CORE MATHEMATICS PI Page 1 of 18 HILTON COLLEGE TRIAL EXAMINATION AUGUST 2014 CORE MATHEMATICS PAPER I GENERAL INSTRUCTIONS

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1 CORE MATHEMATICS PI Page of 8 HILTON COLLEGE TRIAL EXAMINATION AUGUST 04 Time: hours CORE MATHEMATICS PAPER I GENERAL INSTRUCTIONS 50 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY.. This questio paper cosists of 0 pages. There is also a separate yellow formula sheet. Please check that your paper is complete.. Read the questios carefully.. This questio paper cosists of questios. Aswer all questios. 4. Number your aswers eactly as the questios are umbered. 5. You may use a approved o-programmable ad o-graphical calculator, uless a specific questio prohibits the use of a calculator. 6. Roud off your aswers to oe decimal digit where ecessary, uless otherwise stated. 7. All ecessary workig details must be show. 8. It is i your ow iterest to write legibly ad to preset your work eatly. 9. 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 CORE MATHEMATICS PI Page of 8 QUESTION SECTION A (a) Simplify () = = () () + + = 8 = () (b) Solve: () () () 8 = 0 by completig the square, give your aswers to d.p. (4) ( ) 4 = = ± 9 = 4 ± 9 = 8.6 or = to d.p. () + = 9 9 = 4 9 = log = () + 0

3 CORE MATHEMATICS PI Page of 8 (c) If f = ad g = the determie: () ( 5) f () () ( ) = f ( 4) f g () = 0 () p if 97 f p = () p = 97 p =

4 CORE MATHEMATICS PI Page 4 of 8 QUESTION (a) Give the et term i each of the followig sequeces: () ; 7 ; ; ; 8 () () 8 ; 64 ; ; 6 ; 8 for sig for magitude () () ; 5 ; 0 ; 7 ; 6 () (b) Cosider the sequece: ; 7 ; ; 5 ;... () Determie a formula for the th term of the sequece () ( ) T = a + d T = + 4 T = 4 eed ot be simplified () Determie, by meas of a formula, which will be the first term () to eceed = = = so, T will be first to eceed (c) Showig workig, determie if ( i ) = 089 = 089. (4) i= S = a + d 089 = ( ) = [ ] = 089 =

CORE MATHEMATICS PI Page 5 of 8 QUESTION (a) Joh has a spier ad a dodecahedral die ( sided)

$Calculate, givig your aswers as simplified fractios i simplest form, the probability that he:$ () P ( orage 8) = = 5 60 () Gets gree o the spier or a prime umber o the die?

() P ( orage 8) = = 5 60 () Gets gree o the spier or a prime umber o the die?

5 CORE MATHEMATICS PI Page 5 of 8 QUESTION (a) Joh has a spier ad a dodecahedral die ( sided) as show: He spis the spier ad tosses the die. Calculate, givig your aswers as simplified fractios i simplest form, the probability that he: () Gets orage o the spier ad 8 o the die? () P ( orage 8) = = 5 60 () Gets gree o the spier or a prime umber o the die? () P Gree prime = P Gree + P prime P Gree prime 5 5 = + = = () Does t get red o the spier ad does t get o the die? () 4 44 P ( ot red ot ) = = 5 60 (b) How may differet umber plates ca I make for Gauteg Provice (GP) if they have three letters followed by three umbers followed by GP. Letters may ot be repeated but umbers may be repeated. () = for letters for umbers 9

6 CORE MATHEMATICS PI Page 6 of 8 QUESTION 4 (a) Determie f '( ) by first priciples if f ( + h) f ( ) f '( ) = lim h 0 h 0 ( + h) + ( + h) + ( + + ) = lim h 0 h = lim h + 4h + h + + h + 4h + h + h = lim h 0 h h( 4 + h + ) = lim h 0 h = 4 + h f = + + (5) Pealize up to marks for icorrect otatio (b) Differetiate the followig fuctios epressig your aswers with positive epoets where ecessary. Pay careful attetio to otatio. () g ( ) ( )( 5 ) g ( ) = 0 g '( ) = 0 = + () Pealize otatio to the value of mark i total across both () ad () () + y = (4) y = + y = + dy = = d

7 CORE MATHEMATICS PI Page 7 of 8 (c) Fid the equatio of the taget to the curve y = + 4 which is perpedicular to the lie y + = 4 (5) y + = 4 has a slope of so, perpedicular lie must have a slope of dy = + 4 = + 4 d so, + 4 =, so = ad y = 4 if y the y = + c 4 = + c c = y = QUESTION 5 (a) The fuctio y = f ( ) is draw below 6 Say whether each of the followig is positive, egative or zero: f = () () ( 5) 0 () '( 5.8) 0 f = () () ''( 8) 0 f > () f f > () (4) ( ) ' 0

8 CORE MATHEMATICS PI Page 8 of 8 (b) A liear fuctio f satisfies the followig: f = 7 ad f = Determie the equatio of f ( ) i the form f ( ) =... (4) we wat a lie which goes through ( 7; ) ad ( ; ) m = = 7 y = + c = ( 7) + c c = y = + (c) The fuctio g ( ) = f ( + p) + q. What are the values of p ad q? () p = ad q = 0

9 CORE MATHEMATICS PI Page 9 of 8 QUESTION 6 (a) What aual iterest rate, compouded mothly is equivalet to a effective () aual rate of 4%? Give your aswer as a percetage to d.p. cosider R for year i + = i + =.4 i =.4 i =.4 =.7% (b) A ma ivests some moey at % p.a. compouded mothly. () Determie how log, to the earest moth, before his moey doubles i value? ( ) A = P + i 0. = + =.0 = log = 70.0 moths 6 TOTAL FOR SECTION A : 75 MARKS

CORE MATHEMATICS PI Page 0 of 8 SECTION B QUESTION 7 A ma has 5 books, by Grisham, by Smith ad by Bryso. He arrages them o a shelf. (a) I how may ways ca he do this if there are o restrictios? () 5!

10 CORE MATHEMATICS PI Page 0 of 8 SECTION B QUESTION 7 A ma has 5 books, by Grisham, by Smith ad by Bryso. He arrages them o a shelf. (a) I how may ways ca he do this if there are o restrictios? () 5! = 0 (b) What is the probability that the books by Grisham are et to oe aother? (5) if we put the three books by Grisham i a rubber bad the there are effectively books These ca be arraged i!! = 6 ways because of the differet possible orderigs of the Grisham books withi the rubber bad so, the probability that the 6 Grisham books will be together = = or 0 0 0% 6

CORE MATHEMATICS PI Page of 8 QUESTION 8 (a) Cosider the followig patter: I which figure will there be 7 squares without dots? (5) Number of squares without dots follows the patter: ; 6 ; ; 8 ; 7 ;.

11 CORE MATHEMATICS PI Page of 8 QUESTION 8 (a) Cosider the followig patter: I which figure will there be 7 squares without dots? (5) Number of squares without dots follows the patter: ; 6 ; ; 8 ; 7 ;.. Quadratic sequece with secod differece of so T = a + b + c ad 0 a = c = T = T = + b + = so b = 0 T = + = + 7 = 7 (b) Cosider the series ( + ) i= i () Determie the value(s) of for which the series coverges. () i= i ( ) ( ) ( ) ( ) + = r = + covergece whe < r < < + < 4 < < < <

12 CORE MATHEMATICS PI Page of 8 () Determie the value of the series if =.6. () S a r + = + = ( ) + =.6 + =.6 0. =. = or i (c) Solve for if ( ) 4 i= i log = 0 (5) i= 4 log = log + log + log + log = 0 log + log + log + 4 log = 0 0 log = 0 log = = 000 5

CORE MATHEMATICS PI Page of 8 QUESTION 9 Mr de Wet wishes to buy a ew Hilu as pictured. The price of the vehicle is R400 000.

13 CORE MATHEMATICS PI Page of 8 QUESTION 9 Mr de Wet wishes to buy a ew Hilu as pictured. The price of the vehicle is R ABSA bak is prepared to offer Mr de Wet the followig terms o a loa: He will be required to pay i a 0% deposit The balace will be fiaced over 54 moths with equal mothly paymets startig oe moth from the date of purchase Iterest will be charged at 0% per aum compouded mothly (a) Calculate his mothly repaymets (4) ( i) + Pv = i = = = R (b) How much will Mr de Wet still owe immediately after makig his 40 th paymet () Pr eset value of 4 paymets = = R ote : R with full accuracy (c) What percetage of his 4 st paymet will go to repayig iterest? () Epress your aswer as a percetage to d.p = = 0.97%

14 CORE MATHEMATICS PI Page 4 of 8 QUESTION 0 Give the equatios of the followig graphs i stadard form: (a) (4) y = a + + = a + + = a + a = ( ) y = + + y = + 4 (b) (4) y = ab + c 0.5 y = ab ow sub 0; = a a = 0.5 y = 0.5b ow sub ; = b 4 = 0.5b 8 = b b = y =

15 CORE MATHEMATICS PI Page 5 of 8 (c) The hyperbola y = is traslated uits to the left ad 5 uits dow () Give the equatio of the traslated graph () y = 5 + for magitude, for directio () Give the equatio of the aes of symmetry of the traslated graph (4) before traslatio y = ad y = after traslatio, y = ad y = + 5 y = + 7 ad y = 7, (d) The parabola y = + 8 is shifted uits to the right ad uit up. The resultig graph is the reflected i the y-ais. y = + 8 shifted right ad up is y = y = + + y = y = reflected i y ais y = 7 7, Give, i stadard form, the equatio of the parabola which results. (4) 8

CORE MATHEMATICS PI Page 6 of 8 QUESTION (a) The fuctio f ( ) = a + + b 4 has a local miimum at the poit ( ; 9) Determie the values of a ad b. = + + 4 ' = + + f a b f a b f = 9 ad f ' = 0.

16 CORE MATHEMATICS PI Page 6 of 8 QUESTION (a) The fuctio f ( ) = a + + b 4 has a local miimum at the poit ( ; 9) Determie the values of a ad b. = ' = + + f a b f a b f = 9 ad f ' = 0. (5) a + + b 4 = 9 ad a + + b = 0 a + b = 6 ad a + b = subtractig the left had equatio from the right had oe gives : a = 4 a = ad b = 8 (b) A garde has 00 kg of watermelos growig i it. Every day, the total mass of watermelo icreases by 5 kg. However, every day the price per kg of watermelo goes dow by c. () If the curret price is 90c per kg the what is the crop worth if () it is harvested today? = R80 () What will the crop be worth i t days? () V = + t t ( 00 5 )( 90 ) () How much loger should the watermelos be left to grow () i order to maimize the icome? ( 00 5 )( 90 ) V = + t t V = t 5t dv = 50 0t = 0 dt t = 5 days

17 CORE MATHEMATICS PI Page 7 of 8 (c) A bucket has two pipes eterig it. Oe is fillig the tak at a variable rate while the other is draiig it at a variable rate. The volume of water (i litres) i the tak at time t (i t 0; is give by V = t t + 5t hours) with ( [ ]) () What is the average rate of flow i litres/hour i the first hour? () at t = 0, V = 0 at t =, V = average flow = l / hr () Did the volume of water icrease or decrease over that time? () Icrease () What is the istataeous rate of flow at hours? () V = t t + 5t dv = t 4t + 5 dt dv at t =, = 5 l / hr dt (4) At what poit i the three hour time iterval was the bucket fullest? () Give your aswer to the earest miute. dv = t 4t + 5 = 0 dt 5 t = ( silver fo) hour 40 miutes (5) What is the maimum volume the bucket cotaied? () Give your aswer to the earest litre. V = + 5 = 5 litres 0

18 CORE MATHEMATICS PI Page 8 of 8 QUESTION Gettig a feel for BIG umbers! The umber 04 7 is very big! Far too big to display o your calculator. I woder how may digits it has? (a) How may digits does the umber 6 0 have? () 7 (b) How may digits would 5 0 have? () 6 (c) Ca you solve 04 7 = 0? (4) = = log 7 04 = 04log 7 = 70.0 (d) Hece, ca you say how may digits 70 Just for fu, here they are! Cout them if you wish! 04 7 has? () TOTAL FOR SECTION B : 75 MARKS

CORE MATHEMATICS PI Page 1 of 10 HILTON COLLEGE TRIAL EXAMINATION AUGUST 2014 CORE MATHEMATICS PAPER I GENERAL INSTRUCTIONS

CORE MATHEMATICS PI Page 1 of 10 HILTON COLLEGE TRIAL EXAMINATION AUGUST 014 Time: 3 hours CORE MATHEMATICS PAPER I GENERAL INSTRUCTIONS 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. This