University of KwaZulu-Natal. Examinations: 24 May 2010

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1 University of KwaZulu-Natal Examinations: 24 May 2010 Subject, Course and code: Mathematics 133 (Math 133P1) Duration: 3 hours Total Marks for Mathematics Section: 80 Internal Examiner: External Examiner: Mrs. I. Swart, Rev. G. Parish, Mrs D Maqutu Dr. C. Zaverdinos MATHEMATICS SECTION INSTRUCTIONS Answer all questions and show all working. Unless instructed otherwise, round answers off to 3 decimal places. Non-programmable calculators are allowed. This paper consists of 10 pages. Please see that you have them all. - STUDENT NO: QUESTION 1 [Total: 24] (a) Express 12,5% as (i) a fraction in its simplest form and (ii) as a decimal. (1)+(1) (b) Write as a single fraction and simplify: x + 1 2x + 1 x 2x 1. (2)

2 (c) Simplify the following: (i) x 2 (x 3 ) 1 2 page 2 (1) (ii) 3 log a (a 3 ) (1) (d) (i) Solve for x : 1 x 2 x = 1 2 (2) (ii) Solve for x correct to 2 decimal places : e 3x+2 = 17 (2) (e) Calculate 1, , and write the result in scientific notation. (2)

3 page 3 (f) Evaluate 3 a=1 a + 2 2a 1 (2) (g) (i) Write out the binomial expansion of (x y) 5 with simplified coefficients. (3) (ii) Find the coefficient of p 4 in the binomial expansion of (p 1 2p )8 (3) (h) Find all x R such that (x 3)(x + 1) 0 (2) (i) Solve for θ : 2 cos 3θ = 3 for 0 θ 2π 3 (2)

4 page 4 QUESTION 2 [Total: 5] In 32 minutes, 20ml of water flows from a leaking tap into a measuring cylinder. (a) At what rate, in litres per hour, is water leaking from the tap? (2) (b) A large bucket with 2 litres of a solution containing 5g of salt is placed under the leaking tap and left there for exactly one hour. What is the concentration of the salt in the solution at the end of the hour (in g/l)? (3) QUESTION 3 [Total: 5] A small animal is gaining mass exponentially. Each week its mass increases by 2,4%. (a) If its mass, now, is 200g, what will its mass be in 6 weeks time? (3) (b) How long, from now, would it take for its mass to reach 250g? (Give your answer to the nearest day.) (2)

5 page 5 QUESTION 4 [Total: 6] A geyser is warmed by a solar heater during daylight hours. The temperature of the geyser and its contents is given by Φ(t) = sin πt where t is the number of hours after sunrise. 8 (a) When does the temperature reach a maximum? (2) (b) What is the maximum value of Φ? (2) (c) At what times will Φ be a mimimum? (2) QUESTION 5 [Total: 9] (a)differentiate the following functions. Do not simplify your answers. (i) f(t) = t3 3 e t + ln(t + 1) (2) (ii) y = cos(x3 ) x (3)

6 page 6 (b) If y = 2x 2 1, find (i) dy dx = (2) (ii) d2 y dx 2 = (2) QUESTION 6 [Total: 5] (a) From first principles (i.e. using the definition of the derivative) find dy dx if y = x2 2. (3)

7 page 7 (b) Find the equation of the tangent line to the curve y = x2 2 at x = 1. (2) QUESTION 7 [Total: 8] The GDP (gross domestic product) from 1998 to 2008 of a developing country is approximated by the function: G(t) = 0, 2t 3 + 2, 4t (0 t 10) where t is measured in years, G(t) in billions of dollars and t = 0 corresponds to the beginning of (a) Give a formula for the rate of growth of the GDP. (1) (b) Calculate the GDP at the beginning of (i) 1998, (ii) 2001 and (iii) (2 1 2 ) (c) In which year from 1999 to 2008 will maximum growth of the GDP be obtained? (2)

8 page 8 (d) Describe the changes in the growth of the GDP from 1998 to (1) (e) After how many years will the rate of growth of the GDP be a maximum? (1 1 2 ) QUESTION 8 [Total: 4] At a T-shirt sale you purchase 4 large and 5 medium shirts while your friend buys 3 large and 3 medium shirts. If the large and medium shirts cost x and y rands respectively, the transactions can be illustrated by the following equations. Use matrix inversion to find the cost of each size of T-shirt. 4x + 5y = 440 (rands) 3x + 3y = 300 (rands) (4)

9 page 9 QUESTION 9 [Total: 8] Given the matrices A = [ ] [ 1, B = 2 ] [ 5 3, C = 7 6 ], find the following, if defined. If the operation is not defined, say so. a) 1 A b) 2A + C 4 c) A 1 d) BA e) AB (8)

10 page 10 QUESTION 10 [Total: 6] Use Gauss reduction to solve the following set of simultaneous equations: x y 4z = 2 4x 2y 3z = 6 2x + 6y 3z = 4 (6)