Short Questions

1(a)
The complex conjugate of \( \dfrac{i}{1-i} \) 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 \( \int _c \dfrac{e^z}{(z-3)^2}dz \) 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 \( \lim_{z\rightarrow 1}\dfrac{iz}{3}=\dfrac{i}{3} \) by definition.

1 M

1(h)
Show that sin(log i

^{i})=-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 = x

^{2}- y^{2}and \( v=-\dfrac{y}{x^2+y^2} \) are harmonic but u + iv is not regular.
7 M

2(d)
Examine the nature of the function \( f(z)=\left\{\begin{matrix}
\dfrac{x^3y(y-ix)}{x^6+y^2} & ,z\neq 0\\
0 &, z=0
\end{matrix}\right. \) 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 = e

^{x}(x sin y + y cos y).
3 M

3(b)
Evaluate using Cauchy's integral formula \( \int _c \dfrac{3z^2+z+1}{(z^2-1)(z+3)}dz \) where C is the circle |z|=2.

4 M

3(c)
Using contour integration evaluate the real integral \( \int _0 ^{2\pi} \dfrac{\cos 3\theta}{5-4\cos \theta}. \)

7 M

3(d)
If f(z)=u+iv is analytic in domain D then prove that \( \left ( \dfrac{\partial ^2}{\partial x^2}+\dfrac{\partial ^2}{\partial y^2} \right )|Re(f(z)|^2=2|f'(z)|^2 \)

3 M

3(e)
Determine the linear fractional transformation that maps z

_{1}= 0, z_{2}= 1, z_{3}=&infin onto w_{1}, w_{2}=-i, w_{3}=1 repectively.
4 M

3(f)
Expand \( f(z)=\dfrac{1}{(z+2)(z+4)} \) 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 \( A=\begin{bmatrix}
3 & -5\\
-2 & 4
\end{bmatrix} \) by power method.

3 M

4(b)
Evaluate \( \int _0^6\dfrac{dx}{1+x^2} \) by using (i) Trapezoidal rule (ii) Simpson's 1/3 rule taking h=1.

4 M

4(c)
Prove that the transformation \( w=\dfrac{z}{1-z} \) 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

10x

x

x

10x

_{1}+ x_{2}+ x_{3}= 6x

_{1}+ 10x_{2}+ x_{3}= 6x

_{1}+ x_{2}+ 10x_{3}= 6
3 M

4(e)
Evaluate the integral \( \int ^1_0\dfrac{dt}{1+t} \) 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 x_{0}=0.6.
3 M

5(b)
Find the negative root of x

^{3}- 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 \( \dfrac{dy}{dx}=\log (x+y) \) 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 \( \dfrac{3x^2-12x+11}{(x-1)(x-2)(x-3)} \) 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 \( \dfrac{dy}{dx}=x+y, \) y(0) = 1 (Taking h = 0.1).

7 M

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