Solve your IGNOU Doubts

Question:

a) Solve the following ordinary differential equations:

i) $\frac{dy}{dx}+4xy=x$

ii) $\frac{d^2y}{dx^2}+4\frac{dy}{dx}-12=cos2x$

b) Solve the initial value problem:

$\frac{d^2x}{dt^2}-6\frac{dx}{dt}+9x=0,\,x(0)=-1$

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Question:

Use the principle of dominance to reduce the size of the following game. Hence solve the game.

$\begin{bmatrix} 3 & 0 & 4\\ 1 &4 & 2\\ 2 & 2 & 6 \end{bmatrix}.$

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Question:

Show that the set of vectors

$a_1=\begin{bmatrix} 1\\2 \\0 \end{bmatrix} ,a_2=\begin{bmatrix} 2\\0 \\2 \end{bmatrix},a_3=\begin{bmatrix} 0\\2 \\3 \end{bmatrix}.$

from a basis for $E^3.$

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Question:

Solve by simplex method the following linear programming problem: $Max\,z=2x+y+2z$

$s.t$

$3x-y+2z\leq 12$

$-2x+4y\leq 9$

$-x+3y+8z\leq 15$

$x,y,z\geq 0.$

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Question:

The following table is obtained in the intermediate stage while solving an LPP by the simplex method. Image ignouassignments-ignouacademy-com--p-doubts-16278

Discuss whether an optimal solution will exist or not.

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Question:

Find the maximum and minimax values of the following matrix game.

Image ignouassignments-ignouacademy-com--p-solve-67297

Does the matrix have a saddle point. Justify your answer.

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Question:

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

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Question:

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

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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-your-25214

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

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Question:

Consider the epidemic model governed by the following equation

${\frac{\mathrm{d}\mathbf{x}}{\mathrm{dt}}}=-\mathbf{\beta }\mathbf{x}\,\left(\mathbf{n}+1-\mathbf{x}\right)$

with initial condition x = n at t = 0 . Here )t(x is the number of susceptibles at time ,t β 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:

Compare 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.

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Question:

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:

Apply dominance to find the optimum strategies of A and B from the pay-off matrix given below Image ignouassignments-ignouacademy-com--p-ignou-87744

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Question:

Consider the cubic total cost function ${\bf C}=0.06{\bf q}^{3}-0.8{\bf q}^{2}+13{\bf q}+10$

Assume that the price of q is 15 per unit. Find the output which yields maximum profit.

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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 )0 t( = and let )t(n be the number of these individuals surviving t year later. Let )t(x be the number of members of this group who have not had smallpox by year t and are therefore still susceptible. Let β 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 )t( µ 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 ${\frac{\mathrm{d}\mathbf{z}}{\mathrm{dt}}}=-\mathbf{\beta }z\,(1-\mathbf{v}z),\;\mathbf{z}(0)=1$

iii) Find )t(z 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:

When an aeroplane ascends from take-off to an altitude of 10 km, by how much does the gravitational attraction acting on it decrease?

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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 ≤ x ≤ 2 and the initial concentration of phytoplankton as 30moles/ 3 cm , find the concentration distribution of phytoplankton in 0 ≤ x ≤ 2 at any time t .

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Question:

A projectile is fixed with a constant speed v at two different angles of projection α and β such that it gives the same range. Show that $cos\,ec\alpha = sec\,\beta .$

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Question:

A parachutist, whose weight (actually mass) is 64 kg, drops from a helicopter 5000 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 sec $k_1=16 kg/sec$ when the parachute is closed, and is sec $k_2=100\, kg/sec$ when it is open. If the parachute does not open until 1 minute after the parachutist leaves the helicopter, after how many seconds will she hit the ground?