Brief Answer Questions:
[10 × 1 = 10]Integrate: \(\int \left(3x^4 + 2x^2 + 2x + \frac{1}{x}\right) dx\)
Find \(2 \times 2\) matrix if \(a_{ij} = 2i + 3j\).
Evaluate the value of determinant: \(\begin{vmatrix} 4 & 2 & 5 \\ 3 & 1 & 0 \\ 9 & 6 & 7 \end{vmatrix}\)
If \(f(x, y) = x^4 + 4x^2y^2 - 8y^2\), find second order partial derivative with respect to x.
Solve the differential equation: \(\dfrac{dy}{dx} = \dfrac{y}{2}\)
Solve the difference equation: \(y_t - 4y_{t-1} = 0\)
Find the area bounded by the curve \(y = x^2 - 3x + 2\) and \(x\)-axis and two ordinates at \(x = 1\) and \(x = 4\).
Find dual problem from the following primal problem:
Zmin \(= 5x + 3y\)
Subject to constraint \(2x + 3y \geq 6\), \(x + y \geq 2\) and \(x, y \geq 0\)
Test whether Hawkins-Simon condition is satisfied or not when \(A = \begin{bmatrix} 0.8 & 0.7 \\ 0.6 & 0.5 \end{bmatrix}\)
Write down the order and degree of the following differential equation
\(\left(\dfrac{d^2y}{dx^2}\right)^2 + 2\left(\dfrac{dy}{dx}\right)^3 - 6y = 0\)
Short Answer Questions:(Attempt any FIVE Questions)
[5 × 3 = 15]Integrate the following:
(a) \(\int_0^2 2x\sqrt{2x+1}\, dx\)
(b) \(\int \dfrac{dx}{x + x\ln x}\)
If the marginal revenue function \(MR = 3x^2 + 2x + 10\), find the total revenue when \(x = 5\) and \(x = 10\).
Solve the following system of linear equations using matrix method or determinant
\(x + y + z = 5\)
\(2x - y + 3z = 4\)
\(4x + 5y + 10z = 13\)
The marginal cost function is \(MC = 12 + 10x - 6x^2\). The total cost of producing the first 4 units is Rs 64. Find the total cost function.
A firm has a production function \(Q = 10L^{\frac{1}{2}} K^{\frac{1}{2}}\).
(a) Find the marginal productivity of labor \((MPL)\) and the marginal productivity of capital \((MPK)\).
(b) Also show that \(\dfrac{\partial^2 Q}{\partial L \partial K} = 0\)
Solve the differential equation: \(\dfrac{dy}{dx} + y_t - 3y_{t-1} = 7\)
Also solve the difference equation: \(2y_t - 3y_{t-1} = 7, \, y_0 = 7\)
The total revenue for function \(TR = 12x - \dfrac{x^2}{4}\).
The total profit function.
Long Answer Questions:(Attempt any THREE Questions)
[3 × 5 = 15]A watch dealer wishes to buy new watches and has two models \(M_1\) and \(M_2\) to choose from.
Model \(M_1\) costs Rs 100 and \(M_2\) costs Rs 200. In view of the showcase of the dealer he wants to buy watches not more than 30 and can spend upto Rs 4,000. The watch dealer can make a profit of Rs 70 in \(M_1\) and Rs 80 in \(M_2\).
a) Formulate the above given information in the linear programming model.
b) Calculate the maximum profit by using graphical or simplex method.
If the demand and supply functions in a competitive market are \(Q_d = 26 - 0.4P\) and \(Q_s = 4 + 0.6P\) and the rate of adjustment of price when the market is out of equilibrium is \(\dfrac{dP}{dt} = 0.32(Q_d, Q_s)\).
a) Calculate the maximum profit by using graphical or simplex method.
b) Derive and solve the differential equation to get a function for \(p\) in terms of \(t\) given that the price is 40 in time period zero.
Comment on the stability of the market.
The data below are an economy of two industries X and Y. The values are in millions of rupees.
| Product | User X | User Y | Final Demand | Total Output |
| X | 20 | 15 | 22 | 46 |
| Y | X | 10 | 16 | 42 |
Determine the outputs
a. If the final demand changes to 40 for X and 50 for Y.
b. If the final demand changes to 45 for X and 55 for Y.
Two duopolists, firm X and Y, have the following reaction functions
\(P_1 = 500 + 0.65P_2\), \(P_1 = 500 + 0.65P_2\)
If the assumptions of the Bertrand model hold, derive a difference equation for \(P_1\) and calculate what \(P_1\) will be in the time period 8. If firm X starts off in the period 0 by setting a price of 2500.
Comprehensive Answer / Case / Situation Analysis Questions:
[2 × 10 = 20]A joint function of a factory in terms of x and y is given by \(f(x, y) = x^2 + y^2 - 3x - 9y + s\)
24 where x and y representing the output of two goods. Find
a) Stationary points
b) Saddle points
c) Maximum value
d) Minimum value