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VTU Electronics and Communication Engineering (Semester 4)
Engineering Mathematics 4
December 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

1(a) Using Taylor series method, solve$\frac{dy}{dx}=2y+3e^x$, y(0)=0 at x=0.2
6 M
1(b) Using Range - Kutta method of fourth order solve for y(0,
1),
y (0,
2) given that $\frac{dy}{dx}=y(x+y),\ \ y(0)=1$.
7 M
1(c) Given $\frac{dy}{dx}= x^2(1+y)$ and y(1) = 1,
y(1.1) = 1.233,
y(1.2) =1.548,
y(1.3)=1.979, evaluate y(1.4) by Milne's Predictor -Corrector method.
7 M

2(a) Approximate y and z at x =0.1 using Picard's method for the solutionog of the equations $$\frac{dy}{dx} =z, \ \ \frac{dz}{dx}=x^3 (y+z)$$/, given that y(0) =1 and z(0) =$\frac{1}{2}.$
6 M
2(b) Using Range -Kutta fourth order method to solve y"=xy'-y,
y(0) =3,
y(0)=0 find y and z at x = 0.1.
7 M
2(c) Apply Milne's method to compute y(0.4) given that y"+xy'+y=0 and the values
y(0)=1,
y(0.1)=0.995,
y(0.2) =0.9801,
y(0.3) =0.956,
y(0) =0,
y(0.1) = -0.0995,
y(0.2) =0.196,
y(0.3) = -0.2867.
7 M

3(a) Prove that the C-R equations in polar form.
6 M
3(b) Show that f(z) = zn, where n is a positive integer is analytic and hence find its derivative.
7 M
3(c) if φ+iψ represents the complex potential of an electrostatic field where $$\Psi = x^2-y^2+\frac{x}{x^2+y^2}$$/, find φ
7 M

4(a) Find the Bilinear transformation which maps the point 1,
i - 1 into 0,
1, α.
6 M
4(b) State and prove the Cauchy's integral formula.
7 M
4(c) Evaluate $$\int _c\frac{e^{2x}}{(z+1)(z-2)}dz$$/, where e : |z|=3.
7 M

5(a) Find the solution of the Laplace's equation in cylindrical system leading to Bessel's differential equation.
6 M
5(b) Derive Roadrgue's formula $$P_n(x)=\frac{1}{2^0n!}\ \ \frac{d^n}{dx^n}(x^2-1)^n$$/
7 M
5(c) Express $$f(x) = x^4+3x^3-x^2+5x-2$$/ in terms Legendre polynomials.
7 M

6(a) Define the Empherical and Axiomatic definition of probability and and example of each.
6 M
6(b) Of the cigarette smoking population 70% are men and 30% are women. 10% of these men and and 20% of these women smoke wills. What is the probability that person seen smoking a wills will be a man?
7 M
6(c) The chance that a doctor will diagnose a disease correctly is 60%. The chance that a patient will die after correct diagnose is 40% an the chance of death by wrong diagnosis is 70%. If a patient dies. What is the chance that his disease was correctly diagnosed?
7 M

7(a) Derive the mean and variance of Binomial distribution.
6 M
7(b) If x is an exponential distribution with mean4, evaluate i) P(0 ii) P(x>2) and
iii) P(-&infty;
7 M
7(c) The marks of 1000 students on an examination follows a normal distribution with mean 70 and standard deviation 5. Find the number of students whose marks will be
i) less than 65
ii) More than 75 and iii) between 65 and 75.
7 M

8(a) Define the following terms:
i) Type I-error and Type II-error
ii) Level significance.
6 M
8(b) A certain stimulus administered to each of the 12 patients resulted in the following: Change in blood pressure 5,
2,
8,
-1,
3,
0,
6,
-2,
1,
5,
0,
4 can it be concluded that the stimulus will increase the blood pressure ? (t.05 for 11 d.f =2.201).
7 M
8(c) The theory predicts the proportion of beans in the groups G1,
G2,
G3,
G4 should be in the ratio 9:3:3:1. In an experiment with 1600 beans the numbers in the four gorups were 882,
313,
287 and 118. Does the experimental result support the theory? (at 5% LOS for 3 d.f = 7.815).
7 M

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