GRADE 12 NATIONAL SENIOR CERTIFICATE MATHEMATICS P3 PREPARATORY EXAMINATION 2008

Transcription

1 GRADE 12 NATIONAL SENIOR CERTIFICATE MATHEMATICS P3 PREPARATORY EXAMINATION 2008 MARKS: 100 TIME: 2 hours This question paper consists of 11 pages and 4 diagram sheets.

2 Mathematics/P3 2 DoE/Preparatory Examination 2008 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 9 questions. 2. Answer ALL the questions. 3. Clearly show ALL calculations, diagrams, graphs, et cetera, which you have used in determining the answers. 4. An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise. 5. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise. 6. Number the answers correctly according to the numbering system used in this question paper. 7. Diagrams are not necessarily drawn to scale. 8. FOUR diagram sheets for answering QUESTIONS 3.1, 3.2 and QUESTION 4.3 are attached at the end of this question paper. Write your name in the spaces provided and hand them in together with the ANSWER BOOK. 9. Write neatly and legibly.

3 Mathematics/P3 3 DoE/Preparatory Examination 2008 QUESTION 1 Given the following sequence of numbers: 2; -3; -1; -4; -5; If the pattern in the above-mentioned sequence remains unchanged, determine the next 4 terms in the sequence. (2) 1.2 Write down a recursive formula from which the terms in the above-mentioned sequence can be generated. (3) [5] QUESTION 2 Read the following quote and answer the questions below. 'Common assaults increase dramatically in South Africa. No measures taken seem to have any effect on this wave of crime. The police are unsuccessful in applying law and order and no one feels safe anywhere anymore.' Statement: This is the general feeling of the majority of South Africans. The following information was obtained from Statistics South Africa. [Information management South African Police Force] For the police station in Vanderbijlpark the following was recorded for the period 2001 to 2007: Period Common assaults reported April to September 2001 April to September 2002 April to September 2003 April to September 2004 April to September 2005 April to September 2006 April to September Determine the average number of common assaults that were reported over the 7 years at the Vanderbijlpark police station. (2) 2.2 One of the numbers of assaults is noticeably higher than the rest of the entries. If this outlier is not taken into account and the average over the remaining 6 years is determined, what will be the effect of this outlier on the average of the 7 years? (2) 2.3 The statistics of only one police station was taken and the current population of South Africa according to the Census in 2007 was approximately people. Critically express your opinion over the statement at the beginning of the question. Substantiate your answer with clear reference to the conditions for a valid sample and the subsequent generalisation thereof. (5)

4 Mathematics/P3 4 DoE/Preparatory Examination A news magazine drew GRAPH B reflecting the information in the table given. GRAPH A is the real reflection of the information in the table. GRAPH A Assaults in Vanderbijlpark Number of assaults Period April to Sept to 2007 GRAPH B Assaults in SA - Vanderbijlpark Number of assaults Period April to Sept to What is the difference between the two graphs? Why do you think the magazine chose to present the information in this way? (1) (1) [11]

5 Mathematics/P3 5 DoE/Preparatory Examination 2008 QUESTION 3 A group of 220 Grade 12 learners were questioned on how many hours they watch television each week. The results are summarised in the following table: HOURS FREQUENCY The average amount of hours that learners spend in front of the television is 27,95. Determine the standard deviation of the data. Use the table on DIAGRAM SHEET 1 for the calculation. (Round off to 2 decimals.) (4) 3.2 Draw a line graph of the data. Use the diagram on DIAGRAM SHEET 1. (4) 3.3 Fill in the average number of hours on the graph as well as THREE standard deviations on both sides of the average. (4) 3.4 What is your interpretation of the distribution of the data? (1) [13]

6 Mathematics/P3 6 DoE/Preparatory Examination 2008 QUESTION 4 'SPEED KILLS' is the slogan used by the 'ARRIVE ALIVE' campaign every year in an effort to decrease the high death toll on South African roads. Insurance companies have proven statistically that persons under the age of 25 years are more likely to drive above the speed limit and therefore the insurance premiums for this age group are higher than for other age groups. The table below indicates which distance in metres is required by a vehicle to get to a standstill when it travels at a certain speed. x speed km/h y braking distance required The scatter plot of the table looks as follows: SPEED KILLS Braking distance Speed km/h 4.1 Discuss the trend of the scatter plot. (1) 4.2 Determine, with the aid of a method of your choice, the linear regression equation of the line of best fit (4) 4.3 Draw the line of best fit on the scatter plot that is given on DIAGRAM SHEET 2. (2) 4.4 Determine, with the aid of a method of your choice, the correlation coefficient that will indicate the correlation strength. (3) 4.5 Discuss the strength of the correlation between the speed that a vehicle travels and the distance it requires to bring it to a standstill. (2) [12]

7 Mathematics/P3 7 DoE/Preparatory Examination 2008 QUESTION The South African Weather Buro predicts that the winter days will be as follows: COLD, WET or COLD and WET. The chance that the days will be cold is 90%. The chance that the days will be wet is 10%. The chance that the days will be cold and wet is 40%. Calculate the probability of the days being WET or COLD. (3) 5.2 During the time of registration of a student at the University of South Africa, three subject groups are presented to him for the B.Com course that he wants to follow. SUBJECT GROUP A SUBJECT GROUP B SUBJECT GROUP C Accounting Financial Management Applied Mathematics Audit Financial Mathematics Physics Taxation Office Management Chemistry Statistics Psychology Biochemistry Mathematics Physiology If a student decides to choose one subject from Group A and one subject from Group B, how many different combinations are possible? (3) If one subject is chosen from each of the three subject groups and you know that you will definitely not choose Biochemistry and Chemistry, how many combinations will there be to choose from? (2) 5.3 Ping pong balls are numbered 1 to 12 and placed in a hat. Every time a ball is drawn it is placed on a rack, one next to the other How many different arrangements of the 12 balls is possible? (2) Assume that the numbers 8 and 11 must be placed next to each other, in any order. In how many ways can the numbers then be rearranged? (3) The numbers 2, 5, 6 and 9 are taken from the group of balls. Twodigit numbers must be formed out of these four numbers. How many numbers can be formed? (2) 5.4 The probability of getting the first answer in a quiz correct, is 0,6. If the first answer is correct, the probability of getting the next answer correct, rises to 0,7. However, if the first answer is wrong, the probability of getting the next answer correct, is only 0,4. Determine the probability of getting the second answer correct. (4) [19]

8 Mathematics/P3 8 DoE/Preparatory Examination 2008 GEOMETRY QUESTION 6 In the sketch below P, Q, R and S are points on the circle. PR is the diameter of the circle and MRT is a tangent at R. ˆ 2 = 65 0 S and R ˆ 3 = 35 0 Determine, with reasons, the sizes of: 6.1 ˆR 1 (1) 6.2 ˆR 4 (2) 6.3 Ŝ 3 (2) [5]

9 Mathematics/P3 9 DoE/Preparatory Examination 2008 QUESTION Complete the theorem: Angles subtended by a chord at a circle, on the same side of the chord, are. (1) 7.2 In the sketch below AD BC with A, B, C, D and E points on the circumference of the circle. EB, EC and BD are straight lines and EF = EG Prove that: B ˆ ˆ 2 = D2 (1) E ˆ ˆ 1 = E3 (3) Δ AEF Δ CED (3) FBCG is a cyclic quadrilateral (3) [11]

10 Mathematics/P3 10 DoE/Preparatory Examination 2008 QUESTION 8 The sketch below shows Δ PQR : PQR ˆ = 16x ; QPR ˆ = 15 x ; Rˆ = 5 x S and T are the midpoints of PQ and PR respectively. PM is extended to N; SM QN and MN = 6 units. Determine, with reasons: 8.1 The value of x (3) 8.2 The size of T ˆ 1 (5) PM 8.3 The ratio of MN (4) 8.4 The value of PO (4) [16]

11 Mathematics/P3 11 DoE/Preparatory Examination 2008 QUESTION 9 The following three equal triangles show an interesting pattern developing: Triangle 1 Triangle 2 Triangle 3 E is the midpoint of AB and F is the midpoint of BC. B ˆ = 0 90 and AB = BC = 1. EBFD is a square. 9.1 Prove that ΔABC ΔAED (4) 9.2 Prove that AB AC = AE AD (1) 9.3 Determine, with reasons, the value of AD. AC (2) 9.4 If the construction shown in Triangle 2, is repeated in ΔAED and ΔDFC, how many similar triangles are formed in Triangle 3? (1) [8] TOTAL: 100

12 Mathematics/P3 DoE/Preparatory Examination 2008 NAME: DIAGRAM SHEET 1 QUESTION 3.1 To calculate the standard deviation: HOURS x x = 27,95 FREQUENCY f = σ ( x x) 2 2 ( x x) ( x x) f ( x x) n 2 f QUESTION 3.2

13 Mathematics/P3 DoE/Preparatory Examination 2008 NAME: DIAGRAM SHEET 2 QUESTION 4.3 SPEED KILLS Braking distance Speed km/h QUESTION 6

14 Mathematics/P3 DoE/Preparatory Examination 2008 NAME : DIAGRAM SHEET 3 QUESTION

15 Mathematics/P3 DoE/Preparatory Examination 2008 NAME : DIAGRAM SHEET 4 QUESTION 8 QUESTION 9