NATIONAL SENIOR CERTIFICATE EXAMINATION MATHEMATICS P2 SEPTEMBER 2016 GRADE 12. This question paper consists of 13 pages including the formula sheet

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1 NATIONAL SENIOR CERTIFICATE EXAMINATION MATHEMATICS P SEPTEMBER 06 GRADE MARKS: 50 TIME: 3 Hours This questio paper cosists of 3 pages icludig the formula sheet

2 Mathematics/P September 06 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios.. This questio paper cosists of 0 questios.. Aswer ALL the questios. 3. Aswer all the questios i the ANSWER BOOK provided. 4. Clearly show ALL calculatios, diagrams, graphs, etc. which you have used i determiig the aswers. 5. Aswers oly will NOT ecessarily be awarded full marks. 6 You may use a approved scietific calculator (o-programmable ad ographical), uless stated otherwise. 7. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. 8. Diagrams are NOT ecessarily draw to scale. 9. Write eatly ad legibly.

3 Cumulative Frequecy Mathematics/P 3 September 06 QUESTION.. The data i the table ad the ogive below represets the umber of cars passig through a toll gate over a hour peak period durig a log weeked. The coutig of cars started at 3:00. Time i miutes Cumulative frequecy Graph showig the total umber of cars passig through a toll gate over a period of hours Time (i miutes).. How may cars passed through the toll gate i the first hour? ().. How may cars passed through the toll gate betwee 3:30 ad 3:40? ()..3 After how log had 500 cars passed through the toll gate? ()..4 Draw a box ad whisker diagram for the above data. (3)

4 Mathematics/P 4 September 06. The followig data values are give i the table below: p.. What is the value of p if the mea for the data set is 60? ().. Determie the possible values of p if p is the media of the data set. () [] QUESTION The table below shows the amout of moey spet o advertisig ad the correspodig icome of the compay, i thousads of rad, per moth over a 6-moth period. Moth Advertisig 3 4,5 5 7,4 Icome Draw a scatter plot for the data. (3). Determie the equatio of the least squares regressio lie. ().3 Determie the correlatio coefficiet of the data. ().4 Commet o the stregth of the correlatio coefficiet i QUESTION.3. ().5 Predict the compay s icome i a moth where R3 500 is spet o advertisig. () [9]

5 Mathematics/P 5 September 06 QUESTION 3 3. I the followig sketch AMB is a straight lie with AM = MB ad CM AB. y M(5 ; 4) B A( ; 3) 0 C x 3.. Calculate the coordiates of B. () 3.. Determie the equatio of MC. (4) 3..3 Calculate the size of, correct to oe decimal place. (5) 3..4 State with a reaso why the vertices of triagle BCM lie o the circumferece of a circle. () 3. I the figure below, A(7 ; ), B( 4 ; 0) ad C(8 ; ) are the vertices of a triagle ad D is the midpoit of BC. y A(7 ; ) B( 4 ; 0) D 0 C(8 ; ) x 3.. Calculate the legth of AD ad leave the aswer i its simplest surd form. (4) 3.. If AD is produced to the poit E(x ; 4) i the fourth quadrat such that AD = DE calculate the value of x. (5) 3.3 If A( 7 ; 4), B( 3 ; y) ad C(5 ; ) are colliear, calculate the value of y. (3) [5]

6 Mathematics/P 6 September 06 QUESTION 4 I the figure, M is the cetre of two cocetric circles (i.e. the circles have a commo cetre). The larger circle has equatio x y 4y x 44. The lie y x 5 0 is the taget to the smaller circle at A. K is the x-itercept of the taget AK. B is a poit o the larger circle such that MAB is a straight lie. y B M 0 A K x 4. Determie the coordiates of M. (3) 4. Show that the coordiates of A are A( ; 4). (3) 4.3 Determie the equatio of the smaller circle. (3) 4.4 Calculate the legth of AB. () 4.5 The straight lie y x 5 0 meets the straight lie y 0 at poit K. Calculate the area of AMK. (5) [6]

7 Mathematics/P 7 September 06 QUESTION 5 This questio must be aswered without the use of a calculator. 5. If ta 8 ad [ 80;360] determie by usig a sketch, the value of 8 si cos. (5) 5. Simplify to a sigle trigoometric ratio of x: cos 5.ta(80 x).cos(90 x) si( x) (6) [] QUESTION 6 6. Give the equatio cos si( 30). 6.. Show that cos si( 30) is equivalet to 3 si 3cos. (3) 6.. Hece or otherwise, calculate if [ 80 ;80]. (4) 6. Cosider the fuctios f ( ) cos ad g ( ) si( 30) for [ 80;80 ] 6.. Sketch graphs of f ad g o the same set of axes. Show itercepts o the axes clearly. (5) 6.. Use the graph to determie [ 90;90] if: f ( ). g( ) 0 (3) 6..3 For which values of will g ( ) 0, [ 80;80]? (3) [8]

8 Mathematics/P 8 September 06 QUESTION 7 Two observers at A ad B sight a helicopter H hoverig at 50 m directly above C. The agles of elevatio from A ad B to H respectively are i the same horizotal plae. H 6,6 ad,8. A, B ad C are 50 A 6,6 C P,8 B 7. Calculate the distace betwee the observers A ad B if AĈB 04, 5. (6) 7. Calculate C ÂB. (3) [9]

9 Mathematics/P 9 September 06 Provide Euclidea geometry reasos for all statemets i Questio 8 to Questio 0 QUESTION 8 8. I the diagram, the vertices of PNR lie o the circle with cetre O. Diameter SR ad chord NP itersect at T. Poit W lies o NR. OT NP. R 30. ˆ S P 3 T 4 N 3 O W 30 R Determie, statig reasos, the size of: 8.. Ŝ (3) 8.. ˆR (3) 8..3 ˆN (3) 8..4 If it is further give that NW = WR, prove that TNWO is a cyclic quadrilateral. (4)

10 Mathematics/P 0 September I the figure below, PQ is a diameter of the circle with cetre O. PT is a chord of a circle which is produced to R. PQ = 0, QT = 8 ad TR = x P 0 T O 8 Q R 8.. Show that PT = 6 (3) 8.. Show that QR = x x 64 i PQR. () 8..3 Calculate the value of x. (3) []

11 Mathematics/P September 06 QUESTION 9 9. I the accompayig figure, AB is a taget to the circle at B. O is the cetre of the circle. Draw the diagram ad prove the theorem which states that if AB is a taget to the circle at B, the ABC ˆ Dˆ. (5) D C O B A 9. I the figure, AB is a diameter of the circle The taget to the circle at D meets AB produced at C. The bisector of Ĉ cuts DB at E ad meets AD at F. The radius of the circle is 3 uits ad the legth of CD is 4 uits. C B 4 3 E D F A 9.. Prove that DBC ADC. (3) 9.. Calculate the legth of BC. (6) 9..3 If DEC AFC, show that CE = EF. (4) 9..4 If DF calculate the umerical value of the ratio: FA area area of of ΔCDE ΔACD (5) [3]

12 Mathematics/P September 06 QUESTION 0 Circles STP ad RQP touch iterally at P. VP is a commo taget to the circles at P. W is a poit o PQ such that SW RT. Chords RQ, ST, PQ ad PR are draw. Q R S W T P V 0. Prove that RQ ST (4) 0. Prove that PW PT (3) WT TQ [7] TOTAL: 50

13 Mathematics/P 3 September 06 FORMULA SHEET b x A b 4ac a P i A P i A P i A P i a d S a T S r ; r a r a r S ; r P x i i d f x lim h 0 f T F ar x i i x h f x h d x x y y x x y y M ; y mx c y y mx x m y x m ta y x x a y b r I a b c bc. cos A a b ABC : si A si B c sic Area of ABC ab. sic si cos si.cos cos si si si.cos cos si cos.cos si. si cos cos.cos si. si cos A si A cosa si A sia si A. cos A cos A A P A PA or B PA PB PA ad B S x x y a bx b x x y y x x t x x

GRADE 12 JUNE 2016 MATHEMATICS P2

GRADE 12 JUNE 2016 MATHEMATICS P2 NATIONAL SENIOR CERTIFICATE GRADE 1 JUNE 016 MATHEMATICS P MARKS: 150 TIME: 3 hours *MATHE* This questio paper cosists of 11 pages, icludig 1 iformatio sheet, ad a SPECIAL ANSWER BOOK. MATHEMATICS P (EC/JUNE