Solve your IGNOU Doubts

Question:

Plating of gold on wrist watches requires a chemical called AiCl₃. Concentration of the chemical that is added to the solution is an important factor. The objective is to choose that concentration for which plating is uniform. Three concentration of AiCI₃ chosen were 5%, 15% and 20%. These were added in the solution and plating was done and thickness (in units) measured on 5 samples is gives below.

Sample → ------------------------- ↓ Conc. of `A_iCl₃`	1	2	3	4	5
5%	3	2	4	2	4
15%	4	4	3	2	4
25%	3	3	4	4	2

Use ANOVA to comment on whether the concentration of AiCI₃ gives same result or 3 not. (Use $\alpha=0.05$ )

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Question:

In a locality of 18,000 families, a random sample of 840 families was taken. Of these 840 families, 206 families were found to have a monthly income of Rs. 500 or less. Give the confidence interval for the families having income Rs. 500 or less.

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Question:

Two samples of 9 and 8 sizes give the sum of squares of deviations from respective means equal to 160 and 91 inches squares. Test whether these samples have been drawn from same normal population or not? (Use $\alpha = 0.05$ )

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Question:

A bicycle shop sells the following number of bicycles from 1990 to 2000.

Year	Number sold (thousands)
1990	3
1991	3
1992	3
1993	3
1994	6
1995	6
1996	6
1997	6
1998	9
1999	10
2000	12

Compute the first three moving averages of length 3 for the bicycle sales data and place them in line with the corresponding year

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Question:

A bicycle shop sells the following number of bicycles from 1990 to 2000.

Year	Number sold (thousands)
1990	3
1991	3
1992	3
1993	3
1994	6
1995	6
1996	6
1997	6
1998	9
1999	10
2000	12

Compute the first three moving averages of length

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Question:

There are two samples of 1200 and 900 people drawn from populations respectively, which have 30% and 25% of fair-haired people. Test whether the samples drawn from this population maintain difference or not. (Use $\alpha = 0.05$ )

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Question:

Two cards are drawn simultaneously or successively without replacement from a well-shuffled deck of 52 cards. Find the probability distribution of number of aces (x). Give a graphical representation of probability distribution.

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Question:

Following is the distribution of marks (out of 25) obtained by 10 students in Physics and Mathematics.

No.	Physics (X_i)	Mathematics (Y_i)	`X_i²`	`Y_i²`	`X_iY_i`
1	18	21	324	441	378
2	20	23	400	529	460
3	11	14	121	196	154
4	20	23	400	529	460
5	14	17	196	289	238
6	15	18	225	324	270
7	13	16	169	256	208
8	16	19	256	361	304
9	17	20	289	400	480
10	20	23	400	529	460
Total	164	194	2780	3854	3412

Draw a scatter diagram for all the 10 students and calculate the correlation between marks of Physics and Mathematics.

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Question:

Which of the following statements are true? Give reasons for your answer.

(i) F-distribution is always used in goodness of fit.

(ii) Measure of central tendency in a data set refers to the extent to which the observations are scattered.

(iii) A process is said to be under assignable cause when the points are above the VCL link of ( $\overline{X}, R$ ) chart.

(iv) Simple random sampling is done by using random number tables where the probability of drawing a digit is 0⋅ 1.

(v) All time series have a trend.

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Question:

Let $G = \langle a, b \mid a^2, b^3, aba^{-1}b^{-1} \rangle$ . Show that G is the cyclic group of order six.
(b) Solve the following set of congruences:
$$\begin{aligned} x &\equiv 2 \pmod{17} \\ 3x &\equiv 4 \pmod{19} \\ x &\equiv 7 \pmod{23} \end{aligned}$
(c) Show that $\mathbb{Q}(\sqrt{-19})$ is not a UFD by giving two different factorisations of 20.

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Question:

Suppose n is as in the previous part. Find all the Sylow 2-subgroups of D_n. Describe them in terms of x and y.

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Question:

Suppose n is even, $n = 2^k m$ , where $2 \nmid m, k \ge 2$ . Let $N = \langle x^m \rangle$ and $H = \langle y \rangle$ . Show that HN is a subgroup of D_n. What is its order?

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Question:

Find all the Sylow 2-subgroups of D_n when n is odd. Describe them in terms of x and y.

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Question:

) Prove the relation

$$y^i x^j y^k x^l = \begin{cases} y^i x^{j+l} & \text{if } k \text{ is even} \\ y^{i+k} x^{l-j} & \text{if } k \text{ is odd.} \end{cases}$
Further, find all the elements of order 2 in D_n.

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Question:

Let p be an odd prime that divides $n, n = p^r l, p \nmid l$ . Suppose $C = \langle x^l \rangle$ . Show that C is the unique Sylow p-subgroup of D_n.

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Question:

In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group

$$D_n = \langle x, y : x^n, y^2, yxyx \rangle, n \in \mathbb{N}, n \ge 2.$

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Question:

Show that
$\quad$ (i) Show that a matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \in GL_2(\mathbb{R})$ is symplectic if and only if $ad - bc = 1$ .
$\quad$ (ii) Show that, to prove that $SP_2(\mathbb{R})$ acts transitively on $\mathbb{R}^2 \setminus \{\mathbf{0}\}$ , it is enough to show that, for any vector $\begin{bmatrix} a \\ b \end{bmatrix} \neq \mathbf{0} \in \mathbb{R}^2$ , there is a $2 \times 2$ symplectic matrix with $\begin{bmatrix} a \\ b \end{bmatrix}$ as the first column. (Hint: For any matrix A, what is $A \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ ?)
$\quad$ (iii) Complete the proof by showing that, given any non-zero vector $\begin{bmatrix} a \\ b \end{bmatrix}$ , there is always a non-zero vector $\begin{bmatrix} a' \\ b' \end{bmatrix}$ such that $\begin{bmatrix} a & a' \\ b & b' \end{bmatrix}$ is symplectic.

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Question:

The aim of this exercise is to show that $SP_2(\mathbb{R})$ acts transitively on $\mathbb{R}^2 \setminus \{\mathbf{0}\}$

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Question:

Suppose that $M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$ is $2n \times 2n$ matrix where A, B, C and D are $n \times n$ matrices. Show that M is symplectic if and only if the following conditions are satisfied:
$\quad$ $A^t D - C^t B = \mathbf{I}$
$\quad$ $A^t C - C^t A = \mathbf{0}$
$\quad$ $B^t D - D^t B = \mathbf{0}$
$\quad$ (Hint: Use block matrix multiplication.)
$\quad$ Also, check that the matrix $\begin{bmatrix} \mathbf{0} & -A \\ A & \mathbf{0} \end{bmatrix}$ , where A is a $n \times n$ orthogonal matrix, is a symplectic matrix.

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Question:

If $\mathbb{F}$ is a finite field show that there is always an irreducible polynomial of the form x³ - x + a where $a \in \mathbb{F}$ . (Hint: Show that $x \mapsto x^3 - x$ is not a surjective map.)