Introductory Mathematical Methods for Economics — DU B.A. (Hons.) Economics Previous Year Question Papers

1. (a) Use the technique of indirect proof to prove that for two integers x and y: If xy is even then either x is even, or y is even. (b) Show that a tangent to the curve x3 + y = 1 at the point (x, ya) can be written as x0 5x + yo 5y = 1. Use this to show that in the first quadrant the length of any tangent to this curve is 1. (c) The value of a parcel of land bought for speculation is increasing according to the formula V = 2000e0.5VE • If the interest rate is 10 percent, how long should the parcel be held to maximize present value. 2. (a) Solve the following inequality for x : х (x-6) > 0. (b) Approximate the function f(x) = xV3 by a Taylor polynomial of degree 2 at x, = 8. Use it to find the approximate value of 91/3 . Find an upper bound for the error of approximation. (c) An auditorium with a seating capacity of 200 is hired to stage a school play. The ticket price is Rs. 500 for the first 120 tickets. For every additional ticket sold, the ticket price is reduced by Rs. 2 for all the tickets sold. How many tickets should be sold to maximize the total ticket sales revenue? 3. (a) Evaluate lim f(x) and lim f(x) for f(x) = e 5+3x (b) Prove that if'a' and 'b' are strictly positive then the equation a + x-1 - = 0 x-3 has at least one solution in (1,3) (c) If a firm produces output x at a total cost C = ax? + bx, where a > 0, find an expression for the elasticity of total cost with respect to x and comment on its value. (a) For what values of y does the following limit exist: 1im 2ayn. Find the limit for the values of y, where the limit exists. (b) Consider the function f(x) = Is it differentiable everywhere? Is it twice differentiable everywhere? (c) State Jensen's inequality. Use the inequality to show how the log of the mean of numbers x,, x,..., x, compares with the mean of the log of these numbers. (b) Find all the stationary points for the function f(x) = - x4-0.5x2+1 - and classify these points. (c) Define global maximum and local maximum points. Are all global extreme points necessarily local extreme points? 6. (a) Solve for x: 1ğx+4|=14x -6l. Alternative question for visually disabled examinees.