E(3;2) (4 3) (5 2) r r. 10 ( x 4) ( y 5) 10. y D A(4;5) C(10;3) B(2;-1) SECTION A QUESTION 1 In the diagram below:

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1 SECTION A QUESTION In the diagram below: DC CB A is the centre of the circle. E is the midpoint of AB. The equation of line BA is: y 7 DF is a tangent to the circle at F. y D F A(4;5) E B(;-) C(0;) (a) Find the co-ordinates of E. () 4 5 E ; E(;) (b) Determine the equation of the circle, centre A, passing through point E. Give the equation on the form ( a) ( y b) r () r r r (4 ) (5 ) 9 0 ( 4) ( y 5) 0 Page of

2 (c) Find the gradient of line BC. () m BC 0 (d) Hence, or otherwise, determine the equation of line DC. () m DC y c sub (0;) (0) c c y (e) Show, by calculation, that the co-ordinates of D are (6 ;). () y (6) 7 D(6;) (f) Find the size of ˆ EBC. (5) m EB tan 7,6 mbc tan 6,6 EBC ˆ 7,6 6,6 45 Page of

3 (g) Find the length of DF. (6) DFA ˆ 90 (tan chord) FA 0 AD AD DF AD FA pythag DF (6 4) ( 5) DF 0 ( ) [4] Page of

4 QUESTION (a) Simplify: (sin cos ) sin (4) sin sin.cos cos cos sin.cos cos sin cos tan (b) Solve for in the interval [ 80 ;80 ]: cos( 0 ) sin. (7) cos( 0 ) cos(90 ) 0 (90 ) k k90 or 0 (90 ) 60 0 k80 50 ; 60 ; 0 ;0 [] Page 4 of

5 QUESTION (a) Refer to the figure: A E G D B Below are three statements that refer to the above figure. What additional information would be required, if any, to make each individual statement true? A ˆ E ˆ D ˆ () D must be on the circumference () ˆB ˆ C C GB = GC or G is the centre of the circle () C ˆ ˆ 90 C BE is the diameter () Page 5 of

6 (b) Use the figure below to prove that S ˆ Q ˆ 80. Ô is the centre of the circle. (6) Q P R O S Given: Cyclic quad PQRS, O centre of the circle RTP: Sˆ Qˆ 80 Const.: Join PO and OR Proof: Let Sˆ POR ˆ ( at circum) reflepor ˆ 60 ( ' s around a point) Qˆ 80 ( half at centre) Sˆ Qˆ 80 [9] Page 6 of

7 QUESTION 4 In the figure: ABCD is a cyclic quadrilateral. AB = AF A D B E F C (a) Prove that DE = EF. () Let Fˆ Bˆ ( AB AF) Dˆ ( et of cyclic quad) ED EF (b) If it is further given that ED bisects ˆ CDF, prove that FB bisects Dˆ ( bisec ted) Bˆ ( in same seg) Bˆ Bˆ FB bisects ABC ˆ ˆ ABC. () [6] Page 7 of

8 QUESTION 5 In the figure: O is the centre of the circle. AC BC OD CB C O A D B Prove that: (a) ODB /// ACB () In ODB and ACB ) OBD ˆ 90 Cˆ ( given) ) Bˆ Bˆ ( common) ) DOB ˆ CAB ˆ ( rd of triangle) ODB / / / ACB ( AAA) (b) AD. OA BC (4) OD DB OB (/ / / ' s) AC CB AB AD DB ( OD AB) OB OA ( radii) AD OA CB AD AD OA. BC [7] Page 8 of

9 QUESTION 6 A consumer testing company studied three different brands of washing machines to see how much water was used during each wash. Each washing machine was tested 5 times. The bo and whisker plots below show the results of this study. Washing machine A Washing machine B Washing machine C Number of litres used by washing machine. (a) Which brand of machine (A, B or C) used up most water on average? () C (b) Which brand of machine (A, B or C) is the most predictable? () B (c) The results of Washing Machine C are shown below: () Determine the standard deviation of the litres of water used. () 0,4 () Based on this data, how many litres of water would be used in 67% of the washing loads? () [97,6 0, 4 ; 97,6 0, 4] [87, ; 08] [6] Page 9 of

10 Frequency QUESTION 7 The distance () in kilometres that the staff at a certain school in Durban travel to work each day is summarised in the histogram below: Kilometres (a) Is the data positively or negatively skewed? () Skewed to the right, positively skewed (b) Which of the intervals is the modal interval? () Mode 5-0 (c) Use the histogram to complete the given table. () Intervals Frequency Cumulative Frequency 0 < < < < 0 7 Page 0 of

11 0 < < 0 0 (d) Use your frequency table to draw an ogive below. Label your aes. () (e) Determine the following using your ogive: () the interquartile range () 4,- 5,6 = 8,5 () the percentage of these staff members that stayed between 4km and 4km from the school? () Page of

12 % 0 [] 75 marks Page of

13 SECTION B QUESTION 8 (a) In the figure: ABC is a tangent to the circle at B. D ˆ Dˆ A F B C E D Prove that: () DCBF is a cyclic quadrilateral. (4) Bˆ Dˆ (tan AC chord EB) Dˆ ( Dˆ Dˆ ) DCBF cyclic ( et opp int ) () FC is a tangent to the circle FED. (4) Eˆ Bˆ (tan AC chord BD) Bˆ Fˆ ( in same seg) Fˆ Eˆ FC tan to circle FED (tan chord converse) Page of

14 (b) In the figure: D ABCD is a cyclic quadrilateral. AD = DC = AC = y A y C Bˆ 0 Show that y 0 (5) B Dˆ 50 ( opp of cyclic quad) y. cos50 y y ( ) y Page 4 of

15 (c) In the figure (not drawn to scale): P K OM // PL KN : KO = 8 : LM = KL O Q L Calculate: N Area of KMO () Area of NMO KO. h KO ON. h ON 5 M () () LQ QN PO ML ( PL / / OM ) OK MK p.k 9 PO p 4k 4 LQ PO ( PL / / OQ) QN ON LQ QN 9 p p 0 () [9] Page 5 of

16 QUESTION 9 The sketch below shows the graphs of f() = sin and g() = cos for [ 80 ; 80 ]. The coordinates of A, the point of intersection of the two graphs, are (t; ). A (t; ) g f (a) Determine the value of t. () sin t t 60 (b) Use the sketch to solve the following equation: cos = sin cos for [ 80 ; 80 ]. () cos sin.cos sin 80 ; 60 ;80 (c) If f is translated 0 to the left and units up, what will the new equation of f be? () y sin( 0 ) [7] Page 6 of

17 QUESTION 0 (a) Simplify: sin70.cos0 sin(0 ).cos cos(0 ).sin (4) sin0.cos0 sin(0 ) sin 0 sin 0 (b) Prove the following identity: cos cos sin sin tan cos (cos ) LHS sin sin cos cos cos sin ( cos ) cos ( cos ) sin ( cos ) cos sin tan RHS (5) Page 7 of

18 (c) Determine the general solution for : 7sin cos 5 0 (7) 7sin ( sin ) 5 0 7sin sin 5 0 sin 7sin 0 (sin )(sin ) 0 sin or sin 80 0 k60 0 k60 or 60 0 k60 0 k60 (d) If sin a sin and cos b cos, b Prove that tan. (5) a sin sin cos cos a b sin cos sin cos a b sin sin tan cos b b tan a tan b tan a b tan a a cos b cos a sin [] Page 8 of

19 QUESTION The two circles in the diagram represent two interlocking gears, which touch at point Q (4 ; ). The circles have the following equations: + y 5 = 0 and + y 9y + 50 = 0 A y P Q(4;) O B (a) Show that the co-ordinates of P are (6; 4½). () y y ( 6) y P 6;4 5 (b) Determine the equation of common tangent AB. (4) 4 mob mab 4 4 y c sub(4;) 4.4 c c y Page 9 of

20 (c) If the larger gear makes one full revolution, how many times will the smaller gear turn completely? (4) 5 Csmall Cbig 5 C 5 C 0 twice (d) Find the area of AOB. () 5 5 A0; B ; Area units 4 4 (e) Another tangent to circle O, drawn from A touches the circle at C. Determine the length of CQ. () Q(4;) C( 4;) CQ 8units [6] Page 0 of

21 QUESTION C D 6 cm F E B G 64 A The diagram shows a pyramid shaped cone. Each face is an isosceles triangle with base angles of The base is a square of side 6cm. EG is the slant height of the pyramid. EF is the perpendicular height of the pyramid. Volume of a pyramid = area of base perpendicular height (a) Determine the length of edge AE. () BG GA cm ( EG BA in isos) cos64 AE AE 6,8cm (b) Calculate the height EF. (4) EG tan 64 EG 6,cm EF EF EG FG pythag 6, EF 5,4cm ( ) Page of

22 (c) Determine the volume of the pyramid. () V V V.(6 6). EF 6 5,4 64,8cm (d) The pyramid is to be wrapped in a single layer of gold foil, with no overlaps. Calculate the total area of foil that would be needed. () TSA 4 6 EG 6 6 TSA 4 6 6, 6 6 TSA 0,4cm [] 75 marks Total: 50 marks Page of

MATHEMATICS: PAPER II MARKING GUIDELINES

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