LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 B.Sc. DEGREE EXAMINATION – MATHEMATICS FOURTH SEMESTER – NOVEMBER 2012 MT 4205 - BUSINESS MATHEMATICS Date : 05/11/2012 Time : 1:00 - 4:00 Dept. No. Max. : 100 Marks PART A Answer ALL the questions: ( 10 X 2 = 20) 1. Find the equilibrium price and quantity for the functions 𝑄𝑑 = 2 − 0.02𝑃 and 𝑄𝑠 = 0.2 + 0.07𝑃 2. If the demand law is 𝑝 = 10 (𝑥+1)2 find the elasticity of demand in terms of x. 𝑑𝑦 3. Find 𝑑𝑥 if 𝑦 = 𝑥 2 + 𝑦 2 − 2𝑥. 4. Find the first order partial derivatives of 𝑢 = 3𝑥 2 + 2𝑥𝑦 + 4𝑦 2 . 5. Evaluate 3x c 6. Prove that 7. If 𝐴 = (1 3 8. If 𝐴 = (4 2 a 1 4 x 2 3 x 8 dx b b c a f ( x)dx f ( x)dx f ( x)dx 2 1 ) and 𝐵 = ( 4 2 1 2 ), find 𝐴 . 3 . 0 ), find 𝐴 + 𝐵. −3 x 1 A B x 1 2 x 1 x 1 2 x 1 9. If then find A and B 10. Define Linear Programming Problem. PART B Answer any FIVE of the following: (5x 8=40) 2 35 x . Find (i) Cost when output is 4 units 3 2 Average cost of output of 10 units (iii) Marginal cost when output is 3 units. dy log x 12. If x y e x y then prove that . dx 1 log x 2 11. The total cost C for output x is given by C 13. Find the first and second order partial derivatives of log x 2 y 2 . 14. Integrate x 2 e x with respect to x. a 15. Prove that (i) a a f ( x) dx 2 f ( x) dx , if f(x) is an even function. 0 a (ii) f ( x) dx 0 , if f(x) is an odd function. a 1 2 1 16. If A 0 1 1 then show that A3 3 A2 A 9 I 0 . 3 1 1 (ii) 1 0 4 17. Compute the inverse of the matrix A 2 2 5 . 3 1 2 x5 18. Integrate with respect to x. 2 x 1 x 2 PART C Answer any TWO questions: ( 2 X 20 = 40) 19. (a) If AR and MR denote the average and marginal revenue at any output, show that elasticity of AR demand is equal to . Verify this for the linear demand law p a bx . AR MR 6 5 , find the total (b) If the marginal revenue function for output x is given by Rm 2 x 2 revenue by integration. Also deduce the demand function. 1 20. (a) Let the cost function of a firm be given by the following equation: C 300 x 10 x 2 x 3 3 where C stands for cost and x for output. Calculate (i) output, at which marginal cost is minimum (ii) output, at which average cost is minimum (iii) output, at which average cost is equal to marginal cost . 3x 7 dx . (b) Evaluate 2 2 x 3x 2 21. (a) Find the maximum and minimum values of the function x 4 2 x3 3x 2 4 x 4 . (b) Solve the equations x 2 y 3z 4, 2 x y 3z 5, x y 2 z 3 by Crammer’s rule. 22. (a) The demand and supply function under perfect competition are y 16 x 2 and y 2 x 2 4 respectively. Find the market price, consumer’s surplus and producer’s surplus. (b) Food X contains 6 units of vitamin A per gram and 7 units of vitamin B per gram and costs 12 paise per gram. Food Y contains 8 units of vitamin A per gram and 12 units of vitamin B per gram and costs 20 paise per gram. The daily minimum requirements of vitamin A and vitamin B are 100 units and 120 units respectively. Find the minimum cost of the product mix using graphical method. **********