The Gauss-Seidel method is used to solve the system of equations

Determine the rate of convergence of the method.

If , find the values of C₁ and C₂.

The roots of the quadratic equation  are given by  and . Show that the iteration  will converge near  when .

 Find the approximate root of the equation  using Newton-Raphson method. Perform only 3 iterations with .

 Using Classical Runge-Kutta fourth order method, find an approximate value of y(1.2) for the IVP ,  with .

Find the missing values in the following table:

State whether the following statements are true or false. Give a Short proof or a counter-example in support of your answer.
a) The equation  has not root in the interval [3, 5].
b) , where E is the shift operator and  is the forward difference operator.
c) Every  system of linear equations can be solved using the LU decomposition method.
d) For the data (2, 4), (1, 5), (3, 6) the Newton's divided difference f[x₀, x₁, x₂] is .
e) The Newton-Raphson method cannot be used to find a cube root of a positive real number.

A manufacture of rayon wants to compare that the yield strength of 11⋅5 kg/ mm² is met or not at 5% level of significance. The manufacturer draws a sample and calculates the mean to be 12⋅8kg/ mm² and the standard derivation is known to be 2⋅0kg/ mm² . Carry out the statistical test appropriate for this.
b) A box contains 10 screws, 3 of which are defective. Two screws are drawn at random. Find the probability that none of them is defective, if the sample is drawn
(i) with replacement,
(ii) without replacement.
c) A researcher measures the temperature of a solution 5 times. The observations are 25, 32, 27, 29, 30 ^o C. Determine an unbiased estimate of the temperature.

 Suppose from a total of 120 guava trees, 5 clusters of 4 trees each selected and the yield (in kg) is recorded below:

Estimate the average yield per tree and its standard error.

A shop manufacturer is trying to determine whether or not to market a new type of shop. It chooses a random samples of equivalent size in England and France. It asks each of the people in each sample to try the new soap, and see whether he or she likes it better than other soaps. The results are as follows:

Test the hypothesis that there are international difference in the proportion of people who prefer the new soap, using 5% level of significance.

Write three advantage and three disadvantage of using a sampling approach instead of a census approach for studying a characteristic.

At a call centre, callers have to wait till an operator is ready to take their call. To monitor this process, 5 calls were recorded every hour for the 8-hour working day. The data below shows the waiting time in weeks:

Compute the control limits for  and R--charts. Draw the control charts for  and ,R and comments.

) In a television appearance, an MLA from a state asks voters in that state to indicate whether the MLA should vote against or for a particular piece of legislation. The MLA office receives 4100 replies. Since voters from western part tends to be democrats while those from eastern parts tend to e communists, the MLA divides the replies on the basis of their geographical origin. 68% of the replies from the western part are against the legislation and 36% of the replies from the eastern part are against it.

(i) Did the MLA carry out a stratified sampling?

(ii) Did the MLA carry out a stratified random sampling?

(iii) How would you justify the sample design?

(iv) What improvements would you suggest?

The following table gives the hours spent by 50 students in the playground in a week:

Calculate the mean median and standard deviation of the hours required by stud

 Find the probability that at most 5 defective fuses will be found in a box of 200, if experience shows that 20% of such fuses are defective.

Find  and  .
 Check whether A,B and C are independent or not. Give reasons for your answers.

 It is desired to estimate average annual wool yield per sheep for a herd of 150 at a certain farm house using stratified simple random sampling. Sheeps in the herd are to be grouped into three strata on the basis of first shaving in days. Optimum method sample allocation is said to be used for selecting the overall sample of 25 sheeps from the three strata. Determine approximately optimum strata boundaries using the information on first shaving given below.

Find  and  .
 Check whether A,B and C are independent or not. Give reasons for your answers.

: the event of getting a total of seven on two dices.
Then

 Consider the case of tossing two dices. Let

A : be the event of getting an odd total
B : the event of getting on the first dice, and

 Consider a population of 11 schools from which a sample of size 3 is to be selected. List all the possible samples by circular systematic sampling.