NATIONAL SENIOR CERTIFICATE GRADE/GRAAD 12 MATHEMATICS PAPER 2/ WISKUNDE VRAESTEL 2 MEMORANDUM SEPTEMBER 2018

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1 NATIONAL SENIOR CERTIFICATE GRADE/GRAAD MATHEMATICS PAPER / WISKUNDE VRAESTEL MEMORANDUM SEPTEMBER 08 MARKS/PUNTE: 50 TIME/TYD: HOURS/URE This memorandum consists of pages. Hierdie memorandum bestaan uit bladsye

2 Cumulative Frequency/Kumulatiewe frekwensie Mathematics P /Wiskunde V LimpopoDoE/September 08 QUESTION/ ,8 50 answer/. Marks Frequency Cumulative Frequency ;;9 ;50 () (). Marks out of 50/ (0;0) upper limits/ all points correct / (). Q 5 (accept to6)/ (aanvaar van tot 6) Q 9 (accept 8 to 0)/ (aanvaar van 8 tot 0) IQR/IKO 9 5 Q 5 Q 9 answer / () [0]

3 Mathematics P /Wiskunde V LimpopoDoE/September 08 QUESTION/., seconds/ 0,87 seconds/.. 7,9 0,07 y ˆ 0,07 7,9 0,,87 accuracy / () equation/ (). r 0, 69 r 0, 69 (). y 0,07 (80 ) 7, 9 y, seconds. substitute/ answer/ () [0] QUESTION/. QR ( 5 ) ( ) substitution/ ()

4 Mathematics P /Wiskunde V LimpopoDoE/September M( ; ) M (;0 ) 0 P( 5;) and M(;0) : m 5 y 0 ( ) y. r 5; centre (;0) :.5 y 5 PM 5 ( 5) ( ) 5 P lies OUTSIDE the circle. -value/ -value / () m subst of m and point/ equation/ r 5 and/ ( ;0) LHS/ RHS/ PM 5 5 conclusion/.6 S (;9) -value/ value/ ().7 m PQ m PQ - 5 tan tan 08, 80 7,57 08, 7, 57 co-interior angles, PR -ais. 7, 57 () () () () [9]

5 Mathematics P /Wiskunde V 5 LimpopoDoE/September 08 QUESTION/.. C(; ) -value/ -value ().. y 8 0 -value/ y B (0;) -value/ ().. r ( 0) ( ) 5 r 5 r 5 r ().. Let D ( ; y) : 0 ( y ) 5 and subst in distance formula/ y ( ) 5 y replace y/ or 6 -value/ -value/ (6) n / a D(;5) y () 5

6 Mathematics P /Wiskunde V 6 LimpopoDoE/September Let C( ; ) and A( ; 0) ( ) ( ) ( ) ( ( )( 5) 0 or 5 C (; ) or C (5;0 ) 0) coordinates of C and A/ equating two r standard form/ r / factors C (; ) C (5;0 ).. r or r 0 r (6) r 0 () []

7 Mathematics P /Wiskunde V 7 LimpopoDoE/September 08 QUESTION/ y k k y k ( k sin 5 k sin 50 sin 5 cos 5 ) k diagram and/ Pythagoras sin 5 k () double angle epansion/ 5. k k substitution () tan( 07 ) tan sin ( 60 ) sin( 90) cos 5. tan 7 sin tan 7 cos.cos sin cos cos cos cos sin cos sina cosa RHS: cosa sina sinacosa. cosa sina.cosa cos A cos A tan 7 tan 7 sin cos cos sin cos cos answer / (7) replace/ tan A epansion of sin A / sin A cos A cos A cos A LHS: cosa cos A cos A LHS RHS replacing cos A numerator/ cos A cos A cos A (5) in

8 Mathematics P /Wiskunde V 8 LimpopoDoE/September sin cos sin cos cos cos tan tan 5, k80 76,7 k.90, k Z dividing by cos / cos tan 5, k80 76,7 k. 90 k Z (5) [] QUESTION/ 6 6. Graph done/ : intercepts/ turning points/ shape/ () 6.. ( 0 ;0) (0 ; 70 ] ( 0 ;0) ( 0 ; 70 ] notation / () 6.. [ 80 ; 60] [ 0;60 ] [50 ;70 ] [ 80 ; 60 ] 6. y sin( 0 0 ) sin( 90) cos [ 0 ;60 ] ] [ 50 ;70 ] () y sin( 0 0 ) cos answer/ () []

9 Mathematics P /Wiskunde V 9 LimpopoDoE/September 08 QUESTION/ 7 7. h tan BE h BE tan 7. ED tan h ED h tan 7. BD BE ED BE. DE.cos BEˆ D h tan h tan h tan h tan cos 0 h h tan h cos 60 tan h h tan h tan h h tan tan h ratio/ BE h tan tan correct ratio / Correct use of cos rule/ substitution/ simplification/ () () h h tan h tan tan h ( tan tan ) tan h tan tan tan simplify / h taken out as common factor/ h (5) [9]

10 Mathematics P /Wiskunde V 0 LimpopoDoE/September 08 QUESTION/ 8 8. Qˆ (tan-chord theorem/ ) R () 8. Pˆ Qˆ ( s opp equal sides/ ) 8. Mˆ 6 ( at centre is at circumference/ 8. Mˆ 6 ( on a str line / ) Rˆ 58 ( at centre is at circumference/ R () R () R R () [0]

11 Mathematics P /Wiskunde V LimpopoDoE/September 08 QUESTION/ 9 9. Ĵ 90 (line from centre to midpoint/ R (l ) 9. N 90 (tangent to radius/ Ĵ Nˆ ONPJ is a cyclic quad /koordevierhoek (CONVERSE of et. of cyclic quad/ Ô Ô Ĉ ( at centre at circumference 80 ( s on a str line/ ) Pˆ 80 (et. of a cyclic quad/ ) ) R R (6) R R R (6) []

12 Mathematics P /Wiskunde V LimpopoDoE/September 08 QUESTION/ 0 0. h k 0. Construction: Join PQ and PF and draw altitudes h and k/ construction/ areadpq areapeq DP h PE h DP PE DQ k areadpq DQ areapqf QF QF k But the area PQE area PQF (same base and same height/ ) areadpq areadpq areapeq areapqf DP PE DQ QF R (6)

13 Mathematics P /Wiskunde V LimpopoDoE/September Ĉ Bˆ (tan-chord theorem ) Ĉ Dˆ (alternate/ s; BD CE ) Bˆ Dˆ BC DC (sides opposite equal s/ e) 0.. In BAF and/ en DCF: Bˆ Bˆ Dˆ (et. of a cyclic quad/ ) Â Â Ĉ Ĉ (et. of a cyclic quad/ ) Fˆ is common/ BAF DCF ( ) R R () R R () 0.. BA AF (similar/ s) DC CF BA DC AF CF BA BC (BC DC) AF CF BC DE But (BD CE; prop theorem/ CF EF BA AF DE EF Replacing/ R DC (5)

14 Mathematics P /Wiskunde V LimpopoDoE/September In ECD and EAC: Ê s common/ 0..5 Ĉ Â (tan-chord theorem ) Dˆ Ĉ ( s of a / e van n ) Ĉ ECD EAC ( ) EC ED ( s) EA EC EC EA.ED BA.EF But ED AF (from/ 0..) R R () EC EA.BA.EF AF () [6] TOTAL/ : 50