GTU Computer Engineering (Semester 4)
Mathematics 4
May 2016
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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 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 $$v=-\dfrac{y}{x^2+y^2}$$ 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 = ex (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 z1 = 0, z2 = 1, z3=&infin onto w1, w2=-i, w3=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
10x1 + x2 + x3 = 6
x1 + 10x2 + x3 = 6
x1 + x2 + 10x3 = 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 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 $$\dfrac{dy}{dx}=\log (x+y)$$ 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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