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Brief Answer Questions:
[10 × 2 = 20]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 \(\sqrt{5}\) and \(-\sqrt{5}\).
Solve the equation: \(2^{x-2} \times 8^x = 16 + 5\).
Which term of the arithmetic series \(6 + 10 + 14 \ldots\) is \(62?\)
Find the sum of geometric series \(1 + 3 + 9 + \ldots\) to 8 terms.
Find the compound interest on Rs 12500 for 3 years at 12% p.a.
Evaluate: \[\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\]
Find the point of inflection of \(f(x) = x^3 - 12x^2 + 2\).
Find \(\frac{dy}{dx}\) when \(y = \frac{1}{\sqrt{2x+1}}\).
Short Answer Questions:(Attempt any SIX Questions)
[6 × 5 = 30]Solve the following system of linear equations:
\(x + 2y + 3z = 13; \quad 2x + 4y + z = 11\) and \(3x + 2y + 2z = 14\).
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.
Sketch the graph of \(y = x^2 - 5x + 6\). Also, find the minimum value of \(y\).
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?
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?
Find \(\frac{dy}{dx}\) from the following: (i) \(x^2 + y = 9\) (ii) \(y = t^4 + 2\) and \(x = t^3 + 1\)
Long Answer Questions:(Attempt any THREE Questions)
[3 × 10 = 30]The following table shows the yearly income of a company:
| Year | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
| Income (Rs millions) | 52 | 54 | 61 | 59 | 62 | 60 | 65 |
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 \(C = C_0 + 0.8Y_d\), \(C_0 = 90m\), \(I = 500m\), \(G = 700m\), \(X = 250m\), \(M = 0.3Y_d\), \(t = 0.4\), \(Y_d = Y - T\), \(T = tY\).
Find the expenditure equation and hence find the equilibrium level of national income and consumption. Also, calculate the total tax.
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.
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%.
| Project A | Project B | Project C | Project D | |
| Initial outlay (Rs) | 10,000 | 6,000 | 9,000 | 8,000 |
| Return after 1 year | 11,000 | 6,520 | 9,800 | 8,700 |
Comprehensive Answer / Case / Situation Analysis Questions:
[20]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.