Brief Answer Questions:
[10 × 2 = 20](a) Find the equation of straight line passing through the points \((8,-3)\) and \((-7,2)\).
(b) Solve the following quadratic equation: \(x^2 - 7x + 6 = 0\)
(c) 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?
(d) If supply function \(P = 25 + 0.5Q\), then calculate the point elasticity of supply when the price is increased by Rs 90.
(e) Find the sum of \(3 + 5 + 7 + 9 + \ldots\) to 25 terms.
(f) Find the point of inflection of \(y = x^3 - 5x^2 - 15x + 27\)
(g) Solve for \(x\): \(4^{x+3} \times 8^{x-3} = 256\)
(h) Find the value of \(\displaystyle \lim_{x \to \infty}\frac{x^2+9x-11}{7x^2+5x-12}\).
(i) Find the tenth term of geometric series \(1 + 2 + 4 + 8 + \ldots\)?
(j) If \(f(x) = 7x^2 - 3x + 9\) then find \(f(3)\).
Short Answer Questions:(Attempt any SIX Questions)
[6 × 5 = 30]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?
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.
The demand and supply function for goods are given by
Demand function: \(P_d = 80 - 0.3Q_d\)
Supply function: \(P_s = 30 + 0.2Q_s\)
(a) Calculate equilibrium price and quantity.
(b) Find consumer's surplus, producer's surplus and total surplus.
Solve the following system of linear equations:
\(2x - y + 3z = 30,\ 3x + 2y - z = 20\) & \(4x + 2y + 3z = 61\)
Find \(\frac{dy}{dx}\) from the following: (i) \(3x^3 + 2y^2 = 9\) (ii) \(x = t^2 + 2,\ y = t^3 - 3t\)
Draw the graph of parabola of the function \(y = x^2 - 5x + 6\). Find the vertex, domain and range.
Long Answer Questions:(Attempt any THREE Questions)
[3 × 10 = 30]The following table shows the year of service and monthly income (thousand Rs) of the workers in a factory:
| Service year | 5 | 8 | 7 | 9 | 11 | 10 | 12 |
| Income (000 Rs) | 6 | 9 | 8 | 10 | 12 | 11 | 14 |
Obtain the equation of straight line by least square method. Also estimate the income of the workers who have served for 13 years and 15 years.
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 \(I_0 = \text{Rs }90m\), \(G_0 = \text{Rs }140m\), 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.
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 \(11\%\).
| Project A | Project B | Project C | Project D | |
| Initial outlay (Rs) | 9,000 | 7,000 | 9,000 | 8,000 |
| Return after 1 year | 9,950 | 7,320 | 9,400 | 8,500 |
Comprehensive Answer / Case / Situation Analysis Questions:
[20]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.