Solve your IGNOU Doubts

Question:

Let $a_n$ be the number of See Answer →

Question:

Find the number of integer solutions of the equation $x_1+x_2+x_3+x_4=0,x_1 \geq -4,\forall \,i$ using the generating function technique.

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Question:

State whether the following statements are true or false. Justify your answer with a short explanation or a counter-example. (2 × 10 = 20)
(i) The contra- positive of the statement $"x^2+y^2=0\Rightarrow x=0$ and $y=o"$ is $"x\neq 0$ and $"y\neq 0\Rightarrow x^2+y^2\neq 0"."$
(ii) $a^n=\frac{1}{9}[2^{n+1}+(-1)^n]^2$ is the solution of the recurrence relation
$\sqrt{a_n}=\sqrt{a_{n-1}}+2\sqrt{a_{n-2}},(n\geq 1),a_0=a_1=1.$
(iii) The number of integers between 1 and 360 which are relatively prime to 360 are
96.[ Note that 1 is relatively prime to every positive integer.]
(iv) The coefficient of $x^6y^5z^9$ in $(x+y+z)^{20}$ is $C(20,6).C(20,5).C(20,9).$
(v) If a graph has 6 vertices and 10 edges, then it can’t be regular.
(vi) Peterson graph is bipartite.
(vii) Every edge of a Tree is a bridge.
(viii) The independence number of $C_5,i.e.,\alpha (C_5)$ is $2.$
(ix) The DNF of the expression $p(x,y,z)=(x\wedge z)^{'}$ is $(x\vee y\,\vee z^{'})\vee(x\,\vee\,y^{'}\vee\,z^{'}).$
(x) The number 8 has atmost one self-conjugate partition.

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Question:

a) A marketing manager has 5 salespersons and 5 sales districts. Considering the capabilities of the salespersons and the nature of the districts, the marketing manager estimates the sales per month (in thousand ) for each salesperson in each distinct as follows

Salespersons

	Districts
	1	2	3	4	5
A	32	38	40	28	40
B	40	44	28	21	36
C	41	27	33	30	37
D	22	38	41	36	36
E	29	33	40	35	39

Find the assignment of sales persons to districts that will result in maximum sales.

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Question:

For 10 observations on price $(X)$ and supply $(Y)$ the following data were obtained (in appropriate units): $\sum X=130,\sum Y=200,\sum X^2=2288,\sum Y^2=5506 \,$ $\sum XY=3467.$

Obtain the line of regression of Y on X and estimate the supply when price is 16 units.

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Question:

a) A toy company manufactures two types of doll; a basic version-doll A and a deluxe version-doll B. Each doll of type B takes twice as long as to produce as one of type A, and the company would have time to make a maximum 2000 per day if it produce only the basic version. The supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). The deluxe version requires a fancy dress of which there are only 600 per day available. The company makes profit of ₹3 and 5 per doll respectively on doll A and B. How many of each should be produced per day in order to maximize profit? Solve this problem by graphical method.a) A toy company manufactures two types of doll; a basic version-doll A and a deluxe version-doll B. Each doll of type B takes twice as long as to produce as one of type A, and the company would have time to make a maximum 2000 per day if it produce only the basic version. The supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). The deluxe version requires a fancy dress of which there are only 600 per day available. The company makes profit of 3 and 5 per doll respectively on doll A and B. How many of each should be produced per day in order to maximize profit? Solve this problem by graphical method.

b) Find all the basic solutions of the following system: $x_1+2x_2+x_3=4$

$2_1+x_2+5_3=5$

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Question:

A single observation was taken from a population with p.d.f. $f(x,\theta)=\frac{2}{\theta^2}(\theta-x),$ $0\leq x\leq\theta.$

Obtain $100(1-\alpha )%$ confidence interval for θ.

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Question:

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

a) In an LP model, the feasible solution space can be effected when redundant constraints are deleted.

b) If the primal LPP has an optimal solution, then the set of feasible solution to its dual is bounded.

c) In a simplex iteration, an artificial variable can be dropped all together from the simplex table once the variable becomes non basic.

d) The addition of a constant to all the elements of a payoff matrix in a two – person zero sum game can affect only the value of the game, not the optimal mix of the strategies.

e) There may be a balanced transportation problem without any feasible solution.

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Question:

A group of 250 items with mean 15.6 and standard deviation $\sqrt{13.44}$ . has been divided into two groups. The first has 100 items with mean 15 and standard deviation 3. Find the standard deviation of the second group.

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Question:

X , X and X3 is a random sample of size 3 from a population with mean µ and variance $\sigma ^2.$ $T_1,T_2$ . and $T_3$ are the estimators to estimate µ, and are given by ; $T1=X_1+X_2-X_3;T_2=2X_1+3X_2-4X_2$ and $T_3(\lambda X_1+X_2+X_3).$

i) Are T1 and T2 unbiased? Give reason.
ii) Find the value of λ such that T3 is unbiased.
iii) Which is the best estimator? State giving reasons.

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Question:

Draw the cumulative frequency curves for the following distribution:

Marks	No. of Students
0 – 10	4
10 – 20	8
20 – 30	11
30 – 40	15
40 – 50	12
50 – 60	6
60 – 70	3

From the graph, obtain the median.

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Question:

The probability that a student passes a Physics test is $\frac{2}{3}$
and the probability that the student passes both a Physics test and an English test is $\frac{14}{15}$ The probability that the
student passes at least one test is $\frac{4}{5}$ What is the probability that the student passes the English test?

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Question:

b) i) Calculate the third-degree Taylor polynomial abou $x_0=0$ for $f(x)=(1+x)^{1/2}$

ii) Use the polynomial in part (i) to approximate $\sqrt{1.1}$ and find a bound for the error involved.

iii) Use the polynomial in part (i) to approximate $\int _{0}^{0.1}(1+x)^{1/2}dx.$

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Question:

$\frac{}{}$ $\sqrt{\frac{1+x^4}{x^4}}$

1.414

1.031

1007

1.002

1.001

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Question:

Let X be a single observation from the p.d.f. $f(x,\theta )=\theta e^{\theta x},0\leq x<\infty .$

If X ≥1 is the critical region for testing $H_0:\theta=2$ against the alternative hypothesis $H_1:\theta=1,$ obtain the values of type I and type II errors.

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Question:

a) Using the following table of values, find approximately by Simpson’s rule, the arc ength of the graph $y=\frac{1}{x}$ between the points (1, 1) and $\left ( 5,\frac{1}{5} \right )$

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Question:

c) Show that $u_x=c_1e^a^x+c^2e^{-ax}$ is a solution of the difference equation $u_x+1-2u_xcosh\,a+u_{x-1}=0.$

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Question:

a) Find the mean and standard deviation for the following data:

Class interval	Frequency
0 – 10	5
10 – 20	10
20 – 30	14
30 – 40	15
40 – 50	6

b) State Chebychev’s inequality. Hence obtain the lower bound for $P[-1<x<9]$ if the E(x) and $E(x^2)$ of x are 4 and 20 respectively.

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Question:

b) Use modified Euler’s method to find the approximate solution of IVP

$y'=2xy,y(1)=1$ at $x=1.5$ with $h=0.1$

If the exact solution is $y(x)=e^{x^{2-1}},$ find the error.

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Question:

The police plans to enforce speed limits by using radar traps of 4 different locations within the city. The radar traps at each of the locations $L_1,L_2,L_3$ and $L_4$ are operated 40 % 30% 20% and 30%, of time. If a person who is speeding on h, is way to work has probabilities of 0.2, 0.1, 0.5 and 0.2 respectively of passing through these locations. What is the probability that the person will receive a speeding challan.