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 016) INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios. 1. This questio paper cosists of 11 questios.. Aswer ALL the questios i the SPECIAL ANSWER BOOK provided. 3. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig the aswers. 4. Aswers oly will ot ecessarily be awarded full marks. 5. You may use a approved scietific calculator (o-programmable ad ographical), uless stated otherwise. 6. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. 7. Number the aswers correctly accordig to the umberig system used i this questio paper. 8. Write eatly ad legibly. Please tur over
(EC/JUNE 016) MATHEMATICS P 3 QUESTION 1 The table below shows the amout of time (i hours) that learers aged betwee 1 ad 16 spet playig sport durig school holidays. Time (hours) Cumulative Frequecy 0 t < 0 30 0 t < 40 69 40 t < 60 19 60 t < 80 157 80 t < 100 167 100 t < 10 17 1.1 Draw a ogive (cumulative frequecy curve) i the SPECIAL ANSWER BOOK provided to represet the data above. (4) 1. Write dow the modal class of the data. (1) 1.3 How may learers played sport durig the school holidays, accordig to the data above? (1) 1.4 Use the ogive (cumulative frequecy curve) to estimate the umber of learers who played sport more tha 60% of the time. () 1.5 Estimate the mea time (i hours) that learers spet playig sport durig the school holidays. (4) [1] QUESTION Accordig to a official i the quality assurace departmet of a ca maufacturig busiess, the stadard deviatio of a 340 ml ca is,74 ml. Out of a sample of 0 cas the followig cotet was measured. 34 338 336 340 340 345 334 338 339 340 341 337 336 340 335 336 34 340 337 336.1 Determie the mea volume of these 0 cas. (). Determie the stadard deviatio of these 0 cas. ().3 Determie what percetage of the data is withi oe stadard deviatio of the mea. () [6] Please tur over
4 MATHEMATICS P (EC/JUNE 016) QUESTION 3 I the diagram, A, B, C ad D are the vertices of a rhombus. The equatio of AC is x + 3y = 10. 3.1.1 Show that the equatio of BD is 3x y = 0. (3) 3.1. Calculate the coordiates of K, the poit of itersectio of AC ad BD. (4) 3.1.3 Determie the coordiates of B. (3) 3.1.4 Calculate the coordiates of A ad C if AD = 50. (8) 3. P(-3;) ad Q(5;8) are two poits i a Cartesia plae. 3..1 Calculate the gradiet of PQ. () 3.. Calculate the agle that PQ forms with the positive x-axis, correct to oe decimal place. () 3..3 Determie the equatio of the straight lie parallel to PQ that itercepts the x-axis at (8; 0). (3) [5] Please tur over
(EC/JUNE 016) MATHEMATICS P 5 QUESTION 4 I the diagram the two circles of equal radii touch each other at poit D(p;p). Cetre A of the oe circle lies o the y-axis. Poit B(8; 7) is the cetre of the other circle. FDE is a commo taget to both circles. 4.1 Determie the coordiates of poit D. () 4. Hece, show that the equatio of the circle with cetre A is give by x + y y 4 = 0. (5) 4.3 Determie the equatio of the commo taget FDE. (5) 4.4 Poit B(8; 7) lies o the circumferece of a circle with the origi as cetre. Determie the equatio of the circle with cetre O. () [14] QUESTION 5 5.1 Simplify (WITHOUT THE USE OF A CALCULATOR) si(180 x). cos(x 360 ). ta(180 + x). cos( x) ta( x). cos(90 x). si(90 x) (8) 5. Prove the idetity: si x 1+cos x + 1+cos x si x = si x (5) 5.3 Use compoud agles to show that: cos x = cos x 1 () 5.4 Determie the geeral solutio for x if: cos x + 3 si x = (7) 5.5 I ABC: Â + B = 90. Determie the value of si A. cos B + cos A. si B. (3) [5] Please tur over
6 MATHEMATICS P (EC/JUNE 016) QUESTION 6 The fuctios of f(x) = ta 1 x ad g(x) = cos(x + 90 ) for 180 x 180 are give. 6.1 Make a eat sketch, o the same system of axes of both graphs o the grid provided i the SPECIAL ANSWER BOOK. Idicate all itercepts with the axes ad coordiates of the turig poits. (6) 6. Give the value(s) of x for which: cos(x + 90 ) ta 1 x () [8] QUESTION 7 I the figure, ABCD represets a large rectagular advertisig board. A surveyor, stadig at the poit P, is i the same horizotal plae as the bottom of the uprights holdig the board. AM ad DN are perpedicular to PMN. Also, BM = m, PM = 10 m, PN = 1 m, AP M = 35 ad MP N = 16,9. 7.1 Calculate the legth of AB. (3) 7. Calculate how far the surveyor is from the board. (Legth of PT) (4) [7] Please tur over
(EC/JUNE 016) MATHEMATICS P 7 Give reasos for ALL statemets i QUESTIONS 8, 9, 10 ad 11. QUESTION 8 8.1 Complete the followig: The opposite agles of a cyclic quadrilateral are (1) 8. I the diagram, EF ad EG are tagets to circle with cetre O. FH EK, EK itersects FG at J ad meets GH at K. Ĥ = x. Prove that 8..1 FOGE is a cyclic quadrilateral. (5) 8.. EG is a taget to the circle GJK. (5) 8..3 FÊG = 180 x. (3) [14] Please tur over
8 MATHEMATICS P (EC/JUNE 016) QUESTION 9 ABCD is a cyclic quadrilateral. AS is a taget. CBS is a straight lie. AD SC ad AD = BD. 9.1 Name, with reasos, FIVE other agles each equal to x. (5) 9. Prove that ASCD is a parallelogram. (4) 9.3 Name a triagle i the figure similar to ADB. (1) 9.4 Hece prove that: SC.SB = DC. (5) [15] Please tur over
(EC/JUNE 016) MATHEMATICS P 9 QUESTION 10 10.1 I the diagram below, ABC is draw with DE BC. 10.1.1 Complete the followig statemet: A lie draw parallel to oe side of a triagle divides the other two sides i (1) 10.1. Use QUESTION 10.1.1 to prove that: AD = AE DB EC (5) 10. 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 = 15 uits ad WR = x uits. 10..1 Calculate the value of x. (6) 10.. If VR = 18 uits ad x > 1, determie the legth of PV. (3) [15] Please tur over
10 MATHEMATICS P (EC/JUNE 016) QUESTION 11 Give O is the cetre of the circle. BA AC. D is the midpoit of BC. 11.1 Prove that: ABC DOC (3) 11. Show that: OC = DC.BC AC (1) 11.3 Give: DC = 15 cm ad AB = 18 cm. Calculate the legth of OC, rouded off to oe decimal uit. (5) [9] TOTAL: 150 Please tur over
(EC/JUNE 016) MATHEMATICS P 11 INFORMATION SHEET: MATHEMATICS b b 4 ac x a A P( 1 i) A P( 1 i) A P( 1 i) A P( 1 i) i1 1 T ar F f 1 x 1 i 1 i i1 ( 1) i f ( x h) f ( x) '( x) lim h 0 h S r 1 a r 1 T a ( 1) d S a ( 1 d ; r 1 x[1 (1 i) ] P i ( ) ( ) x1 x y1 y d x x1 y y1 M ; S a 1 r ) ; 1 r 1 y mx c y y m x ) 1 ( x1 m y x y x 1 1 m ta x a y b r I ABC: si cos a A si b c a b c 1 bc. cos A area ABC ab. si C si B sic 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 or B) = P(A) + P(B) P(A ad B) yˆ a bx S b x x ( y y) ( x x) Please tur over