Mathematics Grade 12 P3 NSC November 2012 QP (Eng)

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE GRADE 12 MATHEMATICS P3 NOVEMBER 2012 MARKS: 100 TIME: 2 hours This question paper consists of 9 pages, 5 diagram sheets and 1 information sheet. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P3 2 DBE/November 2012 NSC INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 10 questions. Answer ALL the questions. 2. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers. 3. Answers only will not necessarily be awarded full marks. 4. You may use an approved scientific calculator (non-programmable and non- graphical), unless stated otherwise. 5. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise. 6. Diagrams are NOT necessarily drawn to scale. 7. FIVE diagram sheets for answering QUESTION 1.2, QUESTION 3.1, QUESTION 7.1, QUESTION 7.2, QUESTION 8.2, QUESTION 9 and QUESTION 10 are attached at the end of this question paper. Write your centre number and examination number on these sheets in the spaces provided and insert them inside the back cover of your ANSWER BOOK. 8. An information sheet with formulae is included at the end of the question paper. 9. Number the answers correctly according to the numbering system used in this question paper. 10. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P3 3 DBE/November 2012 NSC QUESTION 1 A recording company investigates the relationship between the number of times a CD is played by a national radio station and the national sales of the same CD in the following week. The data below was collected for a random sample of 10 CDs. The sales figures are rounded to the nearest 50. Number of times CD 47 34 40 34 33 50 28 53 25 46 is played Weekly sales of the CD 3 950 2 500 3 700 2 800 2 900 3 750 2 300 4 400 2 200 3 400 1.1 Identify the independent variable. (1) 1.2 Draw a scatter plot of this data on the grid provided on DIAGRAM SHEET 1. (3) 1.3 Determine the equation of the least squares regression line. (4) 1.4 Calculate the correlation coefficient. (2) 1.5 Predict, correct to the nearest 50, the weekly sales for a CD that was played 45 times by the radio station in the previous week. (2) 1.6 Comment on the strength of the relationship between the variables. (1) [13] QUESTION 2 Each of the 200 employees of a company wrote a competency test. The results are indicated in the table below: PASS FAIL TOTAL Males 46 32 78 Females 72 50 122 Total 118 82 200 2.1 Are the events PASS and FAIL mutually exclusive? Explain your answer. (2) 2.2 Is passing the competency test independent of gender? Substantiate your answer with the necessary calculations. (4) [6] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P3 4 DBE/November 2012 NSC QUESTION 3 A company producing television sets decided to check the lifespan (in years) of their most popular model. They selected 50 sets of the most popular model at random for this test. The lifespan of each set was recorded. The information is represented in the table below. LIFESPAN FREQUENCY (IN YEARS) 4,95 ≤ x < 5,65 2 5,65 ≤ x < 6,35 6 6,35 ≤ x < 7,05 18 7,05 ≤ x < 7,75 17 7,75 ≤ x < 8,45 5 8,45 ≤ x < 9,15 2 3.1 Construct a histogram to represent the data. Use the grid provided on DIAGRAM SHEET 2. (3) 3.2 Calculate the estimated mean lifespan of the most popular model of television set. (3) 3.3 The data representing the lifespan of this batch of television sets is normally distributed. This implies that approximately 68% of the data lies within one standard deviation of the mean, approximately 98% of the data lies within two standard deviations of the mean and approximately 100% of the data lies within three standard deviations of the mean. The standard deviation of this data set is 0,76 years. Calculate the lifespan of the most popular model of television set such that 98% of the lifespan of all the sets will exceed this value. (3) 3.4 The company wants to issue a 5-year guarantee with this model of television set. What would you recommend? Justify your recommendation. (2) [11] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P3 5 DBE/November 2012 NSC QUESTION 4 4 During summer in a certain city in South Africa the probability of a sunny day is and the 7 3 probability of a rainy day is . 7 7 • If it is a sunny day, then the probability that Vusi cycles to work is , the probability 10 1 that Vusi drives to work is and the probability that Vusi takes the train to work 5 1 is . 10 1 • If it is a rainy day, then the probability that Vusi cycles to work is, the probability that 9 5 1 Vusi drives to work is and the probability that Vusi takes the train to work is . 9 3 4.1 Draw a tree diagram to represent the above information. Indicate on your diagram the probabilities associated with each branch as well as all the outcomes. (5) 4.2 For a day selected at random, what is the probability that: 4.2.1 It is rainy and Vusi will cycle to work (2) 4.2.2 Vusi takes the train to work (3) 4.3 If Vusi works 245 days in a year, on approximately how many occasions does he drive to work? (4) [14] QUESTION 5 Every client of CASHSAVE Bank has a personal identity number (PIN) which is made up of 5 digits chosen from the digits 0 to 9. 5.1 How many personal identity numbers (PINs) can be made if: 5.1.1 Digits can be repeated (2) 5.1.2 Digits cannot be repeated (2) 5.2 Suppose that a PIN can be made up by selecting digits at random and that the digits can be repeated. What is the probability that such a PIN will contain at least one 9? (4) [8] QUESTION 6 6.1 Write down a recursive formula for the sequence: 1 ; 5 ; 13 ; 29 ; 61; ... (4) 6.2 Write down the next term of the given recursive sequence: 4 ; 7 ; 13 ; 24 ; 44; ... (2) [6] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P3 6 DBE/November 2012 NSC NOTE: Give reasons for all statements made in QUESTION 7, QUESTION 8, QUESTION 9 and QUESTION 10. QUESTION 7 7.1 If in ∆ LMN and ∆ FGH it is given that L̂ = F̂ and M̂ = Ĝ , prove the theorem that LM LN states = . FG FH L M N F G H (7) 7.2 In the diagram below, ∆VRK has P on VR and T on VK such that PT || RK. VT = 4 units, PR = 9 units, TK = 6 units and VP = 2x – 10 units. Calculate the value of x. V 2x – 10 4 2x – 10 T P 66 99 9 K R (4) [11] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P3 7 DBE/November 2012 NSC QUESTION 8 8.1 Complete the following statement: The angle between the tangent and the chord is equal ... (1) 8.2 In the diagram points P, Q, R and T lie on the circumference of a circle. MW and TW are tangents to the circle at P and T respectively. PT is produced to meet RU at U. MP̂R = 75° PQ̂T = 29° QT̂R = 34° Let TP̂W = a , RP̂T = b , MP̂Q = c and RT̂U = d , calculate the values of a, b, c and d. R Q 1 1 2 3 29° M 75° 34° c d b 1 U P a T (9) [10] W Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P3 8 DBE/November 2012 NSC QUESTION 9 O is the centre of the circle CAKB. AK produced intersects circle AOBT at T. AĈB = x C x B 1 T 4 2 3 1 O 2 K 1 2 3 A 9.1 Prove that T̂ = 180° − 2 x . (3) 9.2 Prove AC || KB. (5) 9.3 Prove ∆BKT ||| ∆CAT (3) AC (3) 9.4 If AK : KT = 5 : 2, determine the value of KB [14] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P3 9 DBE/November 2012 NSC QUESTION 10 In the diagram below, O is the centre of the circle. Chord AB is perpendicular to diameter DC. CM : MD = 4 : 9 and AB = 24 units. B D O M C A 10.1 Determine an expression for DC in terms of x if CM = 4x units. (1) 10.2 Determine an expression for OM in terms of x. (2) 10.3 Hence, or otherwise, calculate the length of the radius. (4) [7] TOTAL: 100 Copyright reserved

Downloaded from hlayiso.com Mathematics/P3 DBE/November 2012 NSC CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 1 QUESTION 1.2 Scatter plot showing the number of times a CD was played vs the CD sales in the following week 5 000 4 500 4 000 3 500 3 000 2 500 2 000 1 500 1 000 500 0 0 5 10 15 20 25 30 35 40 45 50 55 Copyright reserved

Downloaded from hlayiso.com Mathematics/P3 DBE/November 2012 NSC CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 2 QUESTION 3.1 Histogram showing the frequency of the lifespan of the most popular television set (years) 20 15 Frequency 10 5 0 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 Lifespan (years) Copyright reserved

Downloaded from hlayiso.com Mathematics/P3 DBE/November 2012 NSC CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 3 QUESTION 7.1 L M N F G H QUESTION 7.2 V 2x – 10 4 2x – 10 T P 66 99 9 K R Copyright reserved

Downloaded from hlayiso.com Mathematics/P3 DBE/November 2012 NSC CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 4 QUESTION 8.2 R Q 1 1 2 3 29° M 75° 34° c d b 1 U P a T QUESTION 9 W C x B 1 T 4 2 3 1 O 2 K 1 2 3 A Copyright reserved

Downloaded from hlayiso.com Mathematics/P3 DBE/November 2012 NSC CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET 5 QUESTION 10 B D O M C A Copyright reserved

Downloaded from hlayiso.com Mathematics/P3 DBE/November 2012 NSC 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 = Tn = a + (n − 1)d Sn = n (2a + (n − 1)d ) i =1 i =1 2 2 Tn = ar n −1 Sn = ( a r n −1 ) ; r ≠1 S∞ = a ; −1 < r < 1 r −1 1− r 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 + x2 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 θ x 2 − x1 ( x − a )2 + ( y − b )2 = r 2 a b c In ∆ABC: = = a 2 = b 2 + c 2 − 2bc. cos A sin A sin B sin C 1 area ∆ABC = ab. sin C 2 sin (α + β ) = sin α . cos β + cos α . sin β sin (α − β ) = sin α . cos β − cos α . sin β cos(α + β ) = cos α . cos β − sin α . sin β cos(α − β ) = cos α . cos β + sin α . sin β cos 2 α − sin 2 α  cos 2α = 1 − 2 sin 2 α sin 2α = 2 sin α . cos α 2 cos 2 α − 1  ( x ; y ) → ( x cos θ − y sin θ ; y cos θ + x sin θ ) n 2 ∑ fx ∑ (xi − x ) x= σ 2 = i =1 n n n( A) P ( A) = P(A or B) = P(A) + P(B) – P(A and B) n(S ) yˆ = a + bx b= ∑ (x − x )( y − y ) ∑ (x − x) 2 Copyright reserved