Mathematics: Paper 1

Transcription

1 GRADE 1 EXAMINATION JULY 013 Mathematics: Paper 1 EXAMINER: Combied Paper MODERATORS: JE; RN; SS; AVDB TIME: 3 Hours TOTAL: 150 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This questio paper cosists of 13 PAGES, 15 questios, two diagram pages ad a Iformatio Sheet.. Read the questios carefully. 3. Aswer all the questios. 4. Number your aswers eactly as the questios are umbered. 5. You may use a approved, o-programmable, ad o-graphical calculator, uless otherwise stated. 6. Make sure your calculator is i degree mode. 7. Roud off your aswers to TWO decimal places where ecessary, uless stated otherwise. 8. All the ecessary workig details must be clearly show. 9. It is i your ow iterest to write legibly ad to preset your work eatly. NAME Mark Allocatio (for educator s use oly) Q1 Q Q3 Q4 Q5 Q6 Sectio A Marks Eared Total Q7 Q8 Q9 Q10 Q11 Q1 Q13 Q14 Q15 Sectio B Marks Eared Total TOTAL % 150

2 Page of 13 SECTION A [68 MARKS] QUESTION Solve for (roudig off to dp if ecessary): (4) (5) (4) Solve for if by completig the square. (6) 1. Give that ad y y 3, fid the values of ad y. (6) [5] QUESTION.1 Fid the derivative from first priciples of f() = 3 1 (5). Use the rules of differetiatio to determie g (), leavig all aswers with positive epoets, if:..1 g ( ) (3) g ( ) (4) [1]

3 Page 3 of 13 QUESTION 3 Give 3.1 Write dow the equatios of the asymptotes of f. () 3. Write dow the equatios of the aes of symmetry of f. () [4] QUESTION 4 The diagram below shows the graphs of itersectio of f ad g f ( ) a ad g ( ) b. The poit 1 ; y 1 P is the poit of f g (0 ; 1) P (1; ½) 4.1 Calculate the values of a ad b. () 4. Eplai why the iverse of g is ot a fuctio. (1) 4.3 Write dow two ways i which the domai of g could be restricted i order that 1 g is a fuctio. () 4.4 Draw a, fully labelled, diagram (usig the grid provided o the diagram page) of the iverse of f(). (3) [8]

4 Page 4 of 13 QUESTION 5 Mrs Va buys a va today for R cash. She decides that she will sell it i eactly four years time ad put the moey towards the purchase of aother va of the same model. She calculates that this va will depreciate o a reducig balace at 18% pa compouded aually ad she will use the moey from the sale of this va towards the cost of her ew car. She aticipates that the iflatio rate will be 14% pa compouded aually. 5.1 How much will her va be worth at the ed of four years? (3) 5. What will be the price of the ew va of the same model be i 4 years time? (3) 5.3 She decides to ivest a set amout of moey each year i a sikig fud to cover the balace of the purchase price, so that by the ed of the four years she will have sufficiet fuds to buy her ew va for cash. The sikig fud pays iterest at the rate 1,5% pa, compouded aually. If she begis paymet immediately after the purchase of her first car, how much will she eed to ivest each year i order to have the required amout of moey at the ed of the four years? (4) 5.4 Fid the omial iterest rate p.a. compouded semi-aually for a ivestmet at 9 % p.a. effective. (3) [13] QUESTION 6 Draw a fully labelled sketch of 3 y 5 7 3, (usig the grid provided o the diagram page). You must show all your calculatios. [6] Total: Sectio A: 68 marks Please tur over for Sectio B

5 Page 5 of 13 SECTION B [8 MARKS] QUESTION 7 Give the costraits below, which of the regios A to J i the diagram, represets the followig feasible regio? y y 0 A 3 y 3 y 0 H I 3 B C J G F D E (4 ; -) [3] QUESTION 8 The feasible regio for a liear programmig problem is show i the diagram, where 0 ad y 0. Aother costrait is y y A Write dow the further three costraits which defie the regio. (6) 8. A objective fuctio for this problem is P s ty where s ad t are positive itegers. Write dow the smallest values of s ad t so that P is miimised at A (10 ; 0). (3) [9]

6 QUESTION Determie (4) Page 6 of The secod term of a geometric sequece is, the fifth term is. Fid the term. (5) [9] QUESTION 10 This picture shows a tower of cards i a 3 storey triagular patter Fid the 4 th term, usig the table below: Number of storeys of the tower Number of cards used 7 15 (1) 10. Followig the patter i the picture, what is the most umber of storeys for a tower built from oe 5 card pack? (1) Give that a playig card is 9cm log ad that each triagle is equilateral, determie the height of the triagular tower four storeys high, roud your aswer off to three decimal places. (3) 10.4 How may 5 card packs are eeded to build a tower 1 storeys high? (4) [9]

7 Page 7 of 13 QUESTION Sam is traiig for a fu ru by ruig every week for 6 weeks. She rus 3km i the first week ad each week after that she rus km more tha the previous week, util she reaches 5km i a week. She the cotiues to ru 5km each week How far does Sam ru i the 9 th week? (1) I which week does she first ru 5km? () What is the total distace that Sam rus i 6 weeks? (3) 11. Data regardig the growth of a certai tree has show that the tree grows to a height of 150cm after oe year. The data further reveals that durig the et year, the height icreases by 18cm. I each successive year, the height icreases by 8 9 of the previous year s icrease i height. The table below is a summary of the growth of the tree up to the ed of the fourth year. First Year Secod Year Third Year Fourth Year Tree Height (cm) Growth (cm) Determie the icrease i the height of the tree durig the seveteeth year. () 11.. Calculate the height of the tree after 10 years. (3) Show that the tree will ever reach a height of more tha 31cm. (3) [14]

8 Page 8 of 13 QUESTION 1 For each questio below draw a sketch graph of the curve which satisfies the coditios specified. Where possible, idicate the cuts o the ais ; the equatio of the ais of symmetry ad/or the coordiates of the turig poit. 1.1 f ( ) a ad a<0 b c f ( ) 0 whe h ( ) a b h( 1) 0 h ( 1) 5 h( ) 0 for all 1 (3) (3) 1.3 If is a quadratic fuctio such that 1, 11, (4) Fid 3. [10]

9 Page 9 of 13 QUESTION 13 3 The graph represets the fuctios f ( ) a b ad g( ). The -ais is a taget to f. D is a commo y-itercept of the two graphs Determie the coordiates of poit A. () 13. Show that a = -1 ad b = 3. (4) 13.3 Fid the coordiates of B, the local maimum turig poit of f. (3) 13.4 For which values of is f '( ). g'( ) 0? () [11] QUESTION 14 I the diagram ABC is a equilateral triagle with sides equal to p uits DEFG is a rectagle with BE = FC = uits A D G B E F C 14.1 Show that the area of the rectagle is A 3 ( p ) uits. (4) 14. Determie, i terms of p, the maimum area of the rectagle. (4) [8]

10 Page 10 of 13 QUESTION Fid the value of a, a>0, where f ( ) a 3 ad f( f()) 3a 4 (4) 15. The diagram below shows part of a straight lie obtaied by plottig 1 y agaist 1, together with the coordiates of two of the poits o the lie. Epress y i terms of. (5) 1 y (4 ; 4) ( ; 3) 1 [9] Total: Sectio B: 8 marks Total: 150 marks

11 Page 11 of 13 STUDENT S NAME: DIAGRAM SHEET: QUESTION 4

12 Page 1 of 13 STUDENT S NAME: DIAGRAM SHEET: QUESTION 6

13 Page 13 of 13 INFORMATION SHEET: MATHEMATICS Grade 11 ad 1 FET b b 4 ac a A P( 1 i) A P( 1 i) A P( 1 i) A P( 1 i) i1 T 1 ar 1 S i1 ( 1) i a r 1 r 1 ; r 1 T a( 1) d S { a( 1) d} a S 1 r ; 1 r 1 F 1 i 1 i [1 (1 i) ] P i f f ( h) f ( ) '( ) lim h 0 h d ( ) ( ) 1 y y1 M 1 y1 y ; y m c y y m ) a y b r 1 ( 1 y y1 m m ta 1 I ABC: si a A b c a b c 1 bc. cos A area ABC ab. si C si B si C si cos si.cos cos. si si si.cos cos. si cos.cos si. si cos cos.cos si. si cos si cos 1 si cos 1 si si. cos ( ; y) ( cos ysi ; ycos si ) ( ; y) ( cos ysi ; ycos si ) PLEASE TURN OVER