NATIONAL SENIOR CERTIFICATE GRADE 12

Size: px

Start display at page:

Download "NATIONAL SENIOR CERTIFICATE GRADE 12"

Eustacia Curtis
5 years ago
Views:

1 NATIONAL SENIOR CERTIFICATE GRADE 1 MATHEMATICS P SEPTEMBER 016 MARKS: 150 TIME: 3 hours This questio paper cosists of 13 pages, 1 iformatio sheet ad a aswer book.

2 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios This questio paper cosists of 11 questios. Aswer ALL the questios i the SPECIAL ANSWER BOOK provided. Clearly show ALL calculatios, diagrams, graphs et cetera that you used to determie the aswers. Aswers oly will NOT ecessarily be awarded full marks. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. Diagrams are NOT ecessarily draw to scale. You may use a approved scietific calculator (o-programmable ad o-graphical), uless stated otherwise. A INFORMATION SHEET with formulae is icluded at the ed of the questio paper. Write eatly ad legibly. Tur over please

3 Gr 1 Mathematics P 3 September 016 QUESTION 1 Learers were asked the time (to the earest miute) that they usually take to get from home to school each morig. The results are show i the table below: Time i miutes Number of learers 5 < t < t < t < t < t < t Complete the cumulative frequecy colum i the ANSWER BOOK provided. () 1. Draw the ogive of these data o the grid provided i the ANSWER BOOK. (4) 1.3 Calculate the mea time that these learers take to get to school i the morig. () 1.4 Calculate the stadard deviatio of this data. () 1.5 Use the ogive to determie approximately how may learers would take more tha oe stadard deviatio from the average time it takes learers to get to school. (3) [13] Please tur over

4 Gr 1 Mathematics P 4 September 016 QUESTION The table shows the umber of athletes ad total amout of medals that some coutries obtaied durig the 01 Summer Olympics.. (Iformatio accordig to: Number of Number of Coutry athletes medals USA Chia Great Britta Russia South Korea 45 8 Germay Frace Australia South-Africa Caada Netherlad New Zealad Use the table to determie the equatio of the least squares regressio lie for the data. (3). Calculate the correlatio coefficiet for the data. (1).3 How may medals (to the earest iteger) ca be predicted, will a coutry with 350 athletes obtai? ().4 Commet o the correlatio betwee the umber of participatig athletes ad the umber of medals obtaied by a coutry. (1) [7] Please tur over

5 Gr 1 Mathematics P 5 September 016 QUESTION 3 I the diagram below, ABC is a isosceles triagle with A(;1) ad B(4 ; 9). AB = BC ad BC is parallel to the y-axis. y C θ K B (4 ; 9) A ( ; 1) x 3.1 Calculate the legth of AB. () 3. Calculate the coordiates of C. () 3.3 Calculate the coordiates of K, the midpoit of AC. () 3.4 Determie the equatio of AC i the form y = mx + c. (3) 3.5 Calculate the size of θ. (4) 3.6 Calculate the area of ABC. (4) [17] Please tur over

Gr 1 Mathematics P 6 September 016 QUESTION 4 The diagram below cosists of two circles, which touch each other exterally at C (1 ; ). The smaller circle has its cetre O at the origi.

1 Give a reaso why the poits O, C ad D lie o a straight lie. () 4. Calculate the gradiet of OC. () 4.3 Hece, show that the value of t = 3. () 4.4 Determie the equatio of the taget AC i the form y = mx + c.

6 Gr 1 Mathematics P 6 September 016 QUESTION 4 The diagram below cosists of two circles, which touch each other exterally at C (1 ; ). The smaller circle has its cetre O at the origi. The other circle has cetre D(t ; 6). CA is a commo taget which itersects the x-axis at A. CDE is the diameter of the larger circle. y O A x C (1 ; ) D (t ; 6) E 4.1 Give a reaso why the poits O, C ad D lie o a straight lie. () 4. Calculate the gradiet of OC. () 4.3 Hece, show that the value of t = 3. () 4.4 Determie the equatio of the taget AC i the form y = mx + c. (3) 4.5 Calculate the coordiates of E. () 4.6 Determie the equatio of a circle passig through the poits A(5 ; 0), C ad E i the form ( x a) ( y b) r. (6) 4.7 If a circle cetre D ad equatio ( x r 3) ( y 6) has to cut the circle cetre O twice, give all possible values of r. (4) [1] Please tur over

7 Gr 1 Mathematics P 7 September 016 QUESTION If si 34 = p, determie the value of each of the followig i terms of p, WITHOUT USING A CALCULATOR si 14 () 5.1. cos 34. cos + cos56. si( ) (4) cos 68 () 5. Determie the value of each of the followig expressios: 5..1 si cos(90 ).si (180 ).cos(70 ) (6) (4) si x ta x [18] Please tur over

8 Gr 1 Mathematics P 8 September 016 QUESTION 6 I the diagram, the graph of f(x) = si x is draw for the iterval x ε [ 90 ; 180 ]. 1 y f x Draw the graph of g, where g(x) = cos(x 60 ), o the same system of axes for the iterval x ε [ 90 ; 180 ] i the ANSWER BOOK. (3) 6. Determie the geeral solutio of f(x) = g(x). (5) 6.3 Use your graphs to solve x if f(x) g(x) for x ε [ 90 ; 180 ] (3) 6.4 If the graph of f is shifted 30 left, give the equatio of the ew graph which is formed. () 6.5 Which trasformatio must the graph of g udergo to form the graph of h, where h(x) = si x? () [15] Please tur over

9 Gr 1 Mathematics P 9 September 016 QUESTION 7 I the diagram below, D, B ad C are poits i the same horizotal plae. AC is a vertical pole ad the legth of the cable from D to the top of the pole, A, is p meters. AC CD. AD C= θ ; DĈB = (90 θ) e CB D = θ. A p C 90 θ θ D θ B 7.1 Prove that: BD p cos si (5) 7. Calculate the height of the flagpole AC if = 30 ad p = 3 meters. () 7.3 Calculate the legth of the cable AB if it is further give that AD B = 70 (5) [1] Please tur over

10 Gr 1 Mathematics P 10 September 016 Give reasos for all statemets ad calculatios i QUESTION 8, 9, 10 ad 11. QUESTION 8 I the diagram, O is the cetre of the circle. A, B, C, D ad E are poits o the circumferece of the circle. Chords BE ad CD produced meet at F. Ĉ = 100, F = 35 ad A ÊB 55 F 35 A 55 E 1 1 D O C B 8.1 Calculate, givig reasos, the size of each of the followig agles: Â () 8.1. Ê 1 (3) Dˆ 1 () 8. Prove, givig reasos, that AB CF. (3) [10] Please tur over

Gr 1 Mathematics P 11 September 016 QUESTION 9 9.

Use DE DF the diagram i the ANSWER BOOK to prove the theorem that states that. AB AC A D (7) E * # F B * # C 9.

11 Gr 1 Mathematics P 11 September 016 QUESTION I the diagram below, ABC ad DEF are give with Â = D ; B = Ê ad Ĉ = F. Use DE DF the diagram i the ANSWER BOOK to prove the theorem that states that. AB AC A D (7) E * # F B * # C 9. I the diagram, ABC is a taget to the circle at B. BDEF is a cyclic quadrilateral with DB = BF. BE is draw ad ED produced meets the taget at A. C F B E D Prove that: A 9..1 Bˆ 1 Ê (3) 9.. BDA EFB (4) 9..3 BD AD. EF () [16] Please tur over

12 Gr 1 Mathematics P 1 September 016 QUESTION 10 I the diagram, P, S, G, B ad D are poits o the circumferece of the circle such that PS DG AC. ABC is a taget to the circle at B. GB C = x. P S D E F G 10.1 Give a reaso why Ĝ x. (1) 10. Prove that: A C B BP.BF BE BS () 10.. BGP BEG (4) BG BP BF (3) BS [10] Please tur over

13 Gr 1 Mathematics P 13 September 016 QUESTION 11 I the diagram, ABC with poits D ad F o BC ad E a poit o AC such that EF AD ad DE BA. Further it is give that AE EC 5 4 ad DF = 0 cm. A E B D 0 cm F C 11.1 Calculate, givig reasos, the legth of FC (3) BD (4) 11. Determie the followig ratio: Area ECF Area ABC = (4) [11] TOTAL: 150 Please tur over

14 Gr 1 Mathematics P September 016 INFORMATION SHEET b b 4 ac x a A P( 1 i) A P( 1 i) A P( 1 i) A P( 1 i) T a ( 1) d S a ( 1 d ) 1 T ar ar 1 S F f '( x) x 1 i 1 i lim h 0 f ( x h) f ( x) h r 1 ; r 1 x[1 (1 i) ] P i ( ) ( ) x1 x y1 y d x x1 y y1 M ; y mx c y y m x ) x a y b r IABC: si cos a b c si A si B si C 1 ( x1 a b c bc. cos A 1 oppervlakte ABC ab.si C m y S y a ; 1 r 1 1 r 1 m ta x x1 si.cos cos. si si si.cos cos. si cos.cos si. si cos cos.cos si. si cos si cos 1 si si si. cos cos 1 xi x i1 fx x (A) P(A) P(A of B) = P(A) + P(B) P(A e B) yˆ a bx S b x x ( y y) ( x x)

NAME OF SCHOOL NATIONAL SENIOR CERTIFICATE GRADE 12 MATHEMATICS ALTERNATE PAPER PAPER 2 SEPTEMBER 2016

NAME OF SCHOOL NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS ALTERNATE PAPER PAPER SEPTEMBER 06 MARKS: 50 TIME: 3 hours This paper cosists of 3 pages ad a formula sheet INSTRUCTIONS Read the followig istructios