BBA Business Mathematics II

\( Integrate: \int \frac{x+2}{x-5} dx \)

Find the area of the region bounded by the curve \( y = \frac{3}{2}x^2 \) , x axis and the two ordinates x=0 and x=2. 

The marginal cost function of manufacturing xxx units of a product is given by MC =\( 3x^2 - 10x + 3 \). The total cost of producing one unit of the product is Rs. 7. Find average cost function.

Integrate the following:

(a)  \( \int \frac{\log x + 5}{x} dx  \)

(b) \( \int (2x + 1)\sqrt{5x + 2} dx \) 

The demand function for a product is \( P_d = 8 - 2x \)and the supply function is  \( P_s = 2x \) under pure competition. Find the consumer's surplus and producer's surplus.

Solve the differential equations:    \(  \frac{dy}{dx} + \frac{1}{x} \cdot y = x^3 \).    

Integrate: \( \int \left( x^2 + e^x + \frac{1}{x} + 1 \right) dx \)

Find the area of curve bounded by x-axis and ordinates of \( y = 7x^2 \) from \( x_1 = 0 \)and \( x_2 = 2. \)

Integrate the following:

(a) \( \int \frac{\log x}{x} dx \)

\( (b) \int \frac{3x + 1}{\sqrt{5x + 2}} dx \)

The annual rate of repair of a machine is given by \( \frac{dc}{dt} = 5t + 12.5 \), where "t" is the age of machine in years, \( \frac{dc}{dt} \) is Rs per year. Find total cost after 4 years.

Solve the following LP problem using the simplex method or graphic method:

Maximize Z = \( 5x_1 + 3x_2  \)

​.Subject to the constraints

\(  2x_1 + x_2 \leq 5 \)

\(  x_1 + x_2 \leq 4      \)

and x1,x2 ≥ 0

ABC factory produces two articles X and Y, each of which processes by two machines P and Q. The total hours available on machine P and machine Q per week are 10 and 6 respectively. The time requirements and profit per unit for each product are listed below:

(a) Find how many units of each of product should be manufactured to maximize profit?
(b) Calculate the maximum profit. (Using simplex method or Graphical method)

\( If A = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix}, find A^2 \)

Evaluate: \( \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix} ​ \)

The input-output coefficient matrix of an economy of two industries is

\( \begin{bmatrix} 0.8 & 0.2 \\ 0.9 & 0.7 \end{bmatrix} \)

Test whether the system is viable as per Hawkins-Simon conditions.

Solve the following system of linear equations using Cramer's rule or matrix method:

4x+6z=100 

3x+6y+z=100 

3x+4y+3z=100

For the following transaction matrix of two sector economy consisting two industries P and Q, calculate the gross output for each industry if the final demand changes to 18 units for P and 44 units for Q.

Find \( (A + B)^T \) where\( A = \begin{bmatrix} 4 & -2 \\ 3 & -1 \end{bmatrix} \) and \( B = \begin{bmatrix} 4 & -2 \\ 7 & -3 \end{bmatrix} \)

Find the value of determinant: \( \begin{vmatrix} 3 & 3 & 1 \\ 2 & 3 & 1 \\ 2 & 3 & 4 \end{vmatrix} \)

Test whether Hawkins-Simon conditions is satisfied or not,\( A = \begin{bmatrix} 0.2 & 0.3 \\ 0.4 & 0.1 \end{bmatrix} \).

Solve the following linear equations (Using inverse matrix method or determinant method):

In an economy of two industries X and Y the following tables gives the supply and demand position in millions of rupees.

Find \( f_{xx} \)when f(x, y) =\( 3x^2y^2 + 5x^2y \)

A firm's production function is given by the equation  Q = \( 100L^{0.5}K^{0.5}  \)  . Use partial differentiation to find the approximate change in output Q when the firm's manager increases capital by 5% and decreases labour by 3%.

The total cost and demand functions for two goods X and Y are given by the equations

TC=10−6x+5y ;\( P_x = 6 - 2x \),\( P_y = 25 - 0.1y \) ,where x and y are the number of units of the first goods and second goods respectively.

a. Write down the equations for total revenue and total profit.

b. Determine the number of units of each good which should be sold to maximize revenue. Calculate the maximum revenue and prices of each good under revenue maximizations.

c. Determine the number of units of each good which should be bought and sold to maximize profits.d. Calculate the maximum profit, prices of each good and total revenue under profit maximization.

Find \( F_x and F_y \)when \( F(x, y) = 2x^3 y^2 + 7x^2 y^3 \)

If \( f(x, y) = x^2 + 3xy + 5 \), find the value of f(2, 3).

 The demand function is \( Q = 200 - 4P + 0.5P_A + Y \), where \( P = 6 \), \( P_A = 10 \), and \( Y = 500 \), find the price elasticity, cross-price elasticity and income elasticity of demand. What happened to demand when

(i) P decreases by 10% (ii) \( P_A \) increases by 10% (iii) Y increases by 10%?

Given the production function \( Q(L, K) = L^2 - 2KL + 2K^2 \), where L and K represent the labour and capital respectively.

 (i) find out the marginal products of labour and capital when the investments on labour and capital are 30 units and 20 units respectively

(ii) Find out marginal rate of technical substitution (MRTS)

 (iii) estimate the increase in capital needed to maintain the current level of output when a unit decrease in labour is 0.08.

A joint function of a factory in terms of x and y is given by \( F(x, y) = x^3 - 3x + xy^2 \), where x and y representing the output of two goods. Find:

a. Stationary points.
b. Saddle points.
c. Maximum value.
d. Minimum value.

Determine the order and degree of the differential equations:  \( \left( \frac{dy}{dx} \right)^2 - \left( \frac{dy}{dx} \right)^5 + 7xy = 0. . \)

Solve the differential equation:   \( \frac{dy}{dx} = \frac{1}{x} \)

In a competitive market price where \( Q_d = 500 - 5P \) and \( Q_s = 40 + 20P \) , the initial price P(0) is Rs 50.

(a) Derive a function for the time-path of P and use it to predict price in time period 5 given that price adjusts in proportion to excess demand at the rate \( \frac{dP}{dt} = 0.01 (Q_d - Q_s) \).

(b) How many time periods would you have to wait for the price to drop by Rs. 30?

Solve the differential equation: \( \frac{dy}{dt} = 7ty \)

Write down the order and degree of differential equation:\( \left(\frac{d^2y}{dx^2}\right)^3 - \left(\frac{dy}{dx}\right)^3 + 2y = 0 \)

Solve the following differential equation: \( \frac{dy}{dt} + \frac{y}{t} = t^2, given y(0) = 1 \)

The demand and supply function under perfect competition are \( Qd = 170 - 8P \)and \( Qs = -10 + 4P \)respectively, where P and Q denote price and quantity, and price at initial time period is Rs. 10. The rate of adjustment of price when the market is out of equilibrium is\( \frac{dp}{dt} = 0.5(Qd - Qs) \), where \( \frac{dp}{dt} \) denotes price in Rs. Per week.

(a) Derive and solve the relevant differential equation to get the function for P in terms of t.
(b) What will be the price after 5 and 10 weeks?

Solve the first order homogeneous difference equations:\( y_{t+1} - 2y_t = 0, y_0 = 3. \)

The reaction functions for the two duopolists, firms X and Y are given by the equations  \(      P_t^X = 45 + 0.8 \, P_{t-1}^Y \)  and   \( P_t^Y = 45 + 0.8P_{t-1}^X \) , respectively. If the assumptions of the Bertrand model hold, derive a difference equation for  \( P_t^X \)    and calculate what  \(  P_t^X \)  will be in time period 10 if firm X starts off in time period 0 by setting a price of 300.

Find the time path of the national income \( y_t \) of the following data and comment on the stability of the time path.\( y_t = C_t + I_t,  \)  \( C_t = 200 + 0.75y_{t-1}, \)​,    \(  I_t = 50 + 0.15y_{t-1} \) ​ and \( y_0 = 3000. \)

Solve the difference equation: ,\( y_t = 0.6y_{t-1} + 21 \),  \( y_0 = 50 \). Find  \( y_5  \) ​ and \(   y_1  \) ​?

Solve the following difference equation: \( Y_t = 15 Y_{t-1} \) when \( Y_0 = 10. \)

Solve the difference equation: \( Y_{t-1} - 0.7Y_t + 7 \), given \( Y_0 = 7 \) and find \( Y_7 \)

Consider a Lagged Keynesian microeconomic national income model is\( Y_t = C_t + I_t \) Where \( C_t = 0.75Y_{t-1} + 200 \), \( I_t = 50 + 0.15Y_{t-1} \), where \( Y_t \) is total national income, \( C_t \) is consumption and \( I_t \) is investment.

(a) Solve for \( Y_t \) when \( Y_0 = \text{Rs } 3000 \)

(b) Find the value of\( Y_5 \) and \( Y_{10} \)