2.
2 M
1 (c)
Discuss the continuity of \[ f(z)= \left\{\begin{matrix}
\frac {\bar z}{0}; & z \ne 0 \\0;
&z=0
\end{matrix}\right. \]
2 M
1 (d)
Find the image of infinite strip 0≤x≤1. Under the transformation ω=iz+1. Sketch the region in ω-plane.
2 M
1 (e)
Find the second divided difference for the argument x=1, 2, 5 and 7 for the function f(x)=x2.
2 M
1 (f)
Evaluate \[ \oint_c \dfrac {e^z}{(z+i)}dz, \] where C:|z-1|=1.
2 M
1 (g)
Expand \( f(z)= \dfrac {z-\sin z} {z^2} \) at z=0, classify the singular point z=0.
2 M
2 (a) (i)
Find the analytic function f(z)=u+iv, if u-v=ex (cos y-sin y).
4 M
2 (a) (ii)
If f(z)=u+iv is analytic in domain D then prove that: \[ \left ( \dfrac {\partial ^2}{\partial x^2}+ \dfrac {\partial ^2}{\partial y^2} \right )\Big \vert Re (f(z))\Big \vert^2 = 2 \Big \vert f'(z) \Big \vert ^2 \]
3 M
Answer any one question from Q2 (b) & Q2 (c)
2 (b)
Using theory of residue, evaluate \[ \int^\infty_{-\infty} \dfrac {\cos x}{x^2+1} dx \]
7 M
2 (c)
Expand \[ f(z)=\dfrac {1}{(z+1)(z-2) \] in Laurent's series in the region (i) |z|<1 (ii) 1<|z|<2 (iii) |z|>2.
7 M
Answer any two question from Q3 (a), (b) & Q3 (c), (d)
3 (a) (i)
Evaluate \(\int_c \overline{z}dz \) where C is along the sides of triangle having vertices z=0.1,i.
4 M
3 (a) (ii)
Determine bilinear transformation which maps the points z=0, i, 1 into ω=i,-1,∞.
3 M
3 (b) (i)
Determine the points where \(\omega = \left ( z+ \dfrac {1}{z} \right ) \) is not conformal mapping. Also Find the image of circle |z|=2 under the transformation \( \omega = \left ( z+ \dfrac {1}{z} \right ) \)
4 M
3 (b) (ii)
Find the radius of convergence of \[ \sum^\infty_{n=1} \left ( \dfrac {6n+1}{2n+5} \right )^2 (z-2i)^{n} \]
3 M
3 (c) (i)
Using residue theorem, evaluate \[ \int_c \dfrac {e^z+z}{z^3-z}dz, \ where \ C:\ |z|=\dfrac {\pi}{2} \]
4 M
3 (c) (ii)
State Cauchy integral theorem. Evaluate \[ \int_c \left (\dfrac {3}{z-i} - \dfrac {6}{(z-i)^z} \right )dz, \ where \ C:|z|=2. \]
3 M
3 (d) (i)
If \(x+\dfrac {1}{x} =2 \cos \theta, \) prove that \((i) \ x^n+\dfrac {1}{x^n} = 2 \cos \ n\theta \ and \\ \dfrac {x^{2n}+1}{x^{2n-1}+x} = \dfrac {\cos n \theta}{\cos (n-1)\theta} \)
4 M
3 (d) (ii)
Determine and sketch the image of region 0≤x≤1, 0≤y≤π under the transformation ω=ez.
3 M
Answer any two question from Q4 (a), (b) & Q4 (c), (b)
4 (a) (i)
Construct Newton's forward interpolation polynomial for the following data:
Hene y for x=5.
Hene y for x=5.
| x: | 4 | 6 | 8 | 10 |
| y: | 1 | 3 | 8 | 16 |
4 M
4 (a) (ii)
Evaluate \( \int^1_0 e^{-3x^2} dx \) by using Gaussian Quadrature formula with n=3.
3 M
4 (b)
Using the power method, find the largest eigen value of the \[ A=\begin{bmatrix} 2 &-1 &0 \\-1 &2 &-1 \\0
&-1 &2 \end{bmatrix} \]
7 M
4 (c) (i)
Find the real root of the equation x \(\log_{10}^x \) =1.2 by Regula false method.
4 M
4 (c) (ii)
A river is 80 meters wide. The depth 'd' in meter at a distance x meters from one bank is given by the following table. Calculate the area of cross section of the river using Simpon's \( \dfrac {1}{3}rd \) rule.
| x: | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
| y: | 0 | 4 | 7 | 9 | 12 | 15 | 14 | 4 | 8 |
3 M
4 (d)
Use Largrange's method to find polynomial of degree three for the data
hence find the value of x=2.
hence find the value of x=2.
| x: | -1 | 0 | 1 | 3 |
| y: | 2 | 1 | 0 | -1 |
7 M
Answer any two question from Q5(a), (b), (c) & Q5 (d), (e), (f)
5 (a)
State diagonal dominant property. Using Gauss-Seided method to solve
6x+y+z=105, 4x+8y+3z=155, 5x+4y-10z=65.
6x+y+z=105, 4x+8y+3z=155, 5x+4y-10z=65.
7 M
5 (b)
Using the Runge- Kutta method of fourth order, Solve 10 \(\dfrac {dy}{dx}=x^2+y^2, \) y(0)=1 at x=0.2 and x=0.4 taking h=0.1.
7 M
5 (c)
Derive Euler's formula for initial value problem \(\dfrac {dy}{dx}=f(x,y); \ y(x_0)=y_0 \) Hence. Use it find the value of y for \(\dfrac {dy}{dx}= x+y; \ y(0)=1 \) when x=0.1, 0.2 with step size h=0.05. Also Compare with analytic solution.
7 M
5 (d) (i)
Use Gauss elimination method to solve the equation.
x+4y-z=-5, x+y-6z=-12, 3x-y-z=4.
x+4y-z=-5, x+y-6z=-12, 3x-y-z=4.
4 M
5 (d) (ii)
Use Newton- Raphson method, derive the iteration formula for \( \sqrt{N} \) . Also find \(\sqrt{28} \) .
3 M
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