NATIONAL SENIOR CERTIFICATE GRADE SEPTEMBER 05 MATHEMATICS P MARKS: 50 TIME: 3 hours *MATHE* This questio paper cosists of 3 pages icludig iformatio sheet, ad a SPECIAL ANSWERBOOK.
MATHEMATICS P (EC/SEPTEMBER 05) INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios.. This questio paper cosists of 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/SEPTEMBER 05) MATHEMATICS P 3 QUESTION The data i the table below represets the marks obtaied by 0 Grade learers for Eglish Home Laguage (HL) ad Afrikaas First Additioal Laguage (FAL). Eglish HL 4 54 85 3 63 7 9 6 58 66 Afrikaas FAL 50 58 80 45 60 65 98 75 7 58. Draw a scatter plot of the data above by makig use of the grid provided i the SPECIAL ANSWER BOOK. (4). Calculate the equatio of the least squares regressio lie for this data. (3).3 Calculate the correlatio coefficiet. ().4 Describe the correlatio betwee Eglish Home Laguage ad Afrikaas First Additioal Laguage. ().5 Predict the fial Eglish Home Laguage mark for the learer who obtaied 74 marks i Afrikaas First Additioal Laguage. () [] Please tur over
4 MATHEMATICS P (EC/SEPTEMBER 05) QUESTION The weights (i kilogram) of the 0 boys i the hockey squad of School A are give below: 69 59 59 66 64 58 63 58 6 6 57 53 60 5 60 48 47 60 40 60. Determie the mea ad variace for the weights of the School A squad. (3). The followig iformatio was obtaied from the School B boys hockey coach, regardig the weights of the boys i his squad... How may boys are i the School B squad? ().. Determie the mea weight for the School B squad. ()..3 Determie the stadard deviatio for the School B squad. ().3 If five boys of equal weight are added to the squad of School A so that the meas of both schools are the same, what must be the weight of each boy? () [0] Please tur over
(EC/SEPTEMBER 05) MATHEMATICS P 5 QUESTION 3 I the figure A(3 ; 5), B(x ; y), C(5 ; 3) ad D( ; ) are the vertices of parallelogram ABCD. AC ad BD, the diagoals of the parallelogram, itersect at E. y A(3; 5) B(x; y) y)y) E C(5; 3) D ; ) O x 3. Determie: 3.. The co-ordiates of E () 3.. The co-ordiates of B (3) 3..3 The co-ordiates of the midpoit F, of CD ad hece the equatio of the lie passig through F, parallel to AD. (5) 3. The poits G(t + ;,5),D( ; ) ad E(4 ; 4) ad are colliear. Calculate the value of t. (4) 3.3 Determie, by calculatios, whether ABCD is a rhombus or ot. Give a reaso for your aswer. (5) [9] Please tur over
6 MATHEMATICS P (EC/SEPTEMBER 05) QUESTION 4 I the diagram below, M(3 ; ), Q ad N lie o the circumferece of circle with cetre P( ; 4) ad form ΔMQN. NPM is a straight lie. y N α P( ;4) M(3 ; ) Q β θ x 4. Determie the equatio of the circle. (4) 4. Why is? () 4.3 Show that the co-ordiates of Q are ( 4; 0). (3) 4.4 Calculate the gradiet of MN. () 4.5 Hece, calculate the size of α. (5) 4.6 Determie the equatio of a taget to the circle at M. (5) [0] Please tur over
(EC/SEPTEMBER 05) MATHEMATICS P 7 QUESTION 5 5. Prove, without the use of a calculator, that, 5. Determie the geeral solutio of: 5.3 Prove the idetity 5.4 Simplify (4) (7) (3) QUESTION 6 (6) [0] Give ad for ; 6. Write dow the period of g. () 6. Use the set of axes provided i the SPECIAL ANSWER BOOK, to draw sketch graphs of f ad g for x [ 90 ;90 ]. Clearly show all itercepts with the axes ad the co-ordiates of all the turig poits ad ed poits of both curves. (6) 6.3 Use the graphs to determie the value(s) of x for x ; ], where: 6.3. () 6.3. () 6.4 Determie the rage of h(x) = 3f(x). () [3] Please tur over
8 MATHEMATICS P (EC/SEPTEMBER 05) QUESTION 7 I the diagram below, C is a poit o oe side of the Buffalo River ad is 3 m above the water. A is a poit o the other side of the river directly opposite C o the higher bak. B is a boat o the river. A, B ad C are i the same vertical plae. The agle of depressio of B from A is 33,7. The agle of depressio C from A is 5, 60 ad B from C is 6,7. A C 3 m D 33,7 o 6,7 B E 7. Calculate the legth of BC. (3) 7. Calculate the legth of AB. (3) 7.3 Calculate the legth of AD. (3) [9] Please tur over
(EC/SEPTEMBER 05) MATHEMATICS P 9 Give reasos for ALL statemets i QUESTIONS 8, 9, 0 ad. QUESTION 8 I the diagram below, O is the cetre of the circle which passes through P, T, R ad S. PTRS is a cyclic quadrilateral ad ST is draw. x S 3 P O R 3 T 8. Express, givig reasos, each of the followig agles i terms of x. 8.. () 8.. () 8..3 () 8. Hece, calculate the value of x if SOTR is a parallelogram. (3) [9] Please tur over
0 MATHEMATICS P (EC/SEPTEMBER 05) QUESTION 9 I the diagram below, M is the midpoit of chord PT of circle with cetre O. OR is a radius passig through M. QR is produced to itersect taget TA at A, such that TA RA. T ad R are joied. P Q R O M 3 T Prove, statig reasos, that: 9. MTAR is a cyclic quadrilateral (4) 9. PR = TR (5) 9.3 = (3) [] Please tur over
(EC/SEPTEMBER 05) MATHEMATICS P QUESTION 0 0. Complete the statemet of the followig theorem: If two triagles are equiagular the their correspodig sides are ad the two triagles are similar. () 0. I the figure below, AB is a taget to the circle with the cetre O. AC = AO ad BA CE. DC produced cuts taget BA at B. E 3 A 4 x F 3 4 O D C 3 B 0.. If A 4 = x, determie with reasos three other agles equal to x. (3) 0.. Prove that ACF ADC. (3) 0..3 Prove that (4) [] Please tur over
MATHEMATICS P (EC/SEPTEMBER 05) QUESTION. Make use of the diagram i the SPECIAL ANSWER BOOK, to prove the theorem which states that if DE BC the,. A D E B C (6). I the diagram below, DE BC, AN DE ad BC. A D M E B Write dow the values of: N C.. ()....3 (4) (3) [5] TOTAL: 50 Please tur over
(EC/SEPTEMBER 05) MATHEMATICS P 3 INFORMATION SHEET: MATHEMATICS b b 4 ac x a A P( i) A P( i) A P( i) A P( i) i i ( ) i T ar ar S r T a ( ) d S a ( d ; r x i F x[ ( i) ] P i i f f ( x h) f ( x) '( x) lim h 0 h ( ) ( ) x x y y d x x y y M ; y mx c y y m x ) x a y b r I ABC: si cos si a A ( x S ) a ; r r y y m m ta x x b c a b c bc. cos A area ABC ab. si C si B si C 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) Please tur over