NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 05 MATHEMATICS: PAPER II MARKING GUIDELINES Time: 3 hours 50 marks These marking guidelines are prepared for use by examiners and sub-examiners, all of whom are required to attend a standardisation meeting to ensure that the guidelines are consistently interpreted and applied in the marking of candidates' scripts. The IEB will not enter into any discussions or correspondence about any marking guidelines. It is acknowledged that there may be different views about some matters of emphasis or detail in the guidelines. It is also recognised that, without the benefit of attendance at a standardisation meeting, there may be different interpretations of the application of the marking guidelines. IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page of 6 SECTION A QUESTION (a) M (6; 3) () (b) ˆ o CMA 90 ( Opp angles of cyclic quad) 6 mab mmc y x + c sub in M (6; 3) 3 (6) + c c 9 y x 9 (5) (c) () Area of MCB ( )( ) b h x 9 0 x 4½ OB OC BC 7,5 h 3 7,5 3 Area of MCB ( )( ) (4) () area of AMCO 36,5 4,75 units () [3] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 3 of 6 QUESTION (a) 4 x+ x x+ 8 x x 8 y x 8 N (8; 8) (3) (b) B(6; 6) () (c) 7 y 0x 7(6) 0x x D. D (,;6) DB 6, 4,8 units (3) [7] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 4 of 6 QUESTION 3 (a) () sinθsinθ sin θ cos θ (4) () > 0 or < 0 θt 90 ; 70 () ( ) (b) () tanθ sinθ + sinθ.sinθ + sin θ + sin θ + cos θ (3) () 3 sinθ sinθ 3 tanθ 3 Reference angle 7.57 o θ 7.57 + k80 k Z (5) [4] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 5 of 6 QUESTION 4 B ˆ + D ˆ 80 Opp angles of a cyclic quadrilateral Dˆ + Dˆ ˆ + D3 80 Angles on straight line D ˆ C ˆ tan chord theorem hence Cˆ ˆ ˆ + D3 B (3) [3] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 6 of 6 QUESTION 5 (a) O A M C Construction: Join AO and OC RTP: AM MC Proof: In 's OAM, OCM OM is a common side OA OC (Radii) ˆ ˆ o M M 90 ( given) Therefore AOM COM ( R, H, S ) Hence AM MC (6) (b) () ˆ o B 90 ( Angle in semi circle) BE 0 ( Pythag) BE 6 units () () BC CE GO 4 or ; line parallel to one side of Δ/mean proportion OE 0 5 theorem () (3) BD 8 units ( Line from centre chord) BC 4 6 4 BC 4,57 units 3 4 OR 4 units 7 7 CD 8 4,57 CD 3,43 units OR 4 units 7 (4) OR Make BE k; BC k and DE k ( line 7 from centre chord) therefore CD 6 7 4 CD 7 [4] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 7 of 6 QUESTION 6 (a) C A E B OR D () (b) Aˆ Dˆ ( Angles in same seg) C ˆ B ˆ ( Angles in same seg ) DAEC / / / D DEB (AAA) or they could give the third angle (3) AE DE (c) ( DAEC /// DDEB) EC EB AE EB DE EC () [6] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 8 of 6 QUESTION 7 (a) () y 4x ( for line equation; for the gradient and for the y intercept) (3) () r perfect correlation () (b) () (4) Water Water bill bill for in the Rands month for the in rand month T Electricity bill in Rands for the month Electricity bill for the month in rand () () Strong/positive relationship or (as the water bill increases so does the elec bill) () (3) It would increase. () (4) B would increase. () (5) No, as you would be extrapolating. OR: the equation is valid for values of x between 00 and 000 OR: Yes, but the result will be innacurate () [0] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 9 of 6 QUESTION 8 (a) For recognising it is a histogram e.g. no spaces (5) (b) The mark distribution is skewed to the left or negatively skewed () (c) TRUE; Data skewed to left () Or they calculate the quartile values and make decision: 75; 85 and 90 [8] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 0 of 6 SECTION B QUESTION 9 (a) Bˆ + Cˆ Dˆ + Fˆ () (b) Bˆ Dˆ ( Alternate angles AB / / DC) Cˆ Aˆ ( Angle subtended from equal chord of equal circle) Bˆ Dˆ ( Angles in a triangle) But these are alternate angles BC // AD and ABCD is a parallelogram (6) OR: B ˆ ˆ D (alternate angles AB//DC) Cˆ A ˆ (angle subtended from equal chord of equal circle) BD is a common side AB CB (A.S.A) AB CD ABCD is a parallelogram (one pair of opposite sides equal and parallel) OR Bˆ Dˆ + Bˆ + Dˆ 80 Cˆ ( co int angles // lines) 80 Aˆ ( co int // lines) but Aˆ Cˆ ( Angle subtended Bˆ + Bˆ Dˆ + Dˆ parm ( Opp angles are equal) by same chord equal circles) [7] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page of 6 QUESTION 0 (a) C ˆ Bˆ 55 ( tan chord theorem) ˆD 8 ( tan chord theorem) Dˆ + Dˆ 55 ( alt angles CD / / tangent) OR (angles in same segment) ˆD 37 ( ) ( ) Ê 80 55 + 37 's in a 88 (6) (b) () PDO ˆ 90 (line from centre tangent) PEO ˆ 90 (line from centre tangent) P ˆ Aˆ ( at centre at circumference) O ˆ 80 Aˆ (Opp 's of quad) (8) () O ˆ ˆ 80 + A ( around a point) K ˆ ˆ 90 + A ( at centre at circumference) Kˆ ˆ C3+ Ê (ext of sum of interior opposite angles) Cˆ + Ê 90 + Aˆ 3 Alternate: Construct line DE. C ˆ 3 + Eˆ K (ext of sum of interior opposite angles) K ˆ ˆ ˆ 80 (CDE + KED) ( 's in ) but E ˆ ˆ CDE (tan chord theorem) K ˆ ˆ ˆ 80 (E + KED) ˆ but ˆ ˆ 80 P E KED (PD PE, tangents from common point) ˆ ˆ ˆP C 3 + E 80 90 90 + Aˆ (6) [0] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page of 6 QUESTION (a) sin x+ cos x+ sin xcos x cos x 0 cos x( sin x ) 0 cos x 0 or sin x x ± 90 + k360 or x 30 + k360 or 50 + k360 A (50 ; 3 + ) sin x cos x OR sin x+ cos x+ sin x cos x sin x sin(90 x) sin x cos x sin x sin(90 + x) x 30 + k.0 x 90 + k.360 x 90 x Note: Calculator Answers: Equ Equ plus answer 7/7 Answer Only /7 Equ Equ x 30 /7 A (50 ; 3 + ) (7) (b) B P 8 units 4 units A o 68 Q o 50 C BC 8 sin 68 sin 50 8sin 68 BC sin 50 3 PC BC 5 sin PQC ˆ sin 50 PC 4 ˆ 8sin 68 3 sin PQC sin 50 sin 50 5 4 ˆ 54sin 68 sin PQC 70 PQC ˆ 45, 66 (6) IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 3 of 6 (c) cos A cos 45 + sin Asin 45 cos A cos 45 sin Asin 45 (cosa + ( cos A sin A ) + + cos A sin A cos A sin A cos A sin A (cosa+ (cosa+ ( cos A sin A) (cosa+ (cosa (cosa+ (cosa OR cos A cos 45 + sin Asin 45 cos A cos 45 sin Asin 45 (cosa + cos A sin A ( ) (cosa+ (cosa (cosa+ (cosa+ (cosa (cosa+ cos A+ sinacosa+ sin A cos A sin A + sin A cos A (6) [9] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 4 of 6 QUESTION (a) x + 0x + 5 + y 6y + 9 30 + 5 + 9 ( x+ 5) + ( y 3) 64 Radius is 8 units (4) (b) PQ ( 7 ( 5) ) + ( 3) PQ 69 PQ 3 (3) (c) P(7; ) Q( 5; 3) 3 M PQ 7 5 ( ) 5 5 y x + c sub in (7; ) 5 ( 7 ) + c c 5x y + y 5x + 5x + y (4) (d) 5k A (x; y) p (7; -) k 5k + 44k 49 k 49 69 k 7 3 7 7 A 7 ; + 5 3 3 6 A 3 IEB Copyright 05 Alternate 7 units Main (x 7) + (y + ) 49 5x + y 69y + 676y 549 0 y 9 3 OR 6 3 9 5x + 3 x 7 3 A 7 ; 9 3 3 Solve simultaneously (5)
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 5 of 6 OR ( 7) ( ) 49 x + y+ 5x+ y (Recognising line and circle) Substitution str line int o cirlce 69y + 676y 549 0 Substituting the correct value into the equation to get other coordinate 7 9 A ; 3 3 (e) ( ) ( ) x 7 + y+ 49 x + y + 0x 6y 30 4 + 49 + + 4 + 4 49 + + 0 6 30 x x y y x y x y 0y 4x 34 x 7 OR y 5 5 (3) (f) 5 5 perpendicular OR PCQD is a kite diagonals are perpendicular to each other. () [] IEB Copyright 05
NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES Page 6 of 6 QUESTION 3 BC +,5 ()(,5) cos 30 BC 0,807479764 Area of triangle ()(, 5) sin 30 0,375m Alternate method 80,7 sin 30 00 sin BCA ˆ of finding height BC h 0,375 h 0,98886934 Therefore Volume of the water tank is: BCA ˆ 38.6 Use trig ratio h sin 38,6 50 h 9,89cm π (3) 0,98886934 m eeeeeeeeeeh 3 6,6 eeeeeeeeeeh [8] Total: 50 marks IEB Copyright 05