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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI β 600 034 B.Sc. DEGREE EXAMINATION β MATHEMATICS FOURTH SEMESTER β 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 ππ = 2 β 0.02π and ππ = 0.2 + 0.07π 2. If the demand law is π = 10 (π₯+1)2 find the elasticity of demand in terms of x. ππ¦ 3. Find ππ₯ if π¦ = π₯ 2 + π¦ 2 β 2π₯. 4. Find the first order partial derivatives of π’ = 3π₯ 2 + 2π₯π¦ + 4π¦ 2 . 5. Evaluate ο² ο¨ 3x c 6. Prove that 7. If π΄ = (1 3 8. If π΄ = (4 2 ο² a ο1 ο« 4 x 2 ο 3x ο« 8 ο© dx b b c a f ( x)dx ο« ο² f ( x)dx ο½ ο² f ( x )dx 2 1 ) and π΅ = ( 4 2 1 2 ), find π΄ . 3 . 0 ), find π΄ + π΅. β3 x ο«1 A B ο½ ο« ο¨ x ο 1ο©ο¨ 2 x ο« 1ο© x ο 1 2 x ο« 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 ο« . 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 ο½ 12. If x y ο½ e x ο y then prove that . dx ο¨1 ο« log x ο©2 11. The total cost C for output x is given by C ο½ 13. Find the first and second order partial derivatives of log ο¨ x 2 ο« y 2 ο© . 14. Integrate x 2 e x with respect to x. a 15. Prove that (i) ο² οa a f ( x) dx ο½ 2 ο² f ( x) dx , if f(x) is an even function. 0 a (ii) ο² f ( x ) dx ο½ 0 , if f(x) is an odd function. οa ο¦1 2 1 οΆ ο§ ο· 16. If A ο½ο§ 0 1 ο1ο· then show that A3 ο 3 A2 ο A ο« 9 I ο½ 0 . ο§ 3 ο1 1 ο· ο¨ οΈ (ii) ο¦ 1 0 ο4 οΆ 17. Compute the inverse of the matrix A ο½ ο§ο§ ο2 2 5 ο·ο· . ο§ 3 ο1 2 ο· ο¨ οΈ xο«5 18. Integrate with respect to x. 2 ο¨ x ο« 1ο©ο¨ x ο« 2 ο© 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 ο½ a ο« bx . AR ο MR 6 (b) If the marginal revenue function for output x is given by Rm ο½ ο« 5 , find the total 2 ο¨ x ο« 2ο© 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 ο½ 300 x ο 10 x 2 ο« x3 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 . ο¨ 3x ο« 7 ο© dx . (b) Evaluate ο² 2 2 x ο« 3x ο 2 21. (a) Find the maximum and minimum values of the function x 4 ο« 2 x3 ο 3x 2 ο 4 x ο« 4 . (b) Solve the equations x ο 2 y ο« 3z ο½ 4, 2 x ο« y ο 3z ο½ 5, ο x ο« y ο« 2 z ο½ 3 by Crammerβs rule. 22. (a) The demand and supply function under perfect competition are y ο½ 16 ο x 2 and y ο½ 2 x 2 ο« 4 respectively. Find the market price, consumerβs surplus and producerβs 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. **********