NATIONAL SENIOR CERTIFICATE GRADE 12

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1 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 06 MARKS: 50 TIME: hours This questio paper cosists of 4 pages, iformatio sheet ad a aswer book of 8 pages.

2 Mathematics/P DBE/November 06 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of 0 questios. Aswer ALL the questios i the ANSWER BOOK provided. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig your aswers. Aswers oly will ot ecessarily be awarded full marks. You may use a approved scietific calculator (o-programmable ad o-graphical), uless stated otherwise. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. Diagrams are NOT ecessarily draw to scale. A iformatio sheet with formulae is icluded at the ed of the questio paper. Write eatly ad legibly.

3 Average umber of times shopped per week Mathematics/P DBE/November 06 QUESTION A survey was coducted at a local supermarket relatig the distace that shoppers lived from the store to the average umber of times they shopped at the store i a week. The results are show i the table below. Distace from the store i km Average umber of times shopped per week SCATTER PLOT Distace from the store i km. Use the scatter plot to commet o the stregth of the relatioship betwee the distace a shopper lived from the store ad the average umber of times she/he shopped at the store i a week. (). Calculate the correlatio coefficiet of the data. (). Calculate the equatio of the least squares regressio lie of the data. ().4 Use your aswer at QUESTION. to estimate the average umber of times that a shopper livig 6 km from the supermarket will visit the store i a week. ().5 Sketch the least squares regressio lie o the scatter plot provided i the ANSWER BOOK. () [9]

4 Mathematics/P 4 DBE/November 06 QUESTION The heights of 60 learers i a school are measured. The height of the shortest learer is,9 m ad the height of the tallest learer is, m. The heights are represeted i the histogram below. Histogram Number of of learers ,,,5,7 4,9 5, 6, 7 Heights (i m). Describe the skewess of the data. (). Calculate the rage of the heights. (). Complete the cumulative frequecy colum i the table give i the ANSWER BOOK. ().4 Draw a ogive (cumulative frequecy curve) to represet the data o the grid provided i the ANSWER BOOK. (4).5 Eighty learers are less tha metres i height. Estimate. ().6 The perso takig the measuremets oly had a,5 m measurig tape available. I order to compesate for the short measurig tape, he decided to mout the tape o a wall at a height of m above the groud. After recordig the measuremets he discovered that the tape was mouted at, m above the groud istead of m. How does this error ifluece the followig:,5 m measurig tape Distace above the groud GROUND Mea of the data set Stadard deviatio of the data set () () []

5 Mathematics/P 5 DBE/November 06 QUESTION I the diagram, A( 7 ; ), B, C(6 ; ) ad D are the vertices of rectagle ABCD. The equatio of AD is y = + 6. Lie AB cuts the y-ais at G. The -itercept of lie BC is F(p ; 0) ad the agle of icliatio of BC with the positive -ais is. The diagoals of the rectagle itersect at M. D y y = + 6 A( 7 ; ) M C(6 ; ) O G F(p ; 0) B. Calculate the coordiates of M. (). Write dow the gradiet of BC i terms of p. (). Hece, calculate the value of p. ().4 Calculate the legth of DB. ().5 Calculate the size of. ().6 Calculate the size of O ĜB. ().7 Determie the equatio of the circle passig through poits D, B ad C i the form ( a) ( y b) r. ().8 If AD is shifted so that ABCD becomes a square, will BC be a taget to the circle passig through poits A, M ad B, where M is ow the itersectio of the diagoals of the square ABCD? Motivate your aswer. () [9]

6 Mathematics/P 6 DBE/November 06 QUESTION 4 I the diagram, M is the cetre of the circle passig through T( ; 7), R ad S(5 ; ). RT is a diameter of the circle. K(a ; b) is a poit i the 4 th quadrat such that KTL is a taget to the circle at T. y L T( ; 7) M R S(5 ; ) O K(a ; b) 4. Give a reaso why TŜR 90. () 4. Calculate the gradiet of TS. () 4. Determie the equatio of the lie SR i the form y = m + c. () 4.4 The equatio of the circle above is ( 9) y Calculate the legth of TR i surd form. Calculate the coordiates of R. Calculate si R. Show that b = a 9. If TK = TR, calculate the coordiates of K. () () () () (6) []

7 Mathematics/P 7 DBE/November 06 QUESTION 5 5. Give: si 6 p Determie the followig i terms of p, without usig a calculator. 5.. si96 () 5.. cos6 () 5. Give: cos(a B) cosacosb siasib Use the formula for cos(a B) to derive a formula for si(a B) () 5. Simplify cos A cos( A).cos(90 A) completely, give that 0 A 90. (5) 5.4 Give: cos B ad 0 B 90 5 Determie, without usig a calculator, the value of EACH of the followig i its simplest form: 5.4. cos B () 5.4. si B () 5.4. cos (B + 45 ) (4) []

8 Mathematics/P 8 DBE/November 06 QUESTION 6 I the diagram the graph of f ( ) si is draw for the iterval [ 80 ; 80 ]. y f O the system of aes o which f is draw i the ANSWER BOOK, draw the graph of g( ) cos for [ 80 ; 80 ]. Clearly show all itercepts with the aes, the coordiates of the turig poits ad ed poits of the graph. () 6. Write dow the maimum value of f ( ). () 6. Determie the geeral solutio of f ( ) g( ). (4) 6.4 Hece, determie the values of for which f ( ) g( ) i the iterval [ 80 ; 0 ]. () []

9 Mathematics/P 9 DBE/November 06 QUESTION 7 E is the ape of a pyramid havig a square base ABCD. O is the cetre of the base. EBˆ A, AB = m ad EO, the perpedicular height of the pyramid, is. Volume of pyramid (area of base) ( height) E D C O A B 7. Calculate the legth of OB. () 7. Show that cos 9 (5) 7. If the volume of the pyramid is 5 m, calculate the value of. (4) []

10 Mathematics/P 0 DBE/November 06 Give reasos for ALL statemets ad calculatios i QUESTIONS 8, 9 ad 0. QUESTION 8 8. I the diagram below PQRT is a cyclic quadrilateral havig RT QP. The taget at P meets RT produced at S. QP = QT ad PTˆ S 70. S P T R Q 8.. Give a reaso why Pˆ 70. () 8.. Calculate, with reasos, the size of: (a) Qˆ () (b) Pˆ ()

11 Mathematics/P DBE/November A, B ad C are poits o the circle havig cetre O. S ad T are poits o AC ad AB respectively such that OS AC ad OT AB. AB = 40 ad AC = 48. C S O A T B 8.. Calculate AT. () If OS OT, calculate the radius OA of the circle. 5 (5) []

12 Mathematics/P DBE/November 06 QUESTION 9 ABC is a taget to the circle BFE at B. From C a straight lie is draw parallel to BF to meet FE produced at D. EC ad BD are draw. Ê Ê ad Ĉ y. F E D A 4 B y C 9. Give a reaso why EACH of the followig is TRUE: 9.. Bˆ () 9.. BĈD Bˆ () 9. Prove that BCDE is a cyclic quadrilateral. () 9. Which TWO other agles are each equal to? () 9.4 Prove that Bˆ Ĉ. () [9]

13 Mathematics/P DBE/November 06 QUESTION 0 0. I the diagram respectively such that ST QR. PQR is draw. S ad T are poits o sides PQ ad PR P S T Q R Prove the theorem which states that PS PT. (6) SQ TR

14 Mathematics/P 4 DBE/November I the diagram HLKF is a cyclic quadrilateral. The chords HL ad FK are produced to meet at M. The lie through F parallel to KL meets MH produced at G. MK =, KF =, ML = y ad LH = HG. M y K L H G F 0.. Give a reaso why GFˆM LKˆ M. () 0.. Prove that: (a) GH = y () (b) MFH MGF (5) (c) GF FH y () 0.. Show that y () [0] TOTAL: 50

15 Mathematics/P DBE/November 06 INFORMATION SHEET b b 4 ac a A P( i) A P( i) A P( i) T a ( ) d S a ( ) d T ar ar S ; r i [ ( i) ] F P i i f ( h) f ( ) f '( ) lim h 0 h ( ) ( ) y y d y y M ; y m c y y m ) a y b r ( a b c IABC: si A si B si C a b c bc. cos A area ΔABC ab.si C si si.cos cos. si cos cos.cos si. si r si m y S y A P( i) a ; r r m ta si.cos cos. si cos cos.cos si. si cos si cos si si si. cos cos i i (A) P(A) P(A or B) = P(A) + P(B) P(A ad B) S yˆ a b b ( y y) ( )

GRADE 12 JUNE 2017 MATHEMATICS P2

GRADE 12 JUNE 2017 MATHEMATICS P2 NATIONAL SENIOR CERTIFICATE GRADE 1 JUNE 017 MATHEMATICS P MARKS: 150 TIME: 3 hours *JMATHE* This questio paper cosists of 14 pages, icludig 1 page iformatio sheet, ad a SPECIAL ANSWER BOOK. MATHEMATICS