BBA Business Mathematics - I

Find the equation of a straight line which passes through the points (2, 5) and (3, 1).

Calculate the price elasticity of demand in the demand function   P = 40 - 0.2Q  when  P=20.

Form a quadratic equation whose roots are \( 5\sqrt{5} \)   and     \( -\sqrt{5} \)

Solve the equation:    \( 2^{x-2} \times 8^x \) = 16 + 5

Solve the following system of linear equations: x+2y+3z=13;  2x + 4y + z = 11 ; 3x+2y+2z=14. 

The following table shows the yearly income of a company:

Obtain the equation of line by least squares method. Also, estimate the income of the company for the years 2024 and 2026.

question_19: | In an economy which engages in foreign trade, it is assumed that Y = C + I + G + X - M, where \( C = C_0 + 0.8Y_d \), \( C_0 = 90\text{m} \), \( I = 500\text{m} \), \( G = 700\text{m} \), \( X = 250\text{m} \), \( M = 0.3Y_d \), \( t = 0.4 \), \( Y_d = Y - T \), T = tY.

Find the expenditure equation and hence find the equilibrium level of national income and consumption. Also, calculate the total tax.

Find the equation of straight line which passes through the points (3, 6) and (-5, 2).

A manufacturing company has an investment of Rs 90,000 to purchase two raw materials X and Y whose prices are Rs 1,500 and Rs 1,800 each respectively. Write down the slope of budget constraint.

The following table shows the yearly profit of a company [in million Rs].

Obtain the equation of trend line by least squares method. Also, estimate the profit of the company for the years 2015.

Find the equation of straight line passing through the points (8, -3) and (-7, 2).

If \( f(x) = 7x^2 - 3x + 9 then find f(3) \)

The following table shows the year of service and monthly income (thousand Rs) of the workers in a factory:

Sketch the graph of y =\( x^2 - 5x + 6 \)   Also, find the minimum value of y.

Find the roots of following quadratic equation: \( x^2 - 3x - 10 = 0 \).

Form the quadratic equation whose roots are \( (1 + \sqrt{3}) and (1 - \sqrt{3}) \).

Sketch the graph of parabola \( y = x^2 - 2x - 8 \). Find the minimum value of y.

Solve the following quadratic equation:

\( x^2 - 7x + 6 = 0 \)

Draw the graph of parabola of the function\( y = x^2 - 5x + 6 \) Find the domain and range.

The demand and supply functions for a goods are given by Demand function: P=60−0.6Q

Supply function: P=20+0.2Q

a. Calculate the equilibrium price and quantity

b. Calculate the consumer surplus and producer surplus.

c. Total surplus.

Calculate the IRR and NPV for the investment of each of the following projects. Decide which of the projects are viable and rank them in order of their profitability if the market rate of interest is 9%.

Determine whether the following system of linear equations have a unique solution or many solution or no solution: x + 4y = 9 and 2x + y = 8.

Solve the following system of linear equations: x + 2y + z = 9, x + y - 2z = -9 and 2x - y + z = -2.

Given the national income model Y = E, E = C + I where C = 600 + 0.24Y and I = 280. Determine the equilibrium level of national income and the equilibrium level of consumption.

The supply and demand equations of goods are\( P_s = 6Q_s + 40 \)and \( P_d = -2Q_d + 120 \) respectively. The government decides to impose a tax of Rs t per unit of goods. Find the value of t that maximizes the government's total tax revenue on the assumption that equilibrium conditions prevail in the market. Also, find the equilibrium price, the equilibrium quantity and the total tax raised.

The demand and supply functions for goods are given by:

\( P_d = 42 - 0.2Q_d and P_s = 2 + 0.2Q_s \)

a. Calculate the equilibrium price and quantity.

b. Find the consumer's surplus, producer's surplus and total surplus.

The total cost function of ABC Company is given by the equation \( TC = 150 + 20Q \) where P and Q are price and quantity of the items produced by the company. The demand function of that company is specified by the equation \( P = 60 - 2Q \). Find

a. The maximum revenue.

b. The maximum profit.

c. The break even points.

d. The price at which the maximum profit is obtained.

The demand and supply function for goods are given by

Demand function: \( Pd = 80 - 0.3Qd \)

Supply function: \( Ps = 30 + 0.2Qs \)

a) Calculate equilibrium price and quantity.

b) Find consumer's surplus, producer's surplus and total surplus.

Solve the following system of linear equations:

2x - y + z = 30, 3x + 2y - z = 20 & 4x + 2y + 3z = 61.

The supply and demand equations of goods are \( P_s = 3Q_s + 12 \) and \( P_d = 92 - 2Q_d \) respectively. The government decides to impose a tax Rs. 't' per unit of goods. Find the value of t that maximizes the government's total tax revenue on the assumption that equilibrium conditions prevail in the market. Also, calculate the maximum tax revenue.

In Keynesian macroeconomic model of an economy, it is assumed that \( Y = E, E = C + I + G_0, C = 60 + 0.7Y_d, Y_d = Y - T \) where \( I_0 = \text{Rs } 90\text{m}, G_0 = \text{Rs } 140\text{m} \) tax revenue \( (T) = 0.4Y, t = 0.4 \). Determine the value of

 (i) equilibrium level of national income

(ii) equilibrium level of consumption

(iii) the tax revenue.

The resale value of a piece of an industrial equipment has been found to behave according to the function  V = \( 250000e^{-0.6t}, \) where t= years since original purchase. What is the expected resale value after 5, 10, 15 and 20 years?

Solve: \( 2^{9-x} \cdot 4^x = 2^{2x-1} \)

A consumption function is modeled by the equation \( C = 800 \left(1 - e^{-0.5y}\right) \), where y represents income level. Estimate the values of C when y\( = 5, 10, 15, 20 \text{ and } 25 \).

Solve for x:\( 4^{x+3} \times 8^{x-3} = 256 \)

A consumption function is modelled by the equation\( C = 700(1 - e^{-0.4y}) \), where Y represents income

(i) find level of consumptions at \( y = 5, 10, 15, 20, 25 \)

 (ii) Plot the consumption values in the graph paper.

 (iii) Find the saturation level.

Which term of the arithmetic series 6 + 10 +14............ is 62 ? 

Find the sum of geometric series 1 + 3 + 9 + … to 8 terms.

Find the compound interest on Rs 12500 for 3 years at 12% p.a.

Calculate the number of years required for the sum of Rs 5000 to grow to Rs 20000 at the rate of 5.5% p.a. compound interest.

Kumar buys a house for Rs 5,000,000. The contract is that Mr. Kumar will pay Rs 2,000,000 immediately and the balance in 15 equal installments with 15% p.a. compound interest. How much has to be paid by him annually?

The supply and demand functions of a good are\( P_s = 3Q_s + 40 \)  and \( P_d = -2Q_d + 80  \) respectively. If the government decides to impose a tax of Rs t per unit of goods. Find the value of t that maximizes the government total tax revenue on the assumption that equilibrium condition prevail in the market. For this level of tax, find:

a. The equilibrium price and quantity b. The total tax raised.

The demand function of a product of a company is P=500−2x     Fixed cost and variable cost per unit of the product are given by 200 and 100+2x  respectively, where P is the price and x is the quantity of the product.

a. Write down the revenue function, cost function and profit function.

b. Find the maximum revenue.

c. Find the maximum profit.

d. Calculate the breakeven points.

Find the \( 10^{\text{th}} \)term of the geometric series:\( 2 + 6 + 18 + \dots.. \)

Find the sum of the first twenty terms of the arithmetic series: \( 3 + 5 + 7 + \dots... \)

Find the compound interest on a sum of Rs 1,000 for 2 years at\( 5\% \) per annum.

Find the amount of immediate annuity where periodical payment is Rs 50,000 for 15 years at 5% compounded per annum.

In how many years a computer costing Rs 50,000 be reduced to its half at the rate of 5% depreciation each year?

Project A and B involved the following net cash flows:

Decide which project is the most profitable by determining the NPV at discount rate of 15%.

Depreciation on a machinery is written off at 5% every year on the diminishing balance. If its value at the end year of 6 year is Rs 6,250, what is the original cost of the machinery?

Find the sum of \( 3 + 5 + 7 + 9 + \dots\dots\dots \) to 25 terms.

Find the tenth term of geometric series

\( 1 + 2 + 4 + 8 + \dots\dots\dots \)?

Find the rate of compound interest required for Rs 20,000 to grow to Rs 40,000 in 5 years.

How much should be paid now to secure annuity of Rs 5,000 for 15 years, the rate of interest being 10% per annum?

Calculate the IRR and NPV for the investment of each of the following projects. Decide which of the projects are viable and rank them in order of their profitability if the market rate of interest is 11%.

The total cost function of a company is given by the equation \( (TC) = 150 + 20Q \), where P and Q are price (in Rs) and quantity (in units) of the items produced by the company. The demand function of that company is specified by the equation \( P = 90 - 5Q \) Find

a. The maximum revenue

b. The maximum profit

c. The breakeven points.

d. The price at which the maximum profit is obtained.

Evaluate:

Find the point of inflection of   f(x) = \( x^3 - 12x^2 + 2 \)  

Find    ​\( \frac{dy}{dx} \) when \( y = \frac{1}{\sqrt{2x+1}} \). 

Find \( \frac{dy}{dx} \)​ from the following: (i) \( x^2 + y = 9 \)   (ii)\( y = t^4 + 2  \)  and x =\( t^3 + 1 \)

Find the point of stationary of \( y = 5x^3 - 3x^2 + 25 \)

Find \( \frac{dy}{dx} \) from the following:

 (i) \( x^2 + y^2 = 25 \)

 (ii)\( x = 3t^2 and y = t^3 \)

If supply function P = 25 + 0.5Q, then calculate the point elasticity of supply when the price is increased by Rs 90.

Find the point of inflection of \( y = x^3 - 5x^2 - 15x + 27 \)

Find the value of

\( \lim_{x \to \infty} \frac{1x^2 + 9x - 11}{7x^2 + 5x - 12} \)

Find \( \frac{dy}{dx} \) from the following:

 (i)\( 3x^3 + 2y^2 = 9 \) (ii) \( x = t^2 + 2, y = t^3 - 3t \)