Solve your IGNOU Doubts

Question:

$\text{Let}R=\mathbb{Z}_{20}.$

(i) Give, with justificaiton, a nilpotent element in R.

(ii) Give, with justification, a zero divisor in 푅 which is not nilpotent.

(iii) What is the order of U(R)?

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Question:

$\text{If}F\text{is a field,show that}U(F[x])=F^*.$

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Question:

$\text{Let}\alpha=(1\;2\;5),\quad\beta=(1\;4\;3\;2)\in S_5.\text{Compute}\sigma=\alpha\cdot\beta^{-1}.\text{Write}\sigma$

as a product of transpositons. What is the signature of $\sigma$ ?

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Question:

$\text{Let}A\text{be a}3\times 4\text{real matrix,}B\text{be a}4\times 2\text{real matrix.}$ and 퐶 be a 2 × 3 real matrix. Which of the following operations are defined?

$(i)\quad CA+B^T\\$

$(ii)\quad AB+C^T$

For those operations that are defined, what is the order of the resulting matrix?

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Question:

$\text{Let}S=\{1,2,3,4\}$ and ∗ be the binary operation defined by $a*b=a.$ Compute the Cayley table fo $(S,*).Is*$ commutative? Is ∗ associative? Justify your answers.

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Question:

State Lagrange’s theorem. What are the possible orders of subgroups of a group of order 12?

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Question:

Define a semigroup. Give an example of an infinite semigroup.

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Question:

Define a subgroup of a group. Check whether

$H=\left\{\begin{bmatrix}0&a&b\\0&c&d\end{bmatrix}\middle|a,b,c,d\in\mathbb{C}\right\}$ is a subgroup of the the group of 2 × 3 matrices over ℂ under addition.

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Question:

Define an abelian group. Give an example of a non-abelian group. (You don't need to prove that your example is a group. You have to only prove that it is non-abelian.)

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Question:

Evaluate the integra $\int_0^{10}\frac{dx}{\sqrt{10x-x^2}}$ using beta and gamma functions.

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Question:

Evaluate the integral $\iint_D e^{4x+5y}\,dx\,dy,\quad\text{where}D=\{(x,y):x\geq 0,y\geq 0,x+y\leq 3\}$

by considering D as a region of Type I and then as a region of Type II.

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Question:

. $\text{If}f:[0,5]\rightarrow\mathbb{R}$ be a function defined by $f(x)=2x^2+3x+5,\quad x\in[0,5].$ Show that f is Riemann integrable using both definitions. Also, verify that the results of both definition match.

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Question:

Find the equation of a line passing through points A(0, 0, 1) and B(1, 1, 0). Also, find the coordinates of a point on this line which is at a distance of 10 units from point A opposite to the side of point B.

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Question:

In R we have a built-in data set “trees”. A screenshot of the last four rows together with the R code to obtain it is given as follows. To get more detail about this data set you can run ?trees command on R console.

Image ignouassignments-ignouacademy-com--p-your-82617

Note that all the three variables of this data set are numeric. So, assuming each row of this data set is a point in 3-dimension. Find the distances between the points corresponding to the 28th and the 32st rows using the Manhattan and Chebyshev distance formula

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Question:

Solve the following problems

(a) If f is a function from set X to a set Y then is it possible $f(3)=5,\quad(3)=-5?$

$\text{where}3\in X\text{and}5,-5\in Y$

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Question:

In a class of Statistics, total number of students is 30. Select the linear and circular systematic random samples of 12 students. The age of 30 students is given below:

Age: 22 25 22 21 22 25 24 23 22 21 20 21 22 23 25 23 24 22 24 24 21 20

23 21 22 20 20 21 22 25

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Question:

he following data is the data pertaining to a feeding trial on sheep. Treatments

A: Grazing only

B: Grazing + Maize Supplements

C: Grazing + Maize + Protein Supplement P₁

D: Grazing + Maize + Protein Supplement P₂

E: Grazing + Maize + Protein Supplement P₃

Layout and Wool Yield (100 gm) is given as:

32(D)	33(E)	30(C)	28(B)	24(A)
51(C)	45(D)	41(A)	45(E)	29(B)
41 (E)	29 (A)	24 (B)	36 (D)	35 (C
38 (B	39(C)	42(E	23(A)	37(D)
38(A	24(B)	21(D)	29(C)	26(E)

Analyse the design with appropriate method and calculate the Critical Difference for the treatment mean yield

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Question:

For the given data the yield of the treatment 2 in 3rd block is missing. Estimate the missing value and analyse the data.

Treatments	Blocks
Treatments	I	II	III	IV
1	105	114	108	109
2	112	113	Y	112
3	106	114	105	109

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Question:

A manurial trial with six levels of Farmyard Manure (FYM) was carried out in a randomised block design with 4 replications at the experimental station Junagarh with a new study the rate of decomposition of organic matters in soil and its synthetic capacity in soil on cotton crop. The yield per plot in kg for different levels of FYM and replications is given below:

Cotton Yield Per Plot (in Kg)

Levels of FYM	Replications
Levels of FYM	I	II	III	IV
1	6.90	4.60	4.40	4.81
2	6.48	5.57	4.28	4.45
3	6.52	7.60	5.30	4.45
4	6.90	6.65	6.75	7.75
5	6.00	6.18	6.50	5.50
6	7.90	7.57	6.80	6.62

Carry out the Analysis of Variance test and draw the conclusions.

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Question:

In order to compare the mileage yields of 3 kinds of Gasoline, several tests were run, and the following results were obtained:

Gasoline A:	19	21	20	18	21	21
Gasoline B:	23	20	22	20	24	23
Gasoline C:	20	17	21	19	20	17

Carry out the Analysis of Variance test and test whether there is significant differences between the average mileage of 3 kinds of gasoline at 5% level of significance

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