MATHEMATICS: PAPER II Page 1 of 11 HILTON COLLEGE TRIAL EXAMINATION AUGUST 2013 MATHEMATICS: PAPER II GENERAL INSTRUCTIONS

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1 MATHEMATICS: PAPER II Page 1 of 11 HILTON COLLEGE TRIAL EXAMINATION AUGUST 013 Time: 3 hours MATHEMATICS: PAPER II GENERAL INSTRUCTIONS 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. This questio paper cosists of 11 pages. You are provided with a separate Aswer Sheet (yellow) ad a Formula Sheet (gree). Please check that your paper is complete.. Read the questios carefully. 3. This questio paper cosists of 13 questios. Aswer all questios. 4. Questio 4 (b) ad 4 (c) ad must be aswered o the Aswer Sheet which must be haded i with your aswer book. 5. Number your aswers eactly as the questios are umbered. 6. You may use a approved o-programmable ad o-graphical calculator, uless a specific questio prohibits the use of a calculator. 7. Roud off your aswers to oe decimal digit where ecessary, uless otherwise stated. 8. All ecessary workig details must be show. 9. It is i your ow iterest to write legibly ad to preset your work eatly. 10. Please ote that the diagrams are NOT ecessarily draw to scale. Please do ot tur over this page util you are asked to do so

2 MATHEMATICS: PAPER II Page of 11 QUESTION 1 SECTION A I the figure below A, B ad C are poits o the Cartesia plae. B(5;4) M A(-1;1) Ɵ C(8;-) (a) Determie the gradiet of the lie BC. () (b) Determie Ɵ, the agle of icliatio of lie BC. () (c) Determie M the midpoit of the lie joiig A ad B. () (d) Determie the legth of AB. () (e) Determie the equatio of the lie parallel to BC ad passig through A (3) (f) Show that ABC is a right-agled triagle. (3) (g) Determie the area of ABC. (3) 17 marks

MATHEMATICS: PAPER II Page 3 of 11 QUESTION (a) Below is a picture of a Agry Bird as it moves from Poit A to poit A to poit A y A A A For this questio (a)(i) to (iii) traslatios: ( ; y) ( a ; y b) +

3 MATHEMATICS: PAPER II Page 3 of 11 QUESTION (a) Below is a picture of a Agry Bird as it moves from Poit A to poit A to poit A y A A A For this questio (a)(i) to (iii) traslatios: ( ; y) ( a ; y b) + + may ot be used. (i) Give a descriptio i words of the trasformatio which maps A to A () (ii) Give a rule for the trasformatio which maps A to A i the form: ; y ; () ( ) ( ) (b) (iii) Give a rule for the trasformatio which maps A to A () y A A (i) Describe i words the trasformatio which maps shape A to shape A () (ii) If the area of Shape A is 147uits determie the area of shape A (3) 11 marks

MATHEMATICS: PAPER II Page 4 of 11 QUESTION 3 (a) Give: 8 si α ad 90 < α < 70 17 0 0 With the aid of a sketch ad without the use of a calculator, determie: (i) taα (3) (ii) si(90 0 α) () (iii) cosα

4 MATHEMATICS: PAPER II Page 4 of 11 QUESTION 3 (a) Give: 8 si α ad 90 < α < With the aid of a sketch ad without the use of a calculator, determie: (i) taα (3) (ii) si(90 0 α) () (iii) cosα (3) (b) If si40 0 p, determie the followig i terms of p (without the use of a calculator): (i) si140 0 (1) (ii) cos(-40 0 ) (3) (iii) si80 0 () (c) Simplify the followig (without the use of a calculator): si150.ta 5 ( ) 0 0 si 30.si (4) 3 0 (d) Determie the geeral solutio to: ta( 10 ) + 1,077 0 θ (4) (e) Below is the pyramid of The pyramid of kig Khafra, at Giza. It is made out of a square base ad 4 equilateral triagles. 30m 30m 30m (i) Show that the height of the pyramid is 16.6m (4) 1 (ii) Determie the volume of the pyramid if: ( ) Volume Area of base Height () 3 8 marks

5 MATHEMATICS: PAPER II Page 5 of 11 QUESTION 4 79 teeagers have cellular phoes purchased o pay-as-you-go packages. The table below reflects the amout of moey, i rads, spet o airtime by these teeagers i a certai moth. AMOUNT OF MONEY SPENT ON AIRTIME (IN RANDS) NUMBER OF TEENAGERS 0 < < < < < < 10 5 (a) Determie the estimated mea for the AMOUNT OF MONEY SPENT (3) (b) Complete the table, o the ANSWER SHEET. (3) (c) Draw a ogive curve of the cumulative frequecy vs. amout of moey spet o the ANSWER SHEET. (4) (d) From your graph determie the value of the Iter Quartile Rage. (3) QUESTION 5 13 marks The graph below shows the dotplots for two differet sets of 35 umbers. The sets of umbers have differet meas ad differet stadard deviatios. Data set A: Data set B: (a) Which of the data sets (A or B) has the smallest variace? () (b) Which of the data sets (A or B) has the smallest mea? () (c) Commet o the skew-ess of date set A () 6 marks SECTION A TOTAL 75marks

MATHEMATICS: PAPER II Page 6 of 11 QUESTION 6 SECTION B (a) The mea price of 1 differet pieces of jewelry is R4 500. After aother piece of jewelry was added the mea price icreased to R5 740.

R1 000 R6 000 R47 000 R3 000 R1 000 R14 000 R9 000 R5 000 R1 000 R17 000 (i) Determie the mea profit of the ivestor. (1) (ii) Determie the stadard deviatio of the ivestor.

6 MATHEMATICS: PAPER II Page 6 of 11 QUESTION 6 SECTION B (a) The mea price of 1 differet pieces of jewelry is R After aother piece of jewelry was added the mea price icreased to R Determie the value of the added piece of jewelry? (3) (b) A ivestor has the followig record of profits ad losses o 10 differet ivestmets. R1 000 R6 000 R R3 000 R1 000 R R9 000 R5 000 R1 000 R (i) Determie the mea profit of the ivestor. (1) (ii) Determie the stadard deviatio of the ivestor. (3) (iii) Commet if you thik the ivestor is successful or ot. () QUESTION 7 9 marks (a) Determie the order of rotatioal symmetry for the uio jack flag above. () y A(5;9) (b) Determie the agle of rotatioal symmetry for the shape above. () (c) Determie the co-ordiate of A the image of A after a rotatio of 30 o clock-wise. (Calculators may ot be used.) (5) 9 marks

MATHEMATICS: PAPER II Page 7 of 11 QUESTION 8 (a) Prove the followig idetity: 1+ si cos + si cos cos si (5) (b) Hece, or otherwise, calculate the followig without the use of a calculator.

7 MATHEMATICS: PAPER II Page 7 of 11 QUESTION 8 (a) Prove the followig idetity: 1+ si cos + si cos cos si (5) (b) Hece, or otherwise, calculate the followig without the use of a calculator. cos15 + si15 cos15 si15 (3) (c) Solve the followig for the iterval ϵ ( 180 ; 360) cos(θ 0) cos(40 θ) (5) QUESTION 9 13 marks Petroas Twi Towers are the fifth highest buildig i the world, reachig a height of 45m. Two persos stadig at A ad B look up to the top (T) ad to the coectig bridge (M) respectively. Agle of elevatio from A to T is 5 o T Agle of elevatio from B to M is 9 o Agle CAB 50 o Agle ACB 100 o M 45m A C 100 o 5 o 9 o 50 o (a) Show that AC 969,3m. () (b) Determie the height above the groud of the coectig bridge at poit M. (Roud off your aswer to the earest meter. ) (6) B 8 marks

8 MATHEMATICS: PAPER II Page 8 of 11 QUESTION 10 (a) Determie the equatio of the followig graphs: (i) () (ii) () (b) Give below are the graphs f ( ) 3 si & g( ) cos for the domai ϵ[-90;360] (i) For which values will: f () > g () () (ii) For which values will: f (). g () > 0 (3) 9 marks

MATHEMATICS: PAPER II Page 9 of 11 QUESTION 11 A proposed way to cut budget is

To prevet them from waderig o the roads they will be tied to poles resultig i

The pla is to put poles at A, B ad C to cover the sports field as i the

A is the poit (0 ; 30) C C is the poit (80 ; 55) ad circle ceter C has a

circle ceter A touches the -ais determie the equatio of the circle.

9 MATHEMATICS: PAPER II Page 9 of 11 QUESTION 11 A proposed way to cut budget is to get sheep to eat the grass o the sports fields. To prevet them from waderig o the roads they will be tied to poles resultig i them beig able to eat a circle of grass. The pla is to put poles at A, B ad C to cover the sports field as i the diagram below. A is the poit (0 ; 30) C C is the poit (80 ; 55) ad circle ceter C has a radius of 34m A B The equatio of circle ceter B is: ( ) 70 + y 0y b (a) If circle ceter A touches the -ais determie the equatio of the circle. () (b) If circle ceter B goes through poit C. Determie the legth of the rope tied to the sheep at B (5) (c) Prove that there is o area o the field where all 3 sheep ca eat. (5) (d) Ca you list 1 problem with the proposal () 14 marks

MATHEMATICS: PAPER II Page 10 of 11 QUESTION 1 Daiel Craig starig as James Bod eeds to ru betwee two cameras for a scee i the et James Bod movie. The two cameras are positioed at A ad B.

10 MATHEMATICS: PAPER II Page 10 of 11 QUESTION 1 Daiel Craig starig as James Bod eeds to ru betwee two cameras for a scee i the et James Bod movie. The two cameras are positioed at A ad B. Ad Daiel eeds to ru o a straight lie i such a way that he is always the same distace from each camera. If A(10 ; 0) ad B(5 ; -5) are the locatios of the cameras, the determie the equatio of the lie Daiel Craig eeds to ru. 6 marks

MATHEMATICS: PAPER II Page 11 of 11 QUESTION 13 Mr Lombard s so asked him to cut up his toast ito 5 parts with equal area. So Mr Lombard decided to cut it as i the picture below.

11 MATHEMATICS: PAPER II Page 11 of 11 QUESTION 13 Mr Lombard s so asked him to cut up his toast ito 5 parts with equal area. So Mr Lombard decided to cut it as i the picture below. C A E B D Note: A is a reflectio of B Ad C a reflectio of D Mr Lombard eeded to fid out where to cut the bread to make the area the same. So he assumed the bread to be perfectly rectagular ad put it o a Cartesia plae y 4 6 Mr Lombard worked out that he eeded to cut: From poit (0 ; 0) to poit (0 ; 8) From poit (0 ; 0) to poit (4,8 ; 8) Work out the other 3 cuts that Mr. Lombard eeds to make i order to grat his so s request. 7 marks SECTION B TOTAL 75marks GRAND TOTAL 150marks

12 MATHEMATICS: PAPER II Page 1 of 11 MATHEMATICS INFORMATION SHEET b ± b a 4ac i 1 1 i T a ( 1) d ( + 1) i 1 + S [ a + ( 1) d ] T ar 1 a( r 1) S ; r 1 r 1 a S ; 1 r 1 1 r < < f ( ) lim h 0 f ( + h) f ( ) h P( 1 i) A P( 1 i) A + P ( 1 i) A P ( 1 i) A + F ( 1 + i) 1 1 ( 1 + i) i P i 1 + y1 + y d ( 1 ) + ( y y1 ) M ; y m + c y y m( ) y 1 m m ta θ y ( a) + ( y b) r

13 MATHEMATICS: PAPER II Page 13 of 11 I ABC : a si A b si B c si C a b + c b c. cos A area ABC 1 a b. si C si ( α + β) si α.cosβ + cos α. si β si ( α β) si α.cosβ cosα. si β cos ( α + β) cosα.cosβ si α. si β cos ( α β) cos α.cosβ + si α. si β cos α cos α si 1 si α cos α 1 α si α si α. cosα ( ; y) ( cosθ + ysi θ ; ycosθ si θ) ( ; y) ( cosθ ysi θ ; ycosθ + siθ) f σ ( ) i 1 i ( A) P ( A) P ( A or B) P ( A) + P ( B) P ( A ad B) ( S) ŷ a + b b ( ) ( y y) ( )

14 MATHEMATICS: PAPER II Page 14 of 11 STUDENT NUMBER ANSWER SHEET FOR Q4b ad Q4c AIRTIME Freq (f) Cumulative frequecy Poits 0 < 0 19 (0;1) 0 < < < < < 10 5 f

MATHEMATICS: PAPER II Page 1 of 11 HILTON COLLEGE TRIAL EXAMINATION AUGUST 2013 MATHEMATICS: PAPER II GENERAL INSTRUCTIONS

MATHEMATICS: PAPER II Page 1 of 11 HILTON COLLEGE TRIAL EXAMINATION AUGUST 01 Time: hours MATHEMATICS: PAPER II GENERAL INSTRUCTIONS 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. This