Business Mathematics II

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 \( x_2 = 2. \)

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

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

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

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

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

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

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

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 linear equations (Using inverse matrix method or determinant method):

 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.

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?

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

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)

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 \( Y_{10} \)

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.