Mathematics Grade 11 P3 Eastern Cape November 2012 QP (Eng)

Downloaded from hlayiso.com Province of the EASTERN CAPE EDUCATION NATIONAL SENIOR CERTIFICATE GRADE 11 NOVEMBER 2012 MATHEMATICS P3 MARKS: 100 TIME: 2 hours This question paper consists of 14 pages, including a formula-sheet and diagram sheets.

Downloaded from hlayiso.com 2 MATHEMATICS P3 (NOVEMBER 2012) INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of NINE questions. Answer ALL the questions. 2. Show clearly ALL calculations, diagrams, graphs, et cetera which you have used in determining the answers. 3. An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise. 4. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise. 5. Number the answers correctly according to the numbering system used in this question paper. 6. Diagrams are NOT necessarily drawn to scale. 7. It is in your own interest to write legibly and to present the work neatly. 8. An information sheet with formulae is attached. 9. Diagram sheets are included to assist in answering QUESTIONS 1.2.1, 7.1, 7.2, 8 and 9. Write your name in the space provided and hand it in with your script.

Downloaded from hlayiso.com (NOVEMBER 2012) MATHEMATICS P3 3 QUESTION 1 1.1 Complete the following statements: 1.1.1 If A and B are independent events, then P(A and B) = … (1) 1.1.2 If A and B are mutually exclusive events, then P(A or B) = … (1) 1.2 Two tests namely a Mathematics test (M) and a Science test (S) are given to a group of 300 learners. It is found that 110 learners failed (f) both tests, 120 learners passed (p) both tests, 30 learners passed the Mathematics test but failed (f) the Science test, and 40 learners passed (p) the Science test but failed (f) the Mathematics test. 1.2.1 Summarise the given information in a two way contingency table below by calculating the values of a, b, c, d, e and f as provided in DIAGRAM SHEET 1. Write the answers on the answer sheet provided on DIAGRAM SHEET 1. (6) TWO TESTS Maths (p) Maths (f) Total Science (p) 120 c e Science (f) a 110 f Total b d 300 1.2.2 What is the probability that a learner chosen at random, passed both tests? (1) 1.2.3 What is the probability that a learner chosen at random, passed the Mathematics test, but failed the Science test? (1) 1.2.4 Determine whether passing the Mathematics test (M) and passing the Science test (S) are independent, or dependent events. Support your answer with appropriate calculations. (5) [15]

Downloaded from hlayiso.com 4 MATHEMATICS P3 (NOVEMBER 2012) QUESTION 2 An independent survey was conducted amongst 160 parents of a particular school to determine their choices for chairman of the School Governing Body (SGB) among three candidates, namely Candidate A, Candidate B and Candidate C. The Venn-diagram below shows the preferred choices of the parents. S A B 10 31 23 x 15 18 5 48 C 2.1 How many parents would like to see that any one of the three candidates become the chairman of the SGB? (2) 2.2 What is the probability of a number of parents selected randomly, being undecided in making a decision of who should be the chairman of the SGB? (1) 2.3 Calculate the probability that a parent selected at random will be in favour of Candidate B to be the chairman of the SGB. Express your answer as a decimal. (3) 2.4 Show that at least 33% of the parents chosen at random would have at least TWO of the candidates selected for chairman of the SGB. (3) [9]

Downloaded from hlayiso.com (NOVEMBER 2012) MATHEMATICS P3 5 QUESTION 3 3.1 The Human Sciences Research Council (HSRC) conducted a survey amongst 1000 people selected randomly in all the nine provinces about who they will support to become the next president of South Africa with a voter population of about 23 million people. 550 of the participants of the survey said they will vote for candidate X. Headlines in the newspaper claimed: “55% of all voters support candidate X”. Discuss the validity of this statement along with any limitations it may have and the misconceptions it could create. (4) 3.2 The swimming team of South Africa consists of EIGHT swimmers, namely three men and five women, for the Olympic Games. They have to swim in two qualifying rounds before they can go through to the finals. As part of their rotation policy they must select a captain for each qualifying round. 3.2.1 Draw a tree diagram to determine all the probabilities of men or women being selected as captain – with a column for the first round, then the second round, then the possible outcomes and then probabilities as fractions. Label each branch with its probability. (7) 3.2.2 What is the probability, expressed as a decimal fraction, that both the people chosen will be women? (2) 3.2.3 Express as a percentage, the probability that the two swimmers selected to be captain, will consist of one man and one woman. (3) [16]

Downloaded from hlayiso.com 6 MATHEMATICS P3 (NOVEMBER 2012) QUESTION 4 The estimated number of HIV-positive infections, as per persons tested by the Department of Health, was released in a news report and is shown in a table below: Age group Estimated infections Estimated uninfected Estimated population 15 – 19 398 514 2 130 128 2 528 642 20 – 24 664 716 1 530 514 2 195 230 25 – 29 720 678 1 315 136 2 035 814 30 – 34 540 165 1 206 248 1 746 413 35 – 39 381 155 1 249 108 1 630 263 40+ 395 635 2 109 973 2 505 608 TOTAL 3 100 863 9 541 107 12 641 970 4.1 If you were to pick a person at random what is the probability that he/she would be HIV-positive? Express your answer as a percentage. (2) 4.2 What is the probability that a person would be between 25 – 29 years old and be HIV-positive? Express your answer as a percentage. (2) 4.3 What is the probability that a person would be between 15 – 19 years old and not infected? Express your answer as a percentage. (2) 4.4 What percentage of the total population that was tested, is HIV-positive? (2) 4.5 Which age group has the highest infection rate? Motivate your answer by expressing the highest infection rate as a percentage. (3) 4.6 By comparing the different infection rates of the different age groups, give THREE important messages that this news report is trying to convey to the broader public. (3) [14]

Downloaded from hlayiso.com (NOVEMBER 2012) MATHEMATICS P3 7 QUESTION 5 The graphs below (Graph A and Graph B) show the number of illegal immigrants (in thousands) from a neighbouring state that has entered the borders of South Africa in a three month period since January 2012 to March 2012. This is according to the Department of Home Affairs. 12 10 8 6 Graph A 4 2 0 January February March 60 50 40 30 Graph B 20 10 0 January February March 5.1 Which graph (Graph A or Graph B) do you think, displays the information more clearly? What is causing the effect that the one graph is clearer than the other graph? (2) 5.2 If you were asked by a television station as the Minister of Home Affairs to discuss the illegal immigrant problem, which graph (Graph A or Graph B) would you use in your discussion? Motivate your answer. (2) 5.3 If you were asked by the same television station as a member of the opposition to discuss the illegal immigrant problem, which graph (Graph A or Graph B) would you use in your discussion? Motivate your answer. (2) [6]

Downloaded from hlayiso.com 8 MATHEMATICS P3 (NOVEMBER 2012) QUESTION 6 The vertices of a certain triangle ∆PQR are P(4;7), Q(4; ) and R(1; ). 6.1 If the vertices of ∆PQR is enlarged by a factor of 2, write down the vertices of ∆PQR in the following order: 6.1.1 P(x;y) (1) 6.1.2 Q(x;y) (1) 6.1.3 R(x;y) (1) 6.2 If the vertices of ∆PQR is reduced by a factor of 2, write down the vertices of ∆PQR in the following order: 6.2.1 P(x;y) (1) 6.2.2 Q(x;y) (1) 6.2.3 R(x;y) (1) 6.3 Is ∆PQR /// ∆PQR? Give a reason for your answer. (2) 6.4 If the area of ∆PQR is 16x units2, then the area of the enlarged ∆PQR with factor 2 would be ...x units2. (2) [10]

Downloaded from hlayiso.com (NOVEMBER 2012) MATHEMATICS P3 9 QUESTION 7 7.1 In the figure below, ∆PMN has V the midpoint of PM. RT VN and = . P R V T W M N Determine the following: 7.1.1 (2) 7.1.2 (2) 7.1.3 (2)

Downloaded from hlayiso.com 10 MATHEMATICS P3 (NOVEMBER 2012) 7.2 In the figure below PQRT is a rectangle with PS QS and P ̂ S = y. Q P 2 1 1 2 2 1 R T S Calculate the following angles in terms of y: 7.2.1 Q ̂R (1) 7.2.2 P̂ (1) Prove that: 7.2.3 ∆PSQ /// ∆STP (3) 7.2.4 PS2 = ST × (RS + ST) (3) [14] QUESTION 8 In the figure below, P is a point such that S ̂ P = T ̂ S, TS = 51 mm, PS = 32,6 mm and TP = 29 mm. T 51 mm 29 mm P 32,6 mm S Q 8.1 Prove that ∆STP /// ∆SQT. (3) 8.2 Show that SQ = 79,79 mm. (3) 8.3 Calculate the length of QT correct to TWO decimal places. (3) [9]

Downloaded from hlayiso.com (NOVEMBER 2012) MATHEMATICS P3 11 QUESTION 9 In the figure below, OBEC is a rectangle with OA = 20 units, OD = 10 units and CD = p units. y A B E x O C D 9.1 If ∆AOD /// ∆ECD, show that OB = 2p units. (3) 9.2 Write down the length of OC in terms of p. (1) 9.3 Hence, show that = . (3) [7] TOTAL: 100

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Downloaded from hlayiso.com 12 MATHEMATICS P3 (NOVEMBER 2012) INFORMATION SHEET: MATHEMATICS  b  b 2  4ac x 2a A  P(1  ni) A  P(1  ni) A  P(1  i) n A  P(1  i) n n n n(n  1) 1  n i 1 i  i 1 2 a  r n  1 Sn  ; r 1 ; 1  r  1 r 1 F  x 1  i   1 n  P x[1  (1  i)n ] i i f ( x  h)  f ( x ) f ' ( x)  lim h 0 h  x  x 2 y1  y 2  d  ( x 2  x1 ) 2  ( y 2  y1 ) 2 M  1 ;   2 2  y 2  y1 y  mx  c y  y1  m( x  x1 ) m m  tan x 2  x1 x  a 2   y  b2  r 2 In ABC: a b c 1   a 2  b 2  c 2  2bc. cos A area ABC  ab. sin C sin A sin B sin C 2 sin      sin  .cos   cos  .sin  sin      sin  .cos   cos  .sin  cos     cos  . cos   sin  . sin  cos     cos  . cos   sin  . sin  cos 2   sin 2   cos 2  1  2 sin 2  sin 2  2 sin  . cos  2 cos 2   1  n 2  fx  x  x  i x   i 1 2 n n n( A) P( A)  P (A or B) = P (A) + P (B) – P (A and B) nS  y  a  bx b   x  x  y  y   x  x  2

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Downloaded from hlayiso.com (NOVEMBER 2012) MATHEMATICS P3 13 NAME AND SURNAME: DIAGRAM SHEET 1 QUESTION 1.2.1 TWO TESTS Maths (p) Maths (f) Total Science (p) 120 c = ______ e = ______ Science (f) a = ______ 110 f = ______ Total b = ______ d = ______ 300 QUESTION 7.1 QUESTION 7.2 P Q P 2 1 1 2 R V T 2 1 W R S T M N

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Downloaded from hlayiso.com 14 MATHEMATICS P3 (NOVEMBER 2012) NAME AND SURNAME: DIAGRAM SHEET 2 QUESTION 8 T 51 mm 29 mm 32,6 mm S Q P QUESTION 9 y A B E x O C D