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

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1 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

2 INSTRUCTIONS Read the followig istructios carefully before aswerig the questios.. This questio paper cosists of questios.. Aswer ALL the questios. 3. Number the aswers correctly accordig to the umberig system used i this questio paper. 4. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig your aswers. 5. Aswers oly will ot ecessarily be awarded full marks. 6. You may use a approved scietific calculator (o-programmable ad o-graphical), 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. A iformatio sheet with formulae is icluded at the ed of the questio paper. 0. Write eatly ad legibly.

3 QUESTION The followig table shows the heartbeat per miute of 00 adults betwee the ages 0 ad 60. Heartbeats/miute itervals Number of idividuals Cumulative Frequecy 50 < x < x < x < x < x < x < x 0 0 < x Complete the cumulative frequecy colum i the ANSWER BOOK provided. (). Draw a ogive of these data o the grid provided i the ANSWER BOOK. (4).3 Calculate the estimated mea heartbeat rate of these adults. ().4 Show o your ogive where you would fid the media, lower quartile ad upper quartile. Write dow the estimated values o the graph. (3) [0] 3

4 Questio The average maximum ad miimum temperatures for Cape Tow, i degrees Celsius, over the period of oe year are give as follows: Moth Miimum Maximum Jauary 8 6 February 6 6 March 4 5 April May 0 0 Jue 9 8 July 7 7 August 8 8 September 9 9 October November 3 3 December 4 6. Determie the equatio of the least squares regressio lie of the average miimum ad maximum temperatures above, correct to two decimal places. (3). Calculate the value of the correlatio coefficiet to two decimal places. ().3 Hece commet o the relatioship betwee the miimum ad maximum temperatures. ().4 Calculate the mea miimum temperature for Cape Tow. ().5 Calculate the stadard deviatio of the miimum temperatures. ().6 Determie the umber of moths for which the miimum temperatures lie withi ONE stadard deviatio of the mea. () [] 4

5 QUESTION 3 I the diagram below A(-;4), B(-;-), C(3; p) ad D(x; y) are four poits i a Cartesia plae. M is the midpoit of AC, B = 90 ad the icliatio of lie AB is q. y A (-, 4) D(x; y) M θ x B(-, ) C(3; p) 3. Determie the size of q. (3) 3. Show that p = -. (3) 3.3 Calculate the coordiates of M. (3) 3.4 Determie the equatio of the circle, with cetre M ad passig through poits A ad C. Write your aswer i the form (x - a) + (y - b) = r. (3) 3.5 Determie the coordiates of D if ABCD is a rectagle. () 3.6 Determie the legth of AB. Give your aswer i surd form. (3) 3.7 Hece, prove aalytically that ABCD is a square. () [9] 5

6 QUESTION 4 The lie y = x cuts the circle (x + ) + (y + ) = 0 at A (3; ) ad B, where B lies i quadrat 3. C is a poit o the circumferece. (x + ) + (y + ) = 0 4. Show that the coordiates of B are ( 3; 5) (4) 4. Determie the distace AB, leavig your aswer i simplest surd form. () 4.3 Write dow the legth OB ad OA. () 4.4 Hece determie θ, BCA. Roud your aswer to TWO decimal places. (6) [3] 6

7 QUESTION 5 5. If si 56 = t, without the use of a calculator ad with the use of a sketch, fid the value of the followig: 5.. ta 34 () 5.. si 4 () 5..3 si (4) 5. Fid the value of the followig without the use of a calculator: cos(90 ).si (6) si (80 ).cos(70 ) 5.3 Prove the idetity: si x cos x si x cos x = cos x si x + (6) [0] 7

8 QUESTION 6 I the diagram, the graph of f(x) = cos (x 60 ) is draw for the iterval x ε [ 90 ; 80 ]. f 6. Draw the graph of g, where g(x) = six, o the same system of axes for the iterval x ε [ 90 ; 80 ] 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 ; 80 ] (3) 6.4 If the graph of g is shifted 30 to the right, give the equatio of the ew graph which is formed. () 6.5 Which trasformatio must the graph of fudergo to form the graph of h, where h(x) = si x? () [5] 8

9 QUESTION 7 I the diagram below, A ad B are two poits i the same horizotal plae as the foot of the tower TC. AB = d metres. From A, the agle of elevatio of T is x.tâb = y ad TB A = z. 7. Write dow the size of AT B i terms of y ad z. 7. Show thattc = d.si x.si z si(z+y) 7.3 If the value of d = 400m, x = 90 y ad y = z, determie the height of the tower, TC. () (5) (4) [0] 9

10 Give reasos for ALL statemets i QUESTIONS 8, 9, 0 ad. QUESTION 8 O is the cetre of the circle ad diameter KL. AB is a taget to the circle at K. ON // LM ad F = 76. F A 76 K O M L B N Calculate, with reasos, the size of each of the followig agles: 8. L () 8. Ô () 8.3 M () 8.4 K () 8.5 N +N () 8.6 Prove why KOMN is ot a cyclic quadrilateral. () [] 0

AB AC (7) A D E * # F B * # C 9. I the diagram, ABC is a taget to the circle at B.

11 QUESTION 9 9. I the diagram below, ABC ad DEF are give with A = D ; B = Ê ad C = F. Use the diagram i the ANSWER BOOK to prove the theorem that states that DE DF. AB AC (7) A D 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 A Prove that: 9.. Bˆ Ê (3) 9.. BDA EFB (4) 9..3 BD AD. EF () [6]

12 QUESTION 0 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 A B C 0. Give a reaso why Ĝ x. () 0. Prove that: 0.. BP.BF BE BS () 0.. BGP BEG (4) 0..3 BG BP BF (3) BS [0]

13 QUESTION I PQR, T is a poit o PQ, V is a poit o PR ad TVRW is a parallelogram. PT = x + uits, QW = x + 4 uits, QT =`5 uits ad WR = x uits. P T V Q W R. Calculate the value(s) of x. (5). If VR =8 uits ad x >, determie the legth of PV. (3).3 Hece, calculate the ratio of Area of TQW : Area of PTV (5) [3] TOTAL: 50 3

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

NATIONAL SENIOR CERTIFICATE GRADE 12

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. INSTRUCTIONS AND INFORMATION Read the