Mathematics HIGHER SCHOOL CERTIFICATE Assessment 4 ABBOTSLEIGH. Student s Name: Student Number: Teacher s Name:

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1 Studet s Name: Studet Number: Teacher s Name: ABBOTSLEIGH 06 HIGHER SCHOOL CERTIFICATE Assessmet Mathematics Geeral Istructios Total marks - 00 Readig time 5 miutes Workig time hours Write usig black pe. Board-approved calculators may be used. A referece sheet is provided. I Questios 6, show relevat mathematical reasoig ad/ or calculatios. Make sure your HSC cadidate Number is o the frot cover of each booklet. Start a ew booklet for Each Questio. Aswer the Multiple Choice questios o the aswer sheet provided. If you do ot attempt a whole questio, you must still had i the Writig Booklet, with the words 'NOT ATTEMPTED' writte clearly o the frot cover. Attempt Sectios ad. 0 marks Attempt Questios 0. Allow about 5 miutes for this sectio. 90 marks Attempt Questios - 6. Sectio I Pages - 6 Sectio II Pages 7-5 Allow about hrs ad 5 miutes for this sectio.

2 Outcomes to be assessed: Mathematics Prelimiary Outcomes: P P P P5 P6 P7 P8 Provides reasoig to support coclusios which are appropriate to the cotext Performs routie arithmetic ad algebraic maipulatio ivolvig surds, simple ratioal expressios ad trigoometric idetities Chooses ad applies appropriate arithmetic, algebraic, graphical, trigoometric ad geometric techiques Uderstads the cocept of a fuctio ad the relatioship betwee a fuctio ad its graph Relates the derivative of a fuctio to the slope of its graph Determies the derivative of a fuctio through routie applicatio of the rules of differetiatio Uderstads ad uses the laguage ad otatio of calculus HSC Outcomes: H H H H H5 H6 H7 H8 H9 Seeks to apply mathematical techiques to problems i a wide rage of practical cotexts Costructs argumets to prove ad justify results Maipulates algebraic expressios ivolvig logarithmic ad expoetial fuctios Expresses practical problems i mathematical terms based o simple give models Applies appropriate techiques from the study of calculus, geometry, trigoometry ad series to solve problems Uses the derivative to determie the features of the graph of a fuctio Uses the features of a graph to deduce iformatio about the derivative Uses techiques of itegratio to calculate areas ad volumes Commuicates usig mathematical laguage, otatio, diagrams ad graphs Mathematics Task

3 SECTION I 0 marks Attempt Questios 0 Use the multiple-choice aswer sheet Select the alterative A, B, C or D that best aswers the questio. Fill i the respose oval completely. Sample + = (A) (B) 6 (C) 8 (D) 9 If you thik you have made a mistake, put a cross through the icorrect aswer ad fill i the ew aswer. (A) (B) (C) (D) If you chage your mid ad have crossed out what you cosider to be the correct aswer, the idicate this by writig the word correct ad drawig a arrow as follows. correct. Evaluate l 06 (A) (B) (C) (D) (A) (B) (C) (D) e correct to sigificat figures. (A) 87.9 (B) (C) 88.0 (D) The equatio of the lie passig through the poit (0, ) ad perpedicular to the lie xy0 is: (A) x y 0 (B) x y 6 0 (C) x y 6 0 (D) x y 6 0 Mathematics Task

4 . Flora otices that her household expeses are icreasig by $0.50 each moth. If i July 06 her expeses were $55, the her aticipated expeses for the moth of August 07 will be? (A) $65.50 (B) $ (C) $58 (D) $ Which set of iequatios represet the shaded regio show below? xy y x x (A) x y ad y x x (B) x y ad y x x (C) x y ad y x x (D) x y ad y x x Mathematics Task

5 5. Cosider the fuctio f( x) x x. Which expressio represets the largest possible domai for f( x )? (A) x (B) x (C) x (D) x 6. A composite shape is made up of a parallelogram ad a isosceles triagle, as show i the diagram. Which of the followig is correct? (A) ab 80 (B) ab 80 (C) ab 80 (D) b a 7. If f ( x ) x x, the f( x ) is equal to? (A) x (B) x (C) x x (D) x x Mathematics Task

6 8. The graph of y kx itersects the graph of of the followig statemets is true? y x x at two distict poits. Which (A) k k6 0 (B) k k6 0 (C) k k 0 (D) k k 0 9. Cosider the taget to the graph y,. x at the poit Which of the followig lies is parallel to the taget? (A) x y 0 (B) x y 0 (C) y x 6 (D) yx 0. The limitig sum of 8... is? 9 7 (A) (B) (C) 5 (D) Ed of Sectio Mathematics Task

7 SECTION II 90 marks Attempt Questios 6 Allow about hours ad 5 miutes for this sectio. Aswer each questio i the appropriate writig booklet. Extra writig booklets are available. I Questios 6, your resposes should iclude relevat mathematical reasoig ad/or calculatios. Questio (5 marks) Use a SEPARATE writig booklet. Marks (a) Factorise fully x 08. (b) Show that the derivative of x x x e is 5 e x. (c) Fid the derivative of y tax. (d) If si x y, show that cos x dy dx cos x. (e) (i) Express 8 i the form k where k is a iteger. (ii) Hece or otherwise, simplify simplified surd form , givig your aswer i (f) Evaluate 0 cos x dx. (g) Fid x x 6 dx. Ed of Questio Mathematics Task

8 Questio (5 marks) Use a SEPARATE writig booklet. Marks (a) The lie l has the equatio 7x y 0, respectively. ad the poits C ad D are, ad C D NOT TO SCALE 7xy 0 (i) Fid the gradiet of lie l. (ii) Fid the equatio of the lie which passes through C ad is parallel to l. (iii) The poit A lies o l ad D is the midpoit of AC. Fid the coordiates of A. (iv) Without fidig the poit of itersectio, fid the equatio of the lie which passes through the poit of itersectio of 7xy 0 ad x y 0 ad also passes through D. (b) The equatio x 6x 0 has roots ad. (i) Write dow the values of ad. (ii) Show that. (iii) Show that. (iv) Explai why x x 0 has the roots ad. Ed of Questio Mathematics Task

9 Questio (5 marks) Use a SEPARATE writig booklet. Marks (a) The curve with the equatio y x x is show i the diagram. NOT TO SCALE The curve cuts the x-axis at the poit A (,0) ad passes through the poit B (, ). (i) Evaluate x x dx. (ii) Hece, fid the area of the shaded regio bouded by the curve y x x ad the lie AB. (b) The displacemet x metres from the origi at time t secods, of a particle travellig i a straight lie is give by the formula x t t. (i) Fid the acceleratio of the particle at time t secods. (ii) Fid the time(s) at which the particle is statioary. (c) The volume, V m, of water i a tak after time t secods is give by the equatio (i) Fid dv dt V l(t ) 5t. (ii) Explai why the volume of water is decreasig at t. Mathematics Task Questio cotiues o the ext page.

10 Questio cotiued. (d) There are plas to costruct a series of straight paths o the flat top of a moutai. A straight path will coect the cable car statio at C to a commuicatio tower at T, as show i the diagram below. The bearig of the commuicatio tower to the cable car statio is 060. The legth of the straight path betwee the commuicatio tower ad the cable car statio is 950 m. Paths will also coect the cable car statio ad the commuicatio tower to the camp site at E. The legth of the straight path betwee the cable car statio ad the camp site is 00 m. The agle TCE is 0. (i) Calculate the legth of the path betwee the commuicatio tower ad the camp site, correct to the earest metre. (ii) Fid the bearig of the camp site from the commuicatio tower, correct to the earest degree. Ed of Questio Mathematics Task

11 Questio (5 marks) Use a SEPARATE writig booklet. Marks (a) The area bouded by the curve y x ad the lies x ad x x is rotated about the x-axis. Fid the volume of the solid of revolutio formed. Leave your aswer i exact form. (b) A particle moves i a straight lie. At time t secods its displacemet x metres from a fixed poit O o the lie is give by x cos t, 0 t. (i) Sketch the graph of x as a fuctio of t, showig all the importat features. (ii) Explai how you ca use your graph to determie the times that the particle is at rest. (iii) Fid the time whe the particle first reaches its maximum speed. (c) A compay is desigig a ew logo i the shape of a circle with a small segmet take out as show to the right. The radius of the circle is cm ad the legth of AB is also cm. NOT TO SCALE (i) Explai why AOB. (ii) Fid the area of the logo correct to sigificat figures. Questio cotiues o the ext page. Mathematics Task

12 Questio cotiued. (d) The cubic fuctio is show i the diagram. The derivative of this fuctio is y ax bx c where a, b ad c are real costats with a 0 f '( x) ax bx. NOT TO SCALE Two tagets are draw to this fuctio such that their equatios are: y x at the poit x at the poit y 9 5, 0 ad,. (i) Show that 9ab ad a b. (ii) Hece fid the values of a, b ad c. Ed of Questio Mathematics Task

13 Questio 5 (5 marks) Use a SEPARATE writig booklet. Marks (a) The rate of icrease of a populatio Pt of people i a certai coutry is dp determied by the equatios kp ad P P 0 e kt dt, where k is a costat, P 0 is the origial populatio ad t is the time i years. The populatio of the coutry doubles every 0 years. (i) Show that k l. 0 (ii) Data is first collected about the populatio of this coutry i the year 000. I which year will the coutry reach a populatio three times that it had at the begiig of 000? (iii) Give that at the begiig of the year 000 the populatio was 5. millio, what will be the populatio of the coutry at the begiig of the year 050? Give your aswer correct to sigificat figures. (b) O the st of Jauary 06 the populatio of a particular coutry tow was At the ed of each year 500 people leave the tow to live i the city. Durig the period betwee Jauary ad the people leavig i December each year, the populatio icreases by 5%. (i) Show that the umber of people i the coutry tow just after the first group of 500 left i December 06 is (ii) Show that the expressio for the umber of people i the coutry tow just after the secod group of 500 left i December 07 is give by P (.05 ). (iii) Show that P, the populatio after the th group left is give by P (.05 ). (iv) Hece, determie i which year the populatio of the tow will be zero. (c) A parabola has its focus at 5, ad vertex at,. Show that the equatio of the parabola is y y x Mathematics Task Ed of Questio 5

14 Questio 6 (5 marks) Use a SEPARATE writig booklet. Marks (a) A irrigatio chael has a cross-sectio i the shape of a trapezium as show i the diagram. The bottom ad sides of the trapezium are metres log. Suppose that the sides of the chael make a agle of with the horizotal where. (i) Show that the cross-sectioal area is give by A 6si cos si. da d. (ii) Show that 6cos cos (iii) Hece, show that the maximum cross-sectioal area occurs whe. (iv) Hece, fid the maximum area of the irrigatio chael, correct to the earest square metre. Questio 6 cotiues o the ext page. Mathematics Task

15 Questio 6 cotiued. (b) Whe a radial arm saw (as show o the right) is used, its cuttig edge (as idicated by the dot) moves forwards ad the backwards alog a straight lie. Durig a particular cuttig procedure, the velocity of the cuttig edge of the saw, i metres per secod, ca be modelled by the fuctio v( t) 0.05t 0.8t 0.6t, where t represets the time i secods from the start of the cuttig procedure ad 0 t 5.. (i) For what values of t is the cuttig edge of the saw at rest? (ii) Calculate 5. 0 v() t dt, correct to decimal places. (iii) Iterpret your aswer to part (ii) i the cotext of the motio of the cuttig edge of the saw. (iv) Write a expressio to fid the total distace travelled by the cuttig edge of the saw durig the cuttig procedure. (There is o eed to evaluate this). Ed of Paper Mathematics Task

16 Mathematics Task BLANK PAGE

17 Abbotsleigh uit Mathematics Task 06 Solutios: Questio Workig Solutio 88.0 C x 0 y m m lie Equ of lie is y x y x 0 a 55, d 0.50, T $59.50 A D Test poitt (0,0) x y y x x D 5 x 0 x A 6 B 80 b 80 a 60 a 80 b ab80

18 Trial ad error for B f x ( -) ( x ) x x x x ( x) ( x) x x x OR sub i ( x ) x Solve y kx - y x x kx x x x x k ( k) ()() 0 k k 6 0 ( ) 0 For two distict roots, 0 k k 0 y x y' x At x, y' m of taget = parallel lie has same gradiet ie, x y 0 a, r a S r 5 B C B A

19 Questio Workig Marks (a) x 08 x 7 x x x 9 (b) (c) (d) (e)(i) RTShow : Proof: d dx x x derivative of ( x - ) e is e (x 5) x x x ( x - ) e ( x ) e e () e as required y ta x dy dx sec x x e x x e x x ( ) 6 (x5) si x dy dy RTShow: If y the cos x dx dx cos x Proof: dy ( cos x)(cos x) si x ( si x) dx cos x cos cos si cos x x x x cos x cos x cos x as required 8 6 (sice cos x si x )

20 (e)(ii) (f) (g) 0 cos x dx si x si( ) si 0 si x 0 x dx x 6 x 6 l x 6 C recogisig log dx

21 Questio Workig Marks (a)(i) 7xy y 7x 7 y x 7 m (a)(ii) 7 Sice parallel to l, gradiet C(,) Equ of lie is 7 y ( x) y 9 7x 7 7x y 0 (a)(iii) D is the midpoit of AC x y.5,, x y.5 ad x y x y 5 A is, 5

22 (a)(iv) Equ of lie is give by 7x y k x y 0 Sub i (.5, ) 7(.5) ( ) k (.5) ( ) k k Equ is 7x y x y 0 9 x 57y 7 x y 0 00x 5y5 0 0x7y 0 (b)(i) x 6x 0 a, b 6, c b a 6 c a (b)(ii) RTShow: Proof: 9 as required

23 (b)(iii) RTShow: Proof: 9 as required 0 0 (b)(iv) c For x x 0, product of roots a From part (iii), ( )( ) Roots of x x 0 are ( ) ad ( ).

24 Questio Workig Marks (a)(i) x x dx x x x () () ( ) ( ) () ( ) (a)(ii) Shaded area = area uder curve (i) - area of triagle (b)(i) x t t. x.. x dx dt 8 uits t 6t t d dx dx dt (b)(ii). Particle is statioary whe x 0 i.e. t t 0 tt ( ) 0 t 0, t Particle is statioary at t 0 ad t secods. (c)(i) V l(t ) 5t dv dt 5 t 6 5 t (c)(ii) dv 6 At t, 5 dt () Sice dv dt 5 0, the volume is decreasig

25 (d)(i) TE (950)(00)cos if used radias TE if used radias 908 m (to earest metre) Distace from commuicatio tower to the camp site is 908 m (d)(ii) Fid : si si TE si 0 si 950 TE ' or equivalet step CTE ' 975' 60 75' Bearig = 80 (to mearest degree) Bearig of the camp site form the commuicatio tower is NOTE if studets use the sie rule to fid CTE, they have to be careful to use the obtuse agle ad justify why. This is a lot trickier. They could use the cosie rule to fid CTE.

26 Questio Workig Marks (a)(i) y x x y x x x x x x V x x dx x x x (b)(i) x cos t, 0 t () () ( () () ) 0 uits for period halved ad vertical traslatio labellig correct max ad mi poits ad itercept (b)(ii) Particle is at rest whe the velocity is 0, that is the gradiet of the tagets is 0. Therefore, the maximum ad miimum turig poits will idicate whe the particle is at rest.

27 (b)(iii) Max speed whe acceleratio =0 x cost v si t a cost Whe a 0, cos t 0 cost 0 t t Max speed occurs at secods OR studets could say this occurs at the poit of iflectio o the graph ad fid their aswer this way. (c)(i) (c)(ii) AB (give) OA OB cm radii of circle OAB is a equilateral triagle AOB Area of logo = area of circle - area of segmet Area of segmet = () () si Area of logo () 8.8 cm (to sig figs)

28 (d)(i) y ax bx c y ' ax bx At (, 0) gradiet of the taget is Sub (, 0) ito ' ' () () y a b a b a b y ax bx At (,) gradiet of the taget is 9 Sub (,) ito ' y ' a() b() 9 7a 6b 9a b y ax bx (d)(ii) Solve a b () 9ab () () () 6a 6 a Sub a ito () () b - b 6 b Sub, ito a b y ax bx c y x x c 0= () c ad sub (,0) c a, b, c for oe of the correct values for all vallues correct

29 Questio Workig Marks 5(a)(i) kt P P e 0 After 0 years P P P P e 0k 0 0 e 0k l 0k k l ( ) 0 0 5(a)(ii) l 0 P P e 0 0 t e l t 0 l l t 0 0l t l t Populatio trebles years later, that is 0 5(a)(iii) P 0 5. millio P 5.e l Populatio is 85. millio (to sig figs) 5(b)(i) 5(b)(ii) st Ja 06, poulatio = Each year pop grows by 5% ad 500 leave RTShow: Pop i Dec 06 = Proof: P as required RTShow: P (.05 ) Proof: P P P (.05 )

30 5(b)(iii) 5(b)(iv) 5(c) RTShow: (.05 ) Proof: P P (.05 ) P P P (.05 ) (.05.05) ( Patter cotiues. ) P ( ) settig up to here ( ) as required Populatio is zero GP a, r.05,.05 S l(.05) l 5 l l Populatio will be zero i the year 05 Focus (5,) Vertex (, ) Parabola is sideways opeig to right Focal legth = Equ of parabola is y () x y g a x h Vertex ( h, g) y y x 8( ) y y x 8 y y x 8 5 0

31 Questio Workig Marks 6(a)(i) RTShow: A 6 si cossi Proof: x cos = x cos h si = h si oe of these statemets 6(a)(ii) Cross-sectio is a trapezium h A ( a b) si ( x) si ( cos ) si 8 8cos 6si 6si cos 6 si sicos RTShow: 6 cos cos Proof: A 6 si si cos da d da 6cos si ( si ) cos cos d 6 cos si cos 6 cos ( cos ) cos 6 cos cos cos 6 cos cos 6 cos cos

32 6(a)(iii) RTShow: Max area whe Proof: da d da For max 0 d 6 cos cos 6 cos cos 0 ( cos ) (cos ) 0 cos or cos ad or = 0 ot possible Test for Max: x da d cos cos 6 cos cos cos cos Chage i Max at da d 6(a)(iv A 6si si cos whe A 6si si cos = m (to earest m )

33 6(b)(i) v( t) 0.05t 0.8t 0.6t 0 t 5. 6(b)(ii) 5. At rest vt ( ) t 0.8t 0.6t 0 50t 80t 6t 0 5t 90t t 0 t t t (5 90 ) 0 t 0 or t 90 ( 90) (5)() , ,. Cuttig saw is a rest at 0,. ad 5. secods t t t dt 0.05t 0.8t 0.6t (5.) 0.8(5.) 0.6(5.) 0.05(0) 0.8(0) 0.6(0) (to decimal places) 6(b)(iii) The saw starts movig ad at. secods it as at rest ad chages directio. At 5. secods it is at rest agai. The itegral of v(t) will give the distace travelled. As the aswer to part (ii) is egative, the saw must have travelled further i the secod stage of its movemet tha the first. 6(b)(iv) To fid the total distace travelled, fid the distace travelled (forward) i the first. secods ad add this to the distace travelled (backwards) i the time from. to 5. secods.. 5. Total distace 0.05t 0.8t 0.6t dt 0.05t 0.8t 0.6t dt 0. For recogisig two itegrals separated by t 0,., 5. Correct absolute value or equivalet