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 P (EC/JUNE 017) INSTRUCTIONS AND INFORMATION 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 which you have used i determiig the aswers. 4. Aswers oly will NOT ecessarily be awarded full marks. 5. If ecessary roud off your aswers to TWO decimal places, uless stated otherwise. 6. Diagrams are ot ecessarily draw to scale. 7. You may use a approved scietific calculator (o-programmable ad o-graphical) uless stated otherwise. 8. A iformatio sheet with formulae is icluded at the ed of the questio paper. 9. Write eatly ad legibly.
(EC/JUNE 017) MATHEMATICS P 3 QUESTION 1 The percetages obtaied by learers i their first Mathematics test is show i the table below. Percetages Frequecy Cumulative Frequecy 30 x < 40 1 40 x < 50 50 x < 60 9 60 x < 70 1 70 x < 80 11 80 x < 90 9 90 x < 100 6 1.1 Complete the cumulative frequecy colum i the table give i the ANSWER BOOK. (3) 1. Draw a ogive (cumulative frequecy curve) to represet the data o the grid provided i the ANSWER BOOK. (4) 1.3 Estimate how may learers obtaied 75% or less for the test. Idicate this by meas of B o your graph. () [9]
4 MATHEMATICS P (EC/JUNE 017) QUESTION The water cosumptio (i kilolitres) of 15 households is as follows: 1,4 0,0 34,5 40,1 18,9 19,7 34,9 15,1 3,8 3,7 31,1 0,9 19,7 36,5 33,6.1 List the five umber summary for the data. (4). Draw a box-whisker diagram to represet the data. (3).3 Commet o the skewedess of the data represeted i QUESTION.. (1).4 Determie the stadard deviatio of the data. ().5 Use the stadard deviatio to commet o the spread of the data. (1) [11]
(EC/JUNE 017) MATHEMATICS P 5 QUESTION 3 I the diagram, A (t ; 1), B (6 ; 9) ad C (8 ; -1) are poits i a Cartesia plae. M is the midpoit of BC. P is a poit o AB. CP itersects AM at F (4 ; 3). R is the x-itercept of lie AC ad S is the x-itercept of lie PC. R S 3.1 Calculate the coordiates of M. () 3. Determie the equatio of the media AM. (4) 3.3 Calculate the value of t. () 3.4 Calculate the gradiet of PC. () 3.5 Determie the size of β. () 3.6 Calculate the size of AĈP. (4) [16]
6 MATHEMATICS P (EC/JUNE 017) QUESTION 4 Quadrilateral ABED, with vertices A (0 ; ), B (7; 1), D (-1 ; -5) ad E is give below. Diagoals AE ad BD itersect at C. 4.1 Calculate the coordiates of C, the midpoit of BD. () 4. Show that CA = CB if the coordiates of C are (3 ; -). (3) 4.3 Why is DA B = 90? (5) 4.4 Hece, write the equatio of the circle with cetre C which is passig through A, B, E ad D. () 4.5 Calculate the gradiet of BC, the radius of the circle. () 4.6 Determie the equatio of the taget to the circle at B i the form y = (3) 4.7 Explai why ABED is a rectagle. (3) [0]
(EC/JUNE 017) MATHEMATICS P 7 QUESTION 5 5.1 If si 58 = k, determie, without the use of a calculator: 5.1.1 si 38 () 5.1. cos 58 () 5. Simplify, without the use of a calculator: 5.3 Give cos(α + β) = cos α cos β + si α si β ta 150. si 300. si 10 cos 5. si 135. cos 80 (7) Use the formula for cos(α + β) to derive a formula for si(α + β). (4) 5.4 Prove the idetity: cos x + 1 si x. ta x = 1 ta x (4) 5.5 5.5.1 Show that ta x = si x ca be writte as si x = 0 or cos x = ½. (3) 5.5. Hece, write dow the geeral solutio of the equatio ta x = si x (4) [6]
8 MATHEMATICS P (EC/JUNE 017) QUESTION 6 Give f(x) = ta x ad g(x) = si(x + 45 ) 6.1 Draw the graphs of f(x) ad g(x) o the same set of axes for x [ 90 ; 180 ], o the grid provided i the ANSWER BOOK. (6) 6. Use your graphs to determie the value(s) of x i the iterval x [ 90 ; 90 ] for which: 6..1 g(x) f(x) = 1 () 6.. g(x) f(x) () 6.3 State the period of y = f(x). (1) [11]
(EC/JUNE 017) MATHEMATICS P 9 QUESTION 7 To fid the height h of a tree CD, the ed of the shadow was marked at poits A ad B i the same horizotal plae as its stem C at differet times of the day. The shadow of the tree rotated z betwee the times of observatio, i.e. AC B = z. AB = d metres, AB C = k ad the agle of elevatio of the su at A was y. 7.1 Fid the legth of AC i terms of z, k ad d. () 7. Fid the legth of AC i terms of y ad h. () 7.3 Hece show that d sik.ta y h. (1) si z 7.4 Calculate the legth of h if z = 15, d = 80m, k = 38 ad y = 40. () [7]
10 MATHEMATICS P (EC/JUNE 017) Give reasos for ALL statemets i QUESTION 8, 9, 10 AND 11. QUESTION 8 I the figure, AB is a diameter of the circle with cetre O. AB is produced to P. PC is a taget to the circle at C ad lie ODE perpedicular to BC itersects BC at D ad PC at E. 8.1 Give a reaso why CD = DB. (1) 8. Show that AC OE. (3) 8.3 If BĈP = x, ame two other agles equal to x. (4) 8.4 Prove that OBEC is a cyclic quadrilateral. () [10]
(EC/JUNE 017) MATHEMATICS P 11 QUESTION 9 I the diagram, BD is the diameter of the circle ABCD with cetre O. AB D = 6 ad BÔC = 98. Calculate: 9.1 Â () 9. B 1 (3) 9.3 Ĉ (3) [8]
1 MATHEMATICS P (EC/JUNE 017) QUESTION 10 I the diagram below PQRS is a parallelogram, with the diagoals itersectig at M. 0 QPˆR 90. QR is produced to U. T is a poit o PS. TU itersects QS at V. PQ 6, PR 8, RU 5 ad VS 13 10.1 Determie with reasos the followig ratios i simplified form: 10.1.1 10.1. UR RQ VM MQ (3) (4) 10. Hece, prove that MR VU () [9]
(EC/JUNE 017) MATHEMATICS P 13 QUESTION 11 11.1 I ΔABC ad ΔDEF, Â = D, B = Ê ad Ĉ = F, respectively. Prove that AB DE AC DF. (7) 11. Tagets PQ ad PR touch the circle at Q ad R respectively. T is a poit o the circle such that QT = QR. QT ad PR are produced ad they meet at S. Q 1 = x. 11..1 Name THREE other agles equal to x. (3) 11.. Determie, i terms of x, the size of Q. () 11..3 Hece show that TR QP. (3) 11..4 Prove that ΔSTR ΔSRQ. (3) 11..5 Hece show that RS = ST SQ. () 11..6 If it is further give that QT : TS 3 :, show that SP 5. (3) PQ 3 [3] TOTAL: 150
14 MATHEMATICS P (EC/JUNE 017) b x b 4 ac a A P( 1 i) A P( 1 i) i1 1 T ar F f 1 i1 INFORMATION SHEET MATHEMATICS ( 1) i S r 1 a r 1 A P( 1 i) x[1 (1 i) ] P i '( d A P( 1 i) T a ( 1) d S a ( 1 d ; r 1 x 1 i 1 i f ( x h) f ( x) x) lim h 0 h x ( ) ( ) 1 x y1 y x x1 y y M ; 1 y mx c y y m x ) 1 ( x1 m x a y b r a b c I ABC: a b c bc. cos A si A si B sic 1 area ABC ab. si C si cos y x S y x a 1 r 1 1 ) ; 1 r 1 m ta 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 x x P ( A) ( A) PAor B PA PB PAad B S x x( y y) yˆ a bx b ( x x)