BEAULIEU COLLEGE PRELIMINARY EXAMINATIONS 08 MATHEMATICS GRADE PAPER Eaminer: Mr J Ruiz Mesa Total marks: 50 Date: 6 July 08 Instructions: Moderator: Mr G Scholefield Time: 3hrs This question paper consists of pages and an Information Sheet of pages (i ii). Please check that your paper is complete. Answer all the questions on the question paper. Clearly show ALL the calculations, diagrams, graphs, et cetera you have used in determining the answers. An approved calculator (non-programmable and non-graphical) may be used, unless stated otherwise. If necessary, answers should be rounded off to ONE decimal place, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. An information sheet is included. It is in your own interest to write legibly and to present the work neatly. Please ensure your calculator is in DEGREE mode. DO YOUR BEST!! Question Section A Area Mark Allocation Question Algebra 5 6 Probability 5 7 Section B Area Functions and Graphs Functions and Graphs Mark Allocation 9 9 3 Finance 5 8 Calculus 6 4 Sequences and Series 9 9 5 Calculus 0 Sequences and Series Functions and Graphs 8 Total 76 Total 74
Question SECTION A a) Solve for : ) 4 () ) 3 (4) 3) 3 (4) f a b 3 b) Given 6 f f, determine the values of a and b and the third factor. (4) If 0 c) Determine the values of and y which satisfy the equations: y 4 and 3 y 4 (5) d) Given the equation: k. k k ) Determine, in terms of k the roots of the equation. (4) ) Determine the value(s) of k for which the roots will be equal. () [5] Page of
Question a) A bo contains 8 red pens and green pens. Joanna takes one pen from the bo and notes its colour. She then takes a second pen from the bo, without replacing the first pen, and notes it colour. The tree diagram below shows some of the outcomes. ) In you answer script, copy and complete the diagram to show all probabilities. (3) ) Using your diagram, or otherwise, determine the probability the Joanna chooses at least one green pen. (4) b) The Venn diagram shows the sports played by a group of students at Beaulieu College. The sets show how many students play soccer (S), rugby (R) and do athletics track (T). ) Determine the number of students in the group. () ) Determine the probability that a student chosen at random plays rugby or soccer but does not do athletics track. (3) 3) Determine the probability that a student chosen randomly does athletics track given that he or she plays soccer. (4) [5] Page 3 of
Question 3 a) Matthew needs to borrow R30 000 to be repaid over 5 years. He considers two financial institutions: Classic Savings Bank charges 8% p.a. simple interest over the loan term. Super Savings Bank charges 0% p.a. compounded monthly over the loan term. ) If Matthew chose Classic Savings Bank, determine the total amount of interest he paid for the loan. () ) If Matthew chose Super Saving Bank, determine the monthly repayments on the loan. (3) 3) What was the outstanding balance of Matthew s loan with Super Savings Bank after the 4 th payment? (3) b) Richard purchased a new car for R40 000. The car depreciates at 5% p.a. on a reducing balance. Richard also decided to insure his new car. Initially, the annual cost of the insurance was R 00. One year later, the annual insurance cost increased to R 60. ) Determine the depreciated value of Richard s car three years after purchase. () ) Determine the annual percentage increase in the insurance premiums. () 3) Assuming that the insurance cost increases by the same percentage each year. Determine after how many whole years the insurance cost will be more than the depreciated value of the car. (3) [5] Page 4 of
Question 4 a) The first four terms of a sequence are: 3 3 5; 5 5 ; 7 5 ; 9 5 4 Write down a formula for the nth term of the sequence. () b) The amount of a certain medication remaining in the body is reduced by 0% each hour after the medication is given. The prescription that Dr Barr gave Kim called for a 00-milligram dose of the medication. Question 5 ) Is the amount of medication in Kim s body each hour after she takes the medication modelled by an arithmetic sequence or a geometric sequence? Eplain. () ) Write down a formula to find the amount of medication in Kim s body at any time after she takes it (nth term). () 3) If Dr Barr wants Kim to have at least 30 milligrams of the medication in her body at all times, after how many hours should Kim be taking a new dose of the medication. (4) [9] a) Given f 5 3, determine f ' from first principles. (5) b) Determine the following, leaving the answer with positive eponents. dy ) if d y (3) 3 4 a ) D (3) c) Write down a function g such that ' 3 g () [] Section A: 76 marks Page 5 of
Question 6 a) Given: f 5 SECTION B ) Determine the equation of the vertical asymptote of f. () ) Determine the y-intercept of f. () 3) Determine if f. () 4) Determine the equation of one of the aes of symmetry of f. () b) The inverse of a function is f 4, what is the original function of f? (3) Question 7 a) A tightrope connects two elevated platforms A and B as shown in the diagram below. The curve of the tightrope is modelled by the equation y 0,008 0,8 50. All units are in metres. [9] ) Determine the height of platform A. () ) Given that platform B has the same height as platform A. Determine the distance between the platforms. (3) 3) Write down the domain for which y 0,008 0,8 50 represent the tightrope. 4) L is the lowest point on the tightrope. Determine the height of L above ground. (3) () Page 6 of
b) The following data was collected. From the graph of this data, it would appear as if the output value represents an eponential function with equation f a b. Question 8 Input - 0,3 6 Output 0,67 6 7 4,9 465 ) Megan used the input values of 0 and and the corresponding output values to determine the values of a and b. Determine Megan s function in the form m (3)... ) Gina used the input values of and and the corresponding output values to determine the values of a and b. Determine Gina s function in the form g (4)... 3) Determine the output value for Megan and Gina s equations if =,3 and = 6. () 4) State which of the two functions, m() or g() is the better approimation of the relationship between the input and output. () [9] a) Consider the graph of f y shown alongside. B is a local minimum, D is a local maimum and the tangent to the graph at C is horizontal. Write your answers in term of a, b, c and d. ) Write down the equation of the tangent at C. () ) Write down the solutions to the equation f '. (3) 0 3) For what values of is f decreasing? () 4) Comment on whether the following epression is positive, negative or zero: f b f b a a () 5) What can be said about the tangent lines at A and E if f a f ' e '. () Page 7 of
5 4 3 b) A and B are points on the curve 3 curve are parallel to the -ais. f at which the tangents to the ) Determine the coordinates of the points A and B. (4) ) For which values of is the function f concave up? () c) A square based prism has an open top, as shown below. The length of the sides on the base is cm and the height is h cm. h ) Given that 4 800 cm of material was used to construct this hollow prism. Show that: 4800 h (3) 4 ) Hence, or otherwise, show that the volume of the prism is given by: 3 V 00 cm 3 4 (3) 3) Determine the dimensions required to maimise the volume of the prism. (4) [6] Page 8 of
Question 9 f 4 and Below are the graphs of 3 have the roots at A and B, and g g has another root at G is one of the turning points of the cubic function. ; a cubic function. The two functions. The length of DE = 6 units. a) Find the equation of the function g. (3) b) There are two values where the two functions are increasing at the same rate. Find these values correct to two decimal places. (5) [8] Page 9 of
Question 0 a) Three consecutive terms of an arithmetic sequence are given as follows: b c ; c a ; a b Show that a, b and c are also three consecutive terms of an arithmetic sequence. (5) b) The graph below shows the graph of the function f and a sequence of points on the -ais {0,,, 4}. The points are drawn such that a sequence of equilateral triangles results. The sides of the triangles form a quadratic sequence. ) Determine the value of. (4) ) Hence, or otherwise, determine the coordinates of the point A. (3) [] Section B: 74 marks Page 0 of
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