Department of Mathematics

WORK-BOOKof

Mathematics- Probability, Numerical Methods and Complex Analysis

(BSC014)

ForB. Tech. 2nd year- 3rd Semester

(Mechanical Engineering)

Name of the Student:

Enrollment Number:

University of Engineering & Management, Jaipur

Course Code: BSC014L-T-P: 3-1-0Credits: 4

Introduction:

The aim is to teach the student various topics in Numerical Analysis such as solutions of

nonlinear equations in one variable, interpolation and approximation, numerical differentiation and

integration, direct methods for solving linear systems, numerical solution of ordinary differential

equations. The aim is to teach the student various topics in Numerical Analysis such as solutions of

nonlinear equations in one variable, interpolation and approximation, numerical differentiation and

integration, direct methods for solving linear systems, numerical solution of ordinary differential

equations. The objective of Probability theory is to familiarize the students with statistical

techniques and tools at an intermediate to advanced level that will serve them well towards tackling

various problems in the discipline.This course is intended both for continuing mathematics students

and for other students using mathematics at a high level in theoretical physics, engineering and

information technology, and mathematical economics.

Course Outcomes:

Students will be able to:

Demonstrate accurate and efficient use of complex analysis techniques

Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining

concepts from complex analysis

Apply problem-solving using complex analysis techniques applied to diverse situations in

physics, engineering and other mathematical contexts. Knowledge and Understanding:

Students are able to understand the nature and operations of Numerical Analysis.

Intellectual Skills: By the end of the course the student is expected to solve real-life and

Engineering applications.

The ideas of probability and random variables and various discrete and continuous

probability distributions and their properties.

The basic ideas of statistics including measures of central tendency, correlation and

regression.

The statistical methods of studying data samples.

Course Contains

Probability and Statistics

Module 1: (5L)

Definition of Probability, sampling theory, Conditional probability. Baye’s theorem.

Module 2: (7L)

Random variable: Continuous and discrete random variables. Distributions: Bernoulli, Binomial,

Poisson & Normal distributions. Descriptive Statistics: Mean, Median, mode, Standard Deviation.

Numerical Methods

Module 3: (7L)

Accuracy and Precision: Error Analysis. Solution of polynomial and transcendental equations –

Bisection method, Newton-Raphson method and Regula-Falsimethod.Ordinary differential

equations: Taylor’s series, Euler and modified Euler’s methods. Runge- Kutta method.

Complex Analysis

Module 4: (7L)

Differentiation, Cauchy-Riemann equations, analytic functions, harmonic functions, finding

harmonic conjugate; elementary analytic functions and their properties; Conformal mappings.

Module 5: (6L)

Contour integrals, Cauchy-Goursat theorem, Cauchy Integral formula, Taylor’s series, Laurent’s

series; Residues, Cauchy Residue theorem.

Reference Books

Engineering Mathematics-III(B.K Pal and K.Das)

Brown J.W and Churchill R.V: Complex Variables and Applications, McGraw-Hill.

Probability, Random variables and Stochastic processes4th edition by Athanasios Papoulis

probability and statistics (schaum's outline series)

Dr.B.S.Grewal:Numerical Methods in Engineering &science

Jain, Iyengar ,& Jain: Numerical Methods (Problems and Solution).

Baburam: Numerical Methods, Pearson Education.

Advanced Engineering Mathematics 8e by Erwin Kreyszig is published by Wiley India

Engineering Mathematics: B.S. Grewal (S. Chand & Co.)

Fundamentals of Mathematical Statistics (A Modern Approach), S.C. Gupta & V.K.

Kapoor, Sultan Chand & Sons.

Module 1:

1. A bag contains 5 whites and 4 black balls. If 3 balls are drawn at random. What are the probabilities of:

(A) 2 of them are white (B) almost one of them is white (C) at least 2 are white.

2. The manufacture process of an article consists of two parts x and y. the probabilities of defect in part x

and y are 10% and 15% respectively. What is the probability that the assembled product will not have any

defect ?

3. A pair of dice is thrown. Find the probability of getting a sum of 7, when it is known that the digit in the

1st dice is greater than that of the second.

4. A can hit a target 4 times in 5 shots; B 3 times in 4 shots; C twice in 3 shots. They fire a target. What’s

the probability that at least two shots hits ?

5. Two urns contain respectively 5 white, 7 black and 4 white, 2 black balls. One of the urns is selected by

the toss of a fair coin and then 2 balls are drawn without replacement from the selected urn. If both balls

drawn are white, what is the probability that the first urn selected ?

6. In a bolt factory, machines A, B and C manufacture 25%, 35% and 40% respectively of the total bolts

5%, 4% and 2% are defective bolts. A bolt is drawn at random from the product and is found to be

defective. What is the probability that it was manufactured by machine A, B and C ?

7. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?

8. Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?

9. Define Exhaustive, Mutually Exclusive and Independent events.

10. A and B take turns in throwing of two dice, the first to throw 9 will be awarded a prize. If A has the first turn, show that their chances of winning are in the ratio 9:8.

11. Three groups of children contain respectively 3 girls 1 boy, 2 girls 2 boys, 1 girl 3 boys. One child is selected at random from each group. Find the probability of selecting 1 girl and 2 boys.

12. If , then prove that (i) (ii) .

13. If A and B are independent events, then prove that are also independent.

Module 2:

1. A manufacturer supplies quarter horsepower motors in lots of 25. A buyer, before taking a lot, tests at random a sample of 5 motors and accepts the lot if they are all good; otherwise he rejects the lot. Find the probability that:(i) he will accept a lot containing 5defective motors;(ii) he will reject a lot containing only one defective motors.

2. In an engineering examination, a student is considered to have failed, secured second class, first class and distinction according as he scores less than 45%, between 45% and 60%, between 60% and 75% and above 75% respectively. In a particular year 10% of the students failed in the examination and 5% of the students got distinction. Find the percentage of students who have got first class and second class. Given that if

, then and .

3. If 10% of the pens manufactured by the company are defective, find the probability that a box 12 pens contain

a) Exactly two defective pensb) At least two defective pensc) No defective pend) At most two defective pens

4. A letter is known to come either from Calcutta or from Tatanagar. In the half printed postal stamp of the coming states only two consecutive letter are readable. Find the chances of the letter coming from (i) Calcutta (ii) Tatanagar.

5. From a lot of 10 items containing 3 defectives, a sample of 4 items is drawn at random. If the sample is drawn without replacement and the random variable denotes the number of defective items in the sample. Find

a) The probability distribution of

b)

c)

d)

6. In a Normal distribution, 31% of the items are under 45 and 8% are over 64. Find the parameters of the distribution.

7. Prove that Poisson distribution is a limiting case of Binomial distribution. Also evaluate mode of Poisson distribution.

9. Define the properties of Normal distribution. Also prove that mean and mode are equal.

9. The probability distribution of a r.v. is: Determine the constant .

10. The kms in thousands of kms which car owners get with a certain kind of trial is a r.v. having pdf

.

Find the probability that one of these trial will at least kms.

Module 3:

1. What is the difference between direct and iterative method and explain rate of convergence.

2.Perform first two steps of Bi-section method to find the roots of

3.Perform first step of Regula-Falsi method to find the roots of

4. For the following algebraic equation perform four iterations of Newton Rapshon’s method

.

5. Find an approximate value(by Euler’s) of

6. Using fourth order Runge-Kutta method find the numerical solution of

at in two steps.

7. Given with initial condition at . Find for by using Taylor’s series method. (divide into four step).

8. Use Euler’s Modified method with step size .2 to find the value of at for the following

differential equation:

Module 4:

1. Determine the analytic function , if .

2. If complex function is analytic, then prove that its real and imaginary part satisfies Laplace equation; also prove that the family of curves formed by its real and imaginary parts is orthogonal to each other.

3. Prove that the function is analytic and find its derivative.

4. Determine the analytic function, whose real part is . Also find its conjugate.

5. If find its corresponding analytic function (by Milne’s Method).

6. Examine that the function is harmonic or not.

7. Prove that the function defined by

function is continuous and that Cauchy- Riemann equation are satisfied at the origin, yetdoes not exist.

8. prove that along any radius vector

but not as along the curve is this function differentiable at z = 0.

9. If f(z) a analytic function of z, prove that

10. Find the analytic function of which the imaginary part is given by

Module 5:1. Evaluate the following integral by using Cauchy’s integral formula

i. , where is the circle .

2. Evaluate the following integral by using Cauchy’s integral formula

i. .

3. Evaluate along the curve

4. Evaluate , where is complex number.

5. Evaluate

6. Statement of the Cauchy’s Residue Theorem.

7. Prove that the function is continuous everywhere but f’(0)

does not exist though Cauchy Riemann Equation satisfied at the origin.

8. Evaluate (a) along the straight line joining z = i and z = 2 - i, (b) along the

curve x = 2t - 2 and y = 1 + t – t2.

9. Evaluate along the path (a) the parabola x = 2t, y = t2 + 3; (b)

along the line joining (0, 3) and (2, 4).

10. Evaluate where t > 0 and C is the circle |z| = 3.

11. Evaluate where C is the circle |z| = 1.