MATHEMATICS: PAPER II MARKING GUIDELINES

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NATIONAL SENI CERTIFICATE EXAMINATION SUPPLEMENTARY EXAMINATION MARCH 07 MATHEMATICS: PAPER II MARKING GUIDELINES Time: hours 50 marks These marking guidelines are prepared for use by examiners and

1 NATIONAL SENI CERTIFICATE EXAMINATION SUPPLEMENTARY EXAMINATION MARCH 07 MATHEMATICS: PAPER II MARKING GUIDELINES Time: 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 07

2 NATIONAL SENI CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES SUPPLEMENTARY Page of SECTION A QUESTION AB (6 0) + ( 4 8) AB ,4 () (b) M ( ; ) () (c) 6 mam mam mab 4 0 mmc 4 () not 90 () [7] QUESTION m yq radius of Q 8 (5, 0) lies on OQ (5 5) + (0 m) 64 m 8 () (b) PQ units PQ ( 8) + ( 5) units () (c) The coordinates of A (0; y) (d) (x ) + (y ) 5 (0 ) + (y ) 5 (y ) 6 y ± 4 y 7 A(0; 7) Alternate x 0 7 A(0; 7) 8 5 mpq 5 m 7 4 AP 0 mtan 4 AP is not parallel to PQ since mtan mpq (5) [] A y A y A y + AP x P + 5 P IEB Copyright 07

3 NATIONAL SENI CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES SUPPLEMENTARY Page of QUESTION x + p x cos4 p p o 4 p p () (b) sinθ sin θ sinθcosθ sin θ( sin θ) sinθcosθ sinθcos θ sinθcosθ cosθ (5) (c) () θ θ + θ + sin θ ( sin θ) + sinθ + 0 sin cos sin 0 sin θ sinθ 0 + () () sin θ(sinθ + ) 0 sinθ Ref: 0 o θ 0 + k 60 o θ 0 + k 60 sinθ 0 θ 0 + k 80 or θ 0 + k 60 θ 80 + k 60 (7) (d) sin( x + x) sin x cos x + sin x cos x sin x cos x cos x + sin x ( sin x) sin x( sin x) + sin x sin x sin x sin x + sin x sin x sin x 4sin x (6) (e) () a 5 and b () () q < 5 or q > 5 () [6] IEB Copyright 07

4 NATIONAL SENI CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES SUPPLEMENTARY Page 4 of QUESTION 4 () O 04 (Angle at centre is twice the angle at circumference) Ô 56 (Angles around a point) () O r r P 5 R ˆR ( ) cos 5 r 5 r cos r 5, () 0 00 r + r ( r)( r)cos56 r ( cos56 ) 00 r ( cos56 ) r OP 5, units (b) () Ê OP 0 sin sin56 OP 5, () ˆB (tan chord theorem) Ê (corresponding angles CD//AE) ˆD ˆD ˆB () Ê ˆ A (tan chord theorem) C ˆ ˆ E (Alternate angles AE//CD) for one of these Therefore D ABE///ΔEDC (A,A,A) statements NOTE: B ˆ Dˆ (proved) (can be used as one of statements) () IEB Copyright 07

5 NATIONAL SENI CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES SUPPLEMENTARY Page 5 of () but AE BE ; D ABE///ΔEDC EC DC AE DC BE EC therefore EC BE EC (given) AE DC [7] QUESTION 5 (b) 0 60 Median age 5 years (refer to graph) () Minimum age is ± 4 (refer to graph) (No marks for 45) () (c) There are 00 people younger than Percentage 00 0, 5% of the people who attended the concert were younger than 0. () (d) () start shape end Number of people Age () () Lower quartile has decreased. () [0] 7 marks IEB Copyright 07

6 NATIONAL SENI CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES SUPPLEMENTARY Page 6 of SECTION B QUESTION 6 r 0,88 ; it is a strong, negative relationship. It is very likely that the following is true: Students with higher incomes buy fewer loaves of bread. () (b) b 0, 00 () (c) y 0, 00x +, 74 y 0,00 (7 500) +,74 y 4, 4 5 or 4 (accept either) This is a good approximation (interpolation) [9] QUESTION 7 Construction Draw a line through O and R Oˆ + Oˆ Rˆ + Rˆ RTP: ( ) Proof: but Rˆ R ˆ +P (ext angle of Δ) Ô Similarly Oˆ Oˆ P (isos Δ) Rˆ Rˆ Rˆ + Rˆ (6) Ô Ô + ( ) (b) () Â 45 (Angle at centre two times angle at circumference) ˆB 4 (Given) Therefore EB is not parallel to AC as alternate angles are not equal. () IEB Copyright 07

7 NATIONAL SENI CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES SUPPLEMENTARY Page 7 of () Ĉ Bˆ (OB OC) Ĉ 45 (angles of Δ; ) Ĉ Aˆ (alternate angles OC//AD) FBA ˆ Cˆ + Cˆ (tan chord) 57 FBE ˆ FBA ˆ Bˆ Alternate: BAD ˆ Aˆ + Aˆ ˆ FBO 90 FB//OC (radius tangent) (alternate angles) FB//AD ˆ FBA BAD ˆ (alternate angles) FBE ˆ 57 Bˆ (8) [7] IEB Copyright 07

8 NATIONAL SENI CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES SUPPLEMENTARY Page 8 of QUESTION 8 Construction: OA OA OB (radii) ˆB 60 (Equilateral triangle) BAE ˆ 00 (Opp angles of cyclic quad) ˆD 0 (Angles in a ) (6) (b) Â 00 (Opposite angles of cyclic quad) BOE ˆ 00 (Angle at centre) Ô 60 Ô 0 (angles on straight line) EO ED (isos ) BAF ˆ 90 ( s in semi-circle) FAE ˆ 0 as BAE ˆ 00 (Opposite angles of cyclic quad) Ô 0 centre) EO ED (isos ) OAE ˆ Ê 40 (OA OE radii) Ô E ˆ ˆ D (Ext angle of ) 0 EO ED (isos ) [0] IEB Copyright 07

9 NATIONAL SENI CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES SUPPLEMENTARY Page 9 of QUESTION 9 Any two of the following: Ĝ is a common angle ˆB C ˆ corresponding angles equal AB//CD Aˆ Dˆ corresponding angles equal AB//CD () (b) GC GB GC GB 5 8 CD 5 8 CD ( D GCD /// GBA) AB CD 6 0 units (c) Ê D ˆ (alt angles CD//EF) Fˆ C ˆ (alt angles CD//EF) CD 0 units EF ΔGCD ΔGFE (A.S.A.) EG GD DC//FE (given) and DE FE 0 CDFE is a parallelogram Its diagonals bisect each other EG GD [0] QUESTION 0 tancdb ˆ 0, 5 CDB ˆ 9, 9 Angle of inclination for line EC is: 80 (79, 9 9, 9) 0 Equation of line EC y (tan0 ) x + y ( tan 60 ) x+ Therefore x coordinate of B is when y 0 0 (tan0 ) x + x x tan0 tan60 6, 9 units 6,9 units (6) [6] IEB Copyright 07

10 NATIONAL SENI CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES SUPPLEMENTARY Page 0 of QUESTION BC o sin(80 θ ) BC BC k sinθ k sinθ sinθ k sinθcosθ sinθ k θ BC cos Alternate BE cosθ k BE k cosθ but BC BE BC k cosθ (b) Height of triangle BDC h sinθ k h k sinθ Height of point A from the ground radius of large semi-circle Height of A BC Height of A k cosθ Height of A k cosθ AD DE k DE AD AD DE + AE (Pythagoras) sinθ k sinθ ( k sin θ ) k k + ( k cos θ ) Alternate Point D is equidistant from point B and point C and is the slant height of a cone. Therefore AD k. Marks: slant height of cone. AD k (6) [0] IEB Copyright 07

11 NATIONAL SENI CERTIFICATE: MATHEMATICS: PAPER II MARKING GUIDELINES SUPPLEMENTARY Page of QUESTION (b) (c) BO mbe OE PA BE (rad tangent) map Equation AP: y ( 5 ) ( x 0) y P x + x P(; 5) () MAIN Equation of BE: y x + c 5 (0) + c c B(0; 5 + 9) 5 ( 5 + 9) m BP tanθ tan BEO ˆ θ, BEO ˆ 6, 4 ABP ˆ 6, 4 57, 7 5,7 (8) [5] 77 marks Total: 50 marks IEB Copyright 07

Department of Mathematics

Department of Mathematics TIME: 3 Hours Setter: DAS DATE: 07 August 2017 GRADE 12 PRELIM EXAMINATION MATHEMATICS: PAPER II Total marks: 150 Moderator: GP Name of student: PLEASE READ THE FOLLOWING INSTRUCTIONS