Solve your IGNOU Doubts

Question:

Determine the largest eigenvalue in magnitude and the corresponding eigenvector of the matrix $\begin{pmatrix} 1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}$ using the power method. Take (1, 0, 0)^T as the initial approximation and perform 4 iterations.

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Question:

Determine the order of convergence of the iterative method

$$x_{n+1} = \frac{x_{n-1}f(x_n) - x_nf(x_{n-1})}{f(x_n) - f(x_{n-1})}$

for finding a simple root of the equation $f(x) = 0$ .

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Question:

Prove that $\mu^2 = 1 + \frac{\delta^2}{4}$ .

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Question:

Obtain the interpolating polynomial in simplest form which fits the following data:

x	-1	0	1	2
f(x)	3	-4	5	-6

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Question:

Using synthetic division method, show that 2 is a simple root of the equation

$p(x) = x^4 - 2x^3 + x^2 - x - 2 = 0$ .

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Question:

Find the interpolating polynomial by Newton’s divided difference formula for the following data:

x	0	1	2	4
y	1	1	2	5

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Question:

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

$$\begin{bmatrix} 4 & 0 & 2 \\ 0 & 5 & 2 \\ 5 & 4 & 10 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 4 \\ -3 \\ 2 \end{bmatrix}$

Determine the rate of convergence of the method.

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Question:

If $\delta^2 f(x_0) = C_1 h^2 f''(x_0) + C_2 h^4 f^{(4)}(x_0) + \dots$ , find the values of C₁ and C₂.

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Question:

The roots of the quadratic equation $x^2 + ax + b = 0$ are given by $\alpha$ and $\beta$ . Show that the iteration $x_{k+1} = \frac{-(ax_k + b)}{x_k}$ will converge near $x = \alpha$ when $|\alpha| > |\beta|$ .

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Question:

Find the approximate root of the equation $2x^3 = 3x + 6$ using Newton-Raphson method. Perform only 3 iterations with $x_0 = 2$ .

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Question:

Using Classical Runge-Kutta fourth order method, find an approximate value of y(1.2) for the IVP $\frac{dy}{dx} = xy$ , $y(1) = 2$ with $h = 0.2$ .

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Question:

Find the missing values in the following table:

x	0	1	2	3	4	5
y	0	2	-	18	-	90

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Question:

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 $x^3 - 4x - 16 = 0$ has not root in the interval [3, 5].
b) $\Delta E = E \Delta$ , where E is the shift operator and $\Delta$ is the forward difference operator.
c) Every $3 \times 3$ 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 $\frac{3}{2}$ .
e) The Newton-Raphson method cannot be used to find a cube root of a positive real number.

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Question:

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.

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Question:

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

Cluster	Trees
Cluster	1	2	3	4
1	5	4	2	15
2	11	1	4	7
3	26	10	19	11
4	7	15	12	10
5	2	22	8	6

Estimate the average yield per tree and its standard error.

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Question:

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:

	United States	England	France
Prefer new soap	81	43	26
Do not prefer it	219	257	274

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

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Question:

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

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Question:

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:

Time	Sample number
Time	1	2	3	4	5
9 am:	8	9	15	4	11
10	7	10	7	6	8
11	11	12	10	9	10
12	12	8	6	9	12
1 p.m.	11	10	6	14	11
2	7	7	10	4	11
3	10	7	4	10	10
4	8	11	11	11	7

Compute the control limits for $\overline{x}$ and R--charts. Draw the control charts for $\overline{x}$ and ,R and comments.

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Question:

) 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?

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Question:

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