HILTON COLLEGE TRIAL EXAMINATION AUGUST 2009 MATHEMATICS: PAPER I Time: 3 hours 150 marks GENERAL INSTRUCTIONS PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. This question paper consists of 10 pages. You are provided with a separate Answer Booklet (pages i - iii) which includes a Formula Sheet. Please check that your paper is complete. 2. Read the questions carefully. 3. This question paper consists of 11 questions. Answer all questions. 4. Question 7 must be answered in the Answer Booklet which must be handed in with your answer book. 5. Number your answers exactly as the questions are numbered. 6. You may use an approved non-programmable and non-graphical calculator, unless a specific question prohibits the use of a calculator. 7. Round off your answers to one decimal digit where necessary, unless otherwise stated. 8. All necessary working details must be shown. 9. It is in your own interest to write legibly and to present your work neatly. 10. Please note that the diagrams are NOT necessarily drawn to scale. Please do not turn over this page until you are asked to do so.
Page 2 of 10 SECTION A QUESTION 1 a) Solve for x: 1) (3) 2) (2) 3) (correct to 2 decimal places) (2) b) Solve the equation for x if: 1) (x is rational number) (2) 2) (x is a natural number) (1) 3) (x is an irrational number) (2) c) Given: 1) Show that is NOT a solution to the equation. (2) 2) Solve for x. (4) d) A Kruger rand is to be made into a brooch by being fixed inside a gold triangular frame. Study the diagram alongside. The coin is placed on a Cartesian plane with its centre at the origin. The outer edge of the coin has the equation the equation The side of the frame labelled AB has Find the coordinates of P. (4) [21]
Page 3 of 10 QUESTION 2 a) Simplify without the use of a calculator: (2) b) Solve for x: (3) c) Determine: 1) if (2) 2) (3) d) Determine the following limit: (3) [13] QUESTION 3 e) Daniel wants to buy a boat costing He takes out a loan for 5 years with interest charged at p.a. compounded monthly. 1) Calculate his monthly instalment on the loan. (4) 2) After paying 45 instalments, Daniel decides to settle the balance on the loan. th Calculate the lump sum he will need to pay to settle the loan after he paid the 45 instalment. (4) f) is invested at p.a. compounded quarterly. After how many years will the investment be worth (5) [13]
Page 4 of 10 QUESTION 4 a) Given: 1) Write down the first 3 terms of the sequence given by (2) 2) Determine the value of (2) b) A sequence of numbers is shown by the patterns as follows: 1) Write down the next term in the sequence of numbers. (2) 2) Find a formula for the general term of this sequence of numbers. (5) [11] QUESTION 5 a) Solve for x if (3) b) Given: where denotes the sum to terms of the sequence. 1) Determine (1) th 2) Find the 4 term of the sequence. (3) [7]
Page 5 of 10 QUESTION 6 Refer to the diagram. The graphs of and the horizontal asymptote of g is given: a) Write down the equations of the asymptotes (dotted lines). (2) b) Determine the values of (3) c) If is the transformation obtained by shifting by 1 unit vertically upwards, determine the equation of in the form (2) d) Write down the equation of the reflection in the (1) [8] QUESTION 7 Consider: a) Draw sketch graphs of on the grid provided in your Answer Booklet, for (4) b) Write down the range of (1) c) What is the period of (1) d) Describe in words the transformation from to (1) [7]
Page 6 of 10 SECTION B QUESTION 8 THIS QUESTION MUST BE ANSWERED IN THE ANSWER BOOKLET. Tsumi is the manager of a small business that manufacture handmade sandals. Two types of pairs of sandals are manufactured, Elegance Sandals and Classic Sandals: The company manufactures between 40 and 150 pairs of Elegance Sandals. The company manufactures at least 50 Classic Sandals. It takes the workforce 1 hour to manufacture a pair of Elegance Sandals and 2 hours to manufacture a pair of Classic Sandals. If the time spend by the entire workforce is taken into account then the company have at least 300 hours per week available. Altogether, no more than 200 sandals can be manufactured per week. The profit on a pair of Elegance Sandals is and on a pair of Classic Sandals is Elegance Sandals Classic Sandals Let the number of pairs of Elegance Sandals be x and the number of pairs of Classic Sandals be y. a) Write down the constraints of the above scenario. (8) b) Sketch the constraints in (a) on the grid below. Clearly indicate the feasible region. (5) c) Write down the Profit equation in the form (1) d) Determine the maximum weekly profit. (3) [17]
Page 7 of 10 QUESTION 9 Refer to the diagram. The curves of and are shown above a) Determine the lengths of OA, OC and OD. (3) b) Show that the coordinates of B is (3) c) For which value(s) of x is (1) d) Determine the equation of the tangent at B. (4) e) Show by completing the square that (4) f) Write down the new equation of f if it shifts horizontally to the right by 3 units. (2) [17]
Page 8 of 10 QUESTION 10 Refer to the figure. The graphs of and are drawn. The curves intersect at P and Q.. L lies on g and K lies on f such that f has turning points at R and ( 1; 0) a) Show that the values of a and b are 3 and 2 respectively. (4) b) Calculate the coordinates of the turning point R. (3) c) For what value(s) of x is. (3) d) Find the maximum value of LK. (4) [14]
Page 9 of 10 QUESTION 11 a) Given: Determine from first principles. (5) b) Given a function that satisfies the following conditions: Determine the values of (4) c) Refer to the diagram. The function nd is of the 2 degree. The straight line in the sketch graph represents the function 1) What is the gradient of the tangent to the curve when (1) 2) For which value(s) of x will f be an increasing function? (1) 3) Write down the x-coordinates of the turning point of f. (1) 4) Is this turning point a maximum or a minimum? Explain your reasoning. (2) [14]
Page 10 of 10 QUESTION 12 Consider the line-segment given by in the first quadrant. Let be a variable point on this segment. O is the origin and P is a point on the x-axis so that and the coordinates of Q are NOTE: a) Explain why RPOQ is a trapezium. (1) b) Determine the coordinates of R if the area of trapezium RPOQ is a maximum. (5) c) Hence, calculate the maximum area of the trapezium. (2) [8] TOTAL: 150
Page i of iv ANSWER BOOKLET SECTION A QUESTION 7 (a) (4)
Page ii of iv SECTION B QUESTION 8 Tsumi is the manager of a small business that manufacture handmade sandals. Two types of pairs of sandals are manufactured, Elegance Sandals and Classic Sandals: The company manufactures between 40 and 150 pairs of Elegance Sandals. The company manufactures at least 50 Classic Sandals. It takes the workforce 1 hour to manufacture a pair of Elegance Sandals and 2 hours to manufacture a pair of Classic Sandals. If the time spend by the entire workforce is taken into account then the company have at least 300 hours per week available. Altogether, no more than 200 sandals can be manufactured per week. The profit on a pair of Elegance Sandals is and on a pair of Classic Sandals is Elegance Sandals Classic Sandals Let the number of pairs of Elegance Sandals be x and the number of pairs of Classic Sandals be y. a) Write down the constraints of the above scenario. (8)
Page iii of iv b) Sketch the constraints in (a) on the grid below. Clearly indicate the feasible region. (5) c) Write down the Profit equation in the form (1) d) Determine the maximum weekly profit. (3) [17]
Page iv of iv