tutorial sheet

BALLARI INSTITUTE OF TECNOLOGY AND MANAGEMENTDEPARTMENT OF MATHEMATICS

Engineering Mathematics-IVTUTORIAL SHEET-1[2012-13]

1. Use modified Euler’s method to solve in the range by taking given that at initially.

2. Using Runge-Kuttaa method of fourth order, find for the equation , taking

3. Use Picard’s method to obtain the third approximation to the solution of

and hence find y at

4. Given that and the data , ,

, compute using Adam’s Bash forth method

5. Use Taylor’s series method, given , by taking step size h=0.1,y=10 at x=0 initially.

6. Apply Milne’s method to solve the equation given that=0, =0.02, =0.0795, =0.1762. Compute.



1. Use Taylor’s series method, given , by taking step size a. h=0.1, y=10 at x=0 initially.

2. Using Euler’s modified method to find given that , taking

3. 3. Use Picard’s method to find y at x=0.25, 0.5 given =, y(0)=1.by taking two approximations

4. Solve, x=0,y=1 at x=0.2 using R-k method, take h=0.2

5. Apply Milne’s method to solve the equation given that=0, =0.02, =0.0795, =0.1762. Compute.

6. Given that and the data , ,

, Compute using Adam’s Bash forth method



1. Use Taylor’s series method, given , by taking step size h=0.1, y=10 at x=0 initially.

2. Using Euler’s modified method to find given that

with taking h=0.2

3. Use Picard’s method to find y at x=0.25, 0.5 given =, y(0)=1.by taking two approximations

4. Solve, x=0,y=1 at x=0.2 using R-k method, take h=0.2.

5. Apply Milne’s method to solve the equation given that =0, =0.02, =0.0795, =0.1762. Compute.

6. Given that and the data , , Compute using Adam’s Bash forth method



1. Use Picard’s method to obtain given that

, , (Carry out two approximations)

2. Use fourth order Range- Kutta method to solve the system of equations :


given that

4. Given, evaluate using four

Order Range-Kutta method

5. Apply Milne’s Predictor-Corrector method compute given that satisfies the equation and are governed by the following initial value:. Apply corrector formula twice



1. Apply Picard’s method to obtain the second approximation to the solution of the differential equation. Given that,& also find

2. Use order Runge-Kutta method to solve & also find.

3. Use order Runge-Kutta method to solve the system of equations , =1, at t=0. Compute.

4. Apply Milne’s method to compute y(0.8) given ,given that =0, =0.02, =0.0795, =0.1762, 0 ,

5. Use Picard’s method to obtain given that

, , .

BALLARI INSTITUTE OF TECNOLOGY AND MANAGEMENT

DEPARTMENT OF MATHEMATICSEngineering Mathematics-IV

TUTORIAL SHEET-2[2014-15]

1. Apply Picard’s method to obtain the second approximation to the solution of the differential equation. Given that,& also find.

2. Use order Runge-Kutta method to solve & also find.

3. Use order Runge-Kutta method to solve the system of equations , =1, at t=0. Compute.

4. 4. Use Picard’s method to obtain given that

, , .

5. Apply Milne’s method to compute y(0.8) given ,given that=0, =0.02, =0.0795, =0.1762, 0

,



1. State and prove Cauchy-Riemann equation in polar form2. Find the analytic function whose imaginary part is )3. If is analytic, show that

4. Show that the real and Imaginary parts of analytic function are harmonic in Cartesian form

5. Show that is analytic and hence find its derivative

6. Show that the function is harmonic and find its harmonic conjugate. Also determine the corresponding analytic function.

7. Find the analytic function as a function of z given that the sum of its real and imaginary parts is



1. 1.State and prove Cauchy-Riemann equation in Cartesian form

2. Show that is a regular function of . Also find

3. Shoe that the real and imaginary parts of an analytic function are harmonic in

polar form.

4. If is a regular function of show that


6. Show that is analytic and hence find its derivative.



1. 1.State and prove Cauchy-Riemann equation in polar form

2. Show that is analytic and hence find its derivative.


4. Show that the real and Imaginary parts of analytic function are harmonic in Cartesian form

5. Find the analytic function whose imaginary part is

6. If is analytic, show that



1. Define the following with examples(a)Mutually exclusive events (b) Independent events (c) Exhaustive events

2. If A and B are non mutually exclusive events , prove that

3. A chartered accountant applies for a job in two firms X and Y. He estimate that the probability of his being selected in firm X is 0.7, and being rejected at Y is 0.5 and the probability of atleast one of his applications being rejected is 0.6. What is the probability that he will be selected in one of the firms?

4. State and prove Baye’s theorem

5. Three machines A, B and C produce respectively 60%, 30% and 10% of the total production of a factory. The percentages of defective output of these machines are respectively 2%, 3% and 4%. An item is select at random and is found defective. Find the probability that the item was produced by the machine B.



1. In a bolt factory there are four machines manufacturing respectively of the

total production. Out of these 5 are defective. If a bolt is drawn at random was found

defective, what is the probability that it was manufactured by

2. If A and B are non mutually exclusive events, prove that

3. If independent events, show that the events (i) are independent (ii) are

independent (iii) are independent

4. Given . Find (i)

(ii) (iii) (iv)




1. If A and B are two independent events, prove that

2. If are two events with probabilities 0.25 and 0.5 corresponding to respectively,

find the probability of if

a) are mutually exclusive

b) are independent.

3. Given

4. State and Prove Baye’s Theorem

5. In a bolt factory there are four machines manufacturing respectively of the total

production. Out of these 5 are defective. If a bolt is drawn at random was found

defective, what is the probability that it was manufactured by



1. Find the value of k such that the following distribution represents a finite probability distribution. Hence find mean and standard deviation. Also find

2. Find the mean and standard deviation of Binomial distribution.

3. The number of accidents per day (x) as recorded in a textile industry over a period of 400 days is given. Fit a Poisson distribution for the data and calculate the theoretical frequencies

0 1 2 3 4 5173 168 37 18 3 1

4. The marks of 1000 students in an examination follow 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 than75 (iii) between 65 and75

5. The probability that a bomb dropped hits the target is 0.2 Find the probability that out of 6 bombs dropped (i ) Exactly 2 will be target (ii) at lest 3 will hit the target




2.Find the mean and standard deviation of Poisson distribution.

3. The marks of 1000 students in an examination follow 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 than75 (iii) between 65 and75


5. A set of 5 similar coins tossed 320 times gives following table.0 1 2 3 4 5

requency 6 27 72 112 71 32




2. Find the mean and standard deviation of Exponential distribution.


requency 6 27 72 112 71 32

4. The marks of 500 students in an examination follow a normal distribution with mean 65 and standard deviation 5. Find the number of students whose marks will be (i) Less than 55 (ii) more than35 (iii) between 65 and75




1. A certain stimulus administered to each of the 12 patients resulted in the following change in the 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? (

2. The marks of 1000 students in an examination follow 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 than75 (iii) between 65 and75

3. Define the following (a) Null Hypothesis (b) Type – I and Type-II error (c) Significance level

4. A coin is tossed 1000 times and head turns up 540 times. Decide on the hypothesis that the coin is unbiased.

5. The mean of two large samples of 1000 and 2000 members are 168.7 cms and 170 cms respectively. Can the samples be regarded as drawn from the same Population of standard deviation 6.25 cms ?

6.A set of 5 similar coins tossed 320 times gives following table.0 1 2 3 4 5

requency 6 27 72 112 71 32 Test the hypothesis that data follows binomial distribution (Given =11.07)



1. Define the following (a) Null Hypothesis (b) Type – I and Type-II error (c) Significance level



4. The mean of two large samples of 1000 and 2000 members are 168.7 cms and a. 170 cms respectively. Can the samples be regarded as drawn from the

same b. Population of standard deviation 6.25 cms ?


requency 6 27 72 112 71 32 Test the hypothesis that data follows binomial distribution (Given =11.07)

6 .A certain stimulus administered to each of the 12 patients resulted in the following change in the 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?

(





3. The mean of two large samples of 1000 and 2000 members are 168.7 cms and a. 170 cms respectively. Can the samples be regarded as drawn from the

same b. Population of standard deviation 6.25 cms ?

4. Certain stimulus administered to each of the 12 patients resulted in the following change in the 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? (


0 1 2 3 4 5173 168 37 18 3 1

Test the hypothesis that data follows binomial distribution (Given =11.07)



1. Use Picard’s method to find y at x=0.25, 0.5 given =, y(0)=1.by taking two Approximation

2. If is analytic, show that

3. Evaluate over (i) (ii)

4. Expand in a Laurent’s series valid for (i) (ii)

5. Apply Milne’s method to compute y(0.8) given ,given that

=0, =0.02, =0.0795, =0.1762, 0

,

6. Given that and the data , , , compute



1. Find the bilinear transformation which maps the points into .

2. Evaluate over (i) (ii)

3. Expand in a Laurent’s series valid for (i) (ii) 3


5. A chartered accountant applies for a job in two firms X and Y. He estimate that the probability of his being selected in firm X is 0.7, and being rejected at Y is 0.5 and the probability of at least one of his applications being rejected is 0.6. What is the probability that he will be selected in one of the firms?





2. Given that and the data , , , compute 3. State and prove Cauchy-Riemann equation in polar form4. If is analytic, show that 5. If A and B are non mutually exclusive events , prove that




1. Use modified Euler’s method to solve in the range

by taking given that at initially.

2. Define the following with examples (a)Mutually exclusive events (b) Independent events (c) Exhaustive events

3. Apply Milne’s method to compute y(0.8) given ,given that

=0, =0.02, =0.0795, =0.1762, 0

,

4. A chartered accountant applies for a job in two firms X and Y. He estimate that the probability of his being selected in firm X is 0.7, and being rejected at Y is 0.5 and the probability of at least one of his applications being rejected is 0.6. What is the probability that he will be selected in one of the firms?

5. Certain stimulus administered to each of the 12 patients resulted in the following change in the 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? (



1. Using Runge-Kutta method of fourth order, find for the equation , taking

2. Given that and the data , , , compute

3. State and prove Cauchy-Riemann equation in polar form



0 1 2 3 4 5173 168 37 18 3 1

Test the hypothesis that data follows binomial distribution (Given =11.07)

6. The marks of 500 students in an examination follow a normal distribution with mean 65 and standard deviation 5. Find the number of students whose marks will be (i) Less than 55 (ii) more than35 (iii) between 65 and75



1. Use Taylor’s series method, given , by taking step size h=0.1,y=10 at x=0 initially.

2. Show that the function is harmonic and find its harmonic conjugate. Also determine the corresponding analytic function.


given that


5. Define the followinga. Null Hypothesisb. Type – I and Type-II errorc. Significance level


requency 6 27 72 112 71 32

tutorial sheet

Documents