Solve your IGNOU Doubts

Question:

he mean I.Q. of a large number of children of age 14 was 100 and standard deviation 16. Assuming that the distribution was normal, find
i) the percentage of children having I.Q. under 80.
ii) the limits in which the I.Q. of the middle 40% of the children will lie.
You may like to use the following values:
$P(Z > 1.25) = 0.1056$
$P(Z < -0.525) = 0.3$

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Question:

Solve the following LPP graphically:

aximize:

$$z = 10x_1 + 10x_2$ $
subject to the constraints:
$$4x_1 + 3x_2 \le 12$ $
$$6x_1 + 18x_2 \le 36$ $
$$x_1, x_2 \ge 0.$ $
b) Find all values of k for which the vectors:
$$\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} \text{ and } \begin{bmatrix} 2 \\ -k \\ 2k \end{bmatrix}$ $
are linearly independent.

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Question:

Solve the following LPP graphically:

Maximize:

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Question:

Write the LPP formulation of the following assignment problem:

Image ignouassignments-ignouacademy-com--p-ignou-79385

b) Solve the following game graphically:

Image ignouassignments-ignouacademy-com--p-doubts-75293

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Question:

Write the LPP formulation of the following transportation problem:

	Destination			Supply
	$D_1$	$D_2$	$D_3$
Source $O_1$	10	18	12	200
Source $O_2$	15	17	9	300
Source $O_3$	13	15	7	500
Requirement	400	200	400

b) Solve the following assignment problem for profit maximization:

Image ignouassignments-ignouacademy-com--p-ignou-39458

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Question:

) For what value of k are the following vectors linearly independent?
$$\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ k \\ 1 \end{bmatrix}, \begin{bmatrix} k \\ 0 \\ 1 \end{bmatrix}$ $
b) Solve the following LP problem using simplex method:
Maximize $z = 6x_1 + 4x_2$
Subject to $2x_1 + 3x_2 \le 30$
$$3x_1 + 2x_2 \le 24$ $
$$x_1 + x_2 \ge 3$ $
$$x_1, x_2 \ge 0$ $

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Question:

) Using matrix – minima method, find the initial basic feasible solution of the following transportation problem:

Image ignouassignments-ignouacademy-com--p-doubts-16480

Hence find the optimal solution.

b) Check whether the following set is convex:

$$S = \{(x, y) : x^2 + y^2 \le 1, y^2 \ge x\}$ $

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Question:

Using matrix – minima method, find the initial basic feasible solution of the following transportation problem:

Image ignouassignments-ignouacademy-com--p-doubts-30704

Hence find the optimal solution.

b) Check whether the following set is convex:

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Question:

A department has five employees with five jobs to be performed. The time (in hours) each employee will take to perform each job is given in the following matrix. How should the jobs be allocated, one per employee, so as to minimize the total man hours?

Image ignouassignments-ignouacademy-com--p-solve-32603

b) For the following pay-off matrix, transform the zero-sum game into an equivalent linear programming problem:

Image ignouassignments-ignouacademy-com--p-ignou-65057

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Question:

wo breakfast food manufacturers ABC and XYZ are competing for an increased market share. The pay-off matrix, shown in the following table, describes the increase in market share for ABC and decrease in market share of XYZ. Determine optimal strategies for both the manufacturers and the value of the game.

Image ignouassignments-ignouacademy-com--p-doubts-56526

b) Find all the basic feasible solutions of the following system of linear equations:

$$2x_1 + x_2 - x_3 + 2x_4 = 2$ $
$$3x_1 + 2x_2 + x_3 + 4x_4 = 3$ $
$$x_1, x_2, x_3, x_4 \ge 0$ $
Check if any of them is degenerate solution. Justify your answer.

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Question:

A company makes two kinds of leather belts. Belts A is high quality belt and belt B is of lower quality. The respective profits on A and B are ₹ 4 and ₹ 3 per belt. The production of each type A requires twice as much time as a belt. The production of each type of type B, and if all belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (both A

and B combined). Belt A require a fancy buckle and only 400 buckles per day are available. There are only 700 buckles a day available for belt B. What should be the daily production of each type of belt? Formulate this problem as an LP model and solve it by the graphical method.

b) Find the initial basic feasible solution of the following transportation problem using North-West Corner method.

Image ignouassignments-ignouacademy-com--p-doubts-64097

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Question:

Reduce the following two person zero sum game to $2 \times 2$ game using principle of dominance. And hence solve the game.

Player B
$$\begin{array}{cc|cc} & & B_1 & B_2 \\ \hline & A_1 & 1 & -3 \\ & A_2 & 3 & 5 \\ \text{Player } A & A_3 & -1 & 6 \\ & A_4 & 4 & 1 \\ & A_5 & 2 & 2 \\ & A_6 & -5 & 0 \end{array}$

b) Obtain the dual of the following primal LP problem:
Maximize $z = x_1 - 2x_2 + 3x_3$
Subject to $-2x_1 + x_2 + 3x_3 = 2$
$$2x_1 + 3x_2 + 4x_3 = 1$ $
$$x_1, x_2, x_3 \ge 0$ $

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Question:

. State which of the following statements are true and which are false. Give reasons for your answer with a short proof or a counter example.
a) The intersection of finite number of convex sets is not convex.
b) If value of the $2 \times 2$ matrix game $\begin{bmatrix} 1 & 2 \\ p & 4 \end{bmatrix}$ is 4, then $p \ge 4$ .
c) If 10 is added to each of the entries of the cost matrix of a $3 \times 3$ assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10.
d) For maximization LP model, the simplex method is terminated when all values $c_j - z_j \ge 0$ .
e) The dummy source or destination in a transportation problem is added to prevent solution from becoming degenerate.

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Question:

For a given set of securities, all their portfolios lie on or within the boundary of the region shown in Fig.1.

Image ignouassignments-ignouacademy-com--p-ignou-75985

In the feasible region, find a portfolio which has maximum return. Also, find a portfolio in this region which has minimum risk.

b) Explain the method of delineating the efficient frontier of a feasible region.

c) Given all the portfolios of n securities what criterion would an investor use to select a good portfolio?

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Question:

ompare the phase diagrams of the systems:

i) $\dot{x} = y, \dot{y} = -x$

ii) $\dot{x} = xy, \dot{y} = -x^2$

by locating the equilibrium points and sketching the phase paths.
b) Consider the epidemic model governed by the following equation

$$\frac{dx}{dt} = -\beta x (n + 1 - x)$

with initial condition $x = n$ at $t = 0$ . Here x(t) is the number of susceptibles at time t, $\beta$ is the contact rate. The population is assumed to be closed and homogeneously mixing. Let the contact rate be 0.002 and the number of susceptibles be 5000 initially

i) Find the density of the population when the rate of appearance of new cases is maximum.

ii) Find the time (in weeks) at which the rate of appearance of new cases is maximum.

iii) Obtain the maximum rate of appearance of new cases.

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Question:

Consider the cubic total cost function

$$C = 0.06q^3 - 0.8q^2 + 13q + 10$

Assume that the price of q is 15 per unit. Find the output which yields maximum profit.
b) Apply dominance to find the optimum strategies of A and B from the pay-off matrix given below

	Player B
		u	v
	x	`6(2)`	`7(3)`
Player A	y	`4(7)`	`9(3)`
	z	`7(1)`	`8(2)`

c) Suppose that the previous forecast was 2090 and the actual value of the variable of interest for the period was 1985 and the oldest value of interest was 1955. Using the moving average technique based upon the most recent four observations find new forecast for the next period.

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Question:

A monopolist sets a price 'p' per unit and the quantity demanded 'q' is given by the following relation:

$$q = 17 - p$

Let there be a fixed cost of Rs.9 and a marginal cost of Rs.1 per unit.
i) Write the profit function of monopolist.
ii) For maximum profit, find the number 'x' of units produced. Also find the maximum profit.
iii) A potential entrant enters into the business of the monopolist. He believes that the monopolist will go on making 'x' units. Write the profit function of the entrant.
iv) For maximum profit of entrant, find the number 'z' of units produced.
v) Find the maximum profit of the entrant. Explain whether he should enter into the business or not.

vi) Find the profit made by the monopolist after the entrant has entered into business.

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Question:

Consider the group of individuals born in a given year ( $t = 0$ ) and let n(t) be the number of these individuals surviving t year later. Let x(t) be the number of members of this group who have not had smallpox by year t and are therefore still susceptible. Let $\beta$ be the rate at which susceptibles contract smallpox and let v be the rate at which people who contract smallpox die from the disease. Finally, let $\mu(t)$ be the death rate from all causes other than smallpox. If dx / dt and dn / dt are, respectively the rates at which the number of susceptibles and entire population decline due to contraction from smallpox and also due to death from all causes then
i) Formulate the above problem by writing equations for dx / dt and dn / dt.
ii) Taking $z = x / n$ , show that z satisfies the initial value problem
$$\frac{dz}{dt} = -\beta z(1 - vz), \quad z(0) = 1$
iii) Find z(t) at any time t.
iv) Bernoulli estimated that $v = \beta = \frac{1}{8}$ . Using these values, determine the proportion of 20 years old who have not had smallpox.

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Question:

Consider a one-dimensional growth c(x, t) of phytoplankton in a water mass. Formulate the model describing the dynamics of growth taking into account the following: D, its diffusion coefficient, r its rate of growth, R its mortality rate due to sinking. Fixing the area of interest as $0 \le x \le 2$ and the initial concentration of phytoplankton as $30 \text{ moles/cm}^3$ , find the concentration distribution of phytoplankton in $0 \le x \le 2$ at any time t.
b) When an aeroplane ascends from take-off to an altitude of $10\text{ km}$ , by how much does the gravitational attraction acting on it decrease?

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Question:

A parachutist, whose weight (actually mass) is $64\text{ kg}$ , drops from a helicopter $5000\text{ m}$ above the ground. She falls towards the earth under the influence of gravity. Assume that the gravitational force is constant. Assume that the force due to air resistance is proportional to the velocity of the parachutist. The proportionality constant is $k_1 = 16\text{ kg/sec}$ when the parachute is closed, and is $k_2 = 100\text{ kg/sec}$ when it is open. If the parachute does not open until $1\text{ minute}$ after the parachutist leaves the helicopter, after how many seconds will she hit the ground?
b) A projectile is fixed with a constant speed v at two different angles of projection $\alpha$ and $\beta$ such that it gives the same range. Show that $\operatorname{cosec} \alpha = \sec \beta$