UNIVERSITY OF KWAZULU-NATAL

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1 UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: June 006 Subject, course and code: Mathematics 34 (MATH34P Duration: 3 hours Total Marks: 00 INTERNAL EXAMINERS: Mrs. A. Campbell, Mr. P. Horton, Dr. M. Banda EXTERNAL EXAMINER: Dr. P. Pillay INSTRUCTIONS Answer sections A, B and C on the multiple choice answer sheet in pencil. You may use the back pages of the green answer book for rough work. Each multiple choice question is worth 3 marks. There is no negative marking. Answer section D in the green answer book, showing all working. This paper consists of 7 pages. Please see that you have them all. Section A - Linear Algebra. When the price of a certain commodity is R0, the quantity supplied per time period will be 60 units. When the price drops to R5, the producers will be willing to supply only 5 units per time period. Assuming that the supply function (Q S is linear, the supply function will be given by: (a Q S = 9 p (b Q S = 9p 30 (c Q S = 9p + 50 (d Q S = 9p + 30 (e Q S = 9 p The demand and supply functions for a certain commodity are given by Q D = 5 3 p and Q S = 0 p, respectively. The quantity exchanged at equilibrium is (a 3 7 (b 4 (c 7 (d 33 (e Let A = (a undefined 4 0 (b, B = ( 4 0 ( 0 0 (c The product AB is (d ( (e

2 UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS JUNE 006 PAGE 4. The solution to the matrix equation Ax = b is given by x = A b. Let A = ( 0 b =. The solution x is given by (a ( 4 (b (c ( (d ( and ( (e 5. The figure shows the graphs of the lines 4x y = 4, y x = and x = 3. The region satisfying the linear inequalities y is (a A (b B (c C (d D (e none of these 4x y 4 y x x 3 x 0 y 0 6. Consider the following linear programming problem: maximise P = 3x + 4y subject to x + y 0 x + y 8 x y x, y 0 A sketch of the lines corresponding to the constraints is shown below D A B 0 x C 0 y (4, (0/3,4/3 0 x The maximum value of P is (a 7 (b 6 3 (c 40 (d 5 (e 7

3 UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS JUNE 006 PAGE 3 7. The Simplex method is being used to solve a maximisation problem. The following tableau has been reached (x 3 is a surplus variable, x 4 is an artificial variable and x 5 is a slack variable: c j 8 0 M 0 c B Basis x x x 3 x 4 x 5 Solution Ratio M x x z j M M M M 0 4M c j z j + M 8 + M M 0 0 Which one of the following statements is true? (a x must enter the basis and x 4 must leave the basis. (b x must enter the basis and x 5 must leave the basis. (c x must enter the basis and x 4 must leave the basis. (d x must enter the basis and x 5 must leave the basis. (e None of these. Section B - Calculus 8. Let y = f(x = (a (d y. The difference quotient (x + 3 x is: x + 6 (x + 3 (x + 3 (x + x + 3 (b (x (c x + 6 (x + (x + x + (e None of these. x 6 x (x + 3 (x + x If f(x = (x + (x x + 3 8, then the derivative f (x equals: (a (x + (x x (x + (x (b (x + (x (c (x + (x x (x + (x 8 (d (x + + (x 8 (e (x x (x + (x 0. If f(x = (x + 5x 7, then the derivative f (x equals: (a 8 (x + 5x 8 (4x + 5x (b 7(x + 5x 6 (4x + 5 (c 7(x + 5x 6 (d 7(x + 5x 6 (4x + 5 (e none of these.

4 UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS JUNE 006 PAGE 4. If f(t = t, then the equation of the tangent line at (, is: + 4 (a y = 5 t + (b y = 5 t + (c y = 5 t (d y = 5 t (e none of these.. Let the demand curve of a commodity be given by p = p(q D. The price elasticity of demand is: (a p(q D p (Q D Q D (d Q Dp (Q D p(q D p(q D (b p (Q D Q D (e none of these. (c Q D (p p Q D 3. Which statement applies to the function y = 3 x3 5x when x = 5? (a y is at a maximum and concave down (b y is at a point of inflection (c y = 0 (d y is at a minimum and concave up (e None of these 4. If z = 4x y + xe y, what is z x? (a 8xy + xe y (b 8x + (c (8xy + e y (d 8y + xe y (e None of these Section C - Financial Mathematics You are given the formulae: s n i = ( + in i and a n i = ( + i n. i Where necessary, round off to the nearest Rand. 5. What is the effective rate of interest of % p.a. compounded quarterly? (a 5.8% (b.46% (c.5% (d.8% (e.57% 6. Paul invests R500 in an account earning 6% p.a. compounded monthly. To what amount would it accumulate at the end of 4 years? (a R598 (b R53 (c R63 (d R50 (e R635

5 UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS JUNE 006 PAGE 5 7. Easy-Loans offers you a loan of R After years you must pay Easy-Loans R What is the nominal interest rate that Easy-Loans are charging, if interest is compounded annually? (a 6.5% (b.% (c.50% (d 6.07% (e None of these 8. A student buys a car for R 800. She makes a deposit of R 000 and agrees to pay the balance with equal monthly payments over years at an interest rate of 5% p.a. compounded monthly. How much are the monthly payments? (a R (b R57. 4 (c R (d R (e R A small business will yield R at the end of each quarter for 6 years after which it will be sold for R What should an investor pay for the business if a 40% p.a. return on capital is required, and the sinking fund, into which quarterly payments are made, earns 8% p.a. compounded quarterly? (a R (b R (c R (d R (e None of these 0. Rent of R900 per month is payable at the beginning of each month. If the interest rate is 5% p.a. compounded monthly, what is the cash equivalent of two years of rent paid in advance? (a R0 600 (b R 667 (c R0 55 (d R 690 (e R 577

6 UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS JUNE 006 PAGE 6 Section D - Long Questions Answer these questions in the green answer books.. Use the Gauss reduction technique to solve the following system of linear equations: x + x + x 3 = 3 x + x 3 = 0 x 4x 5x 3 = [5 Marks]. Use the Simplex algorithm to solve the following LP: maximise P = 3x + x subject to x + x 0 x + x x, x 0 [5 Marks] 3. A small manufacturer produces two products, A and B. The production costs per unit of A and B are R6 and R3, respectively, and the corresponding selling prices are R7 and R4. In addition, there are transportation costs of 0 cents and 30 cents for each unit of product A and B respectively. There is R700 available to cover production costs and R0 to cover transportation costs. Set up an LP to help the manufacturer decide how many units of each product must be produced in order to maximise profit. [Do Not Solve the LP - no extra marks will be given for solving the LP.] [5 Marks] 4. Complete the first two lines of an amortization schedule for a debt of R being paid off over 6 years by monthly paymens and with an interest rate of 8% p.a. compounded monthly? Period Outstanding Balance Payment Interest Paid Principle Paid [6 Marks]

7 UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS JUNE 006 PAGE 7 5. The demand function for electric toothbrushes is p = 00 4q. The cost, in Rands, of producing q electric toothbrushes is C = q 0q q3 (i Find the total revenue function. (ii Find the profit function. (iii How many toothbrushes should be made to maximise profit? [ Mark] [ Mark] [5 Marks] 6. Evaluate the following integrals: (i (ii (iii 5 (4 x dx x (x 3 dx 3 log e x dx (iv 3 x (x dx