Sample Examination Papers

55 Maths I Focus Mathematics Etesio HSC Course Sample Eamiatio Papers This chapter cotais two sample Mathematics HSC papers ad two sample Etesio papers They are desiged to give you some practice i worig through a eamiatio These papers cotai wor from the Year Prelimiary Course, as up to % of questios are allowed to cotai this wor You may eed to revise this wor before you try these papers If you ca, set yourself a time limit ad wor uder eamiatio coditios Give yourself hours to do each Mathematics paper ad hours for each Etesio paper Try ot to loo up ay otes while worig through these papers MATHEMATICS PAPER Time allowed Three hours ( Plus 5 miutes readig time ) (e) Two cities are 95 m apart Write this umber correct to sigificat figures (f) Solve + < 7 QUESTION (i) Copy the diagram ito your eamiatio boolet (ii) Show that AC = CE (iii) Fid the legth of DE (b) DIRECTIONS TO CANDIDATES All questios may be attempted All questios are of equal value All ecessary worig should be show i every questio, as mars are awarded for this Badly arraged or careless wor may ot receive mars QUESTION Fid, correct to decimal places, the 8 value of 5 7 4 (b) Factorise - + 6 (c) Solve the equatio = 5 (d) The volume of a coe is give by V = r If a coe has volume m, fid its radius correct to decimal places (i) Copy the diagram ito your eamiatio boolet (ii) Use the sie rule to calculate MO correct to decimal place (iii) Fid MP to the earest metre QUESTION Differetiate (i) + 5 + (ii) l + (iii) ( + ) 5 (b) Fid (i) ( e ) d # # (ii) ( si θ + ) dθ

SAMPLE EXAMINATION PAPERS 55 (c) (i) Ratioalise the deomiator of 5 (ii) Fid itegers a ad b such that 5 = a + b QUESTION 4 (i) Plot poits A (, ) ad B (-, 5) o a umber plae (ii) Show that lie AB has equatio + 4 y - 7 = (iii) Fid the perpedicular distace from the origi to lie AB (iv) Fid the area of triagle OAB where O is the origi (b) QUESTION 6 A plat has a probability of 7 of producig white flowers If plats are chose at radom, fid the probability that (i) plats will produce white flowers (ii) at least plat will produce white flowers (b) Solve si = for 6 (c) A particle P moves so that it is iitially at rest at the origi, ad its acceleratio is give by a = 6 t + 4 cms - (i) Fid the velocity of P whe t = 5 (ii) Fid the displacemet of P whe t = QUESTION 7 Fid the legth of side BC usig the cosie rule ad give your aswer correct to the earest cetimetre QUESTION 5 Kate ears $4 6 pa I the followig year she receives a pay icrease of $8 Each year after that, she receives a pay icrease of $8 (i) What percetage icrease did Kate receive i the first year? (ii) What will be Kate s salary after years? (iii) What will Kate s total earigs be over the years? (b) A fuctio is give by y = - - 9 + (i) Fid the coordiates of the statioary poits ad determie their ature (ii) Fid the poit of ifleio (iii) Draw a setch of the fuctio ABCD is a parallelogram with AD =, AB = ad + ADX = 6 AX is draw so that it bisects DC (i) Copy the diagram ito your eamiatio boolet (ii) Show that ADX is a equilateral triagle (iii) Show that AXB is a right-agled triagle (iv) Fid the eact legth of side BX (b) O a umber plae, shade i the regio give by the two coditios y ³ ad + y (c) The umber of people P with chicepo is icreasig but the rate at which the disease is spreadig is slowig dow over t wees (i) Setch a graph showig this iformatio dp dp (ii) Describe ad for this dt dt iformatio

554 Maths I Focus Mathematics Etesio HSC Course QUESTION 8 Cosider the fuctio give by y = log (i) Copy ad complete the table, to decimal places, i your eamiatio boolet 4 5 y (ii) Apply the trapezoidal rule with 4 subitervals to fid a approimatio, to decimal 5 places, of # log d (b) A populatio of mice at time t i wees is give by P = P e t, where is a costat ad P is the populatio whe t = (i) Give that mice icrease to after 6 wees, calculate the value of, to decimal places (ii) How may mice will there be after wees? (iii) After how may wees will there be 5 mice? QUESTION 9 The surface area of a cylider is give by the formula S = r( r + h) A cylider is to have a surface area of 6 cm (i) Show that the volume is give by V = 8r r (ii) Fid the value of r, to decimal places, that gives the maimum volume (iii) Fid the maimum volume, to decimal place (b) Kim ivests $5 at the begiig of each year i a superauatio fud The moey ears % iterest per aum If she starts the fud at the begiig of 996, what will the fud be worth at the ed of 5? QUESTION (i) Fid the eact poit of itersectio of the curve y = e ad the lie y = 4 (ii) Fid the eact shaded area eclosed betwee the curve ad the lie (b) (i) The quadratic equatio + ( - ) + = has real ad equal roots Fid the eact values of i simplest surd form (ii) If = 5, show that + ( - ) + > for all (c) A parabola has equatio y = + p + q (i) Show that the coordiates of its verte are (- p, q - p ) (ii) Fid the coordiates of its focus (iii) Fid the distace betwee the poit ( m, m + q ) ad poit P vertically below it o the parabola = 8 y whe q >

SAMPLE EXAMINATION PAPERS 555 y = + p + q y (m, m + q) (iv) Fid the miimum distace betwee these two poits whe m + q = 5 MATHEMATICS PAPER Time allowed Three hours ( Plus 5 miutes readig time ) P = 8y QUESTION Solve = 5 (b) If f ( ) = 5 -, fid whe f ( ) = -4 5 (c) Fid the eact value of si 6 (d) Factorise fully a - a - 4 a + 8 (e) Simplify 4 5 (f) Evaluate log a 5 if log a 5 = ad log a = 4 (g) Fid the midpoit of (-, 4) ad (, -) QUESTION Plot poits A e5, o, B e, o ad C (-4, -) o a umber plae Show that the equatio of lie AB is give by + 4 y - 9 = (b) Fid the equatio of the straight lie l through C that is perpedicular to AB (c) Fid the poit of itersectio P of the two lies, AB ad l (d) Fid the area of triagle ABC (e) Fid the coordiates of D such that ADCB is a rectagle QUESTION Differetiate (i) cos (ii) e 5 (iii) log ( ) e (b) Fid the idefiite itegral (primitive fuctio) of (i) ( - ) 4 (ii) si (c) Evaluate # ( e e ) d dy (d) For a certai curve, = 8 6 If d there is a statioary poit at (, ), fid the equatio of the curve QUESTION 4 Mar plays a game of chace that has a probability of 5 of wiig the game Hog plays a differet game i which there is a probability of 8 of wiig Fid the probability that (i) both Mar ad Hog wi their games (ii) oe of them wis the game (iii) at least oe of them wis the game (b) (i) O a umber plae, shade the regio where y ³, ad y + (ii) Fid the area of this shaded regio (iii) This area is rotated about the -ais Fid the volume of the solid formed (c) (i) Setch the graph of y = o a umber plae (ii) Hece solve < QUESTION 5 For the curve y = - 9 + - 7 (i) fid ay statioary poits o the curve ad determie their ature (ii) fid ay poits of ifleio (iii) setch the curve i the domai -

556 Maths I Focus Mathematics Etesio HSC Course (b) A plae leaves Bastow airport ad flies for 85 m o a bearig of It the turs ad flies for m o a bearig of 4 (i) Draw a diagram showig this iformatio (ii) How far from the airport is the plae, to the earest m? (c) Simplify ( cosec θ + cot θ)( cosec θ cot θ) QUESTION 6 (i) Fid the equatio of the ormal to the curve y = at poit P (-, 4) (ii) This ormal cuts the parabola agai at poit Q Fid the coordiates of Q (iii) Fid the shaded area eclosed betwee the parabola ad the ormal, to sigificat figures (b) The graph below shows the displacemet of a particle over time t (ii) Differetiate l (cos ) 4 (iii) Hece evaluate # ta dto decimal places (b) A bridge is 4 metres log, held by wires at A ad C with agles of elevatio of 47 ad as show (i) Fid the legth of AD to sigificat figures (ii) Fid the height of the bridge BD to decimal place (c) Triagle BEC is isosceles with BC = CE Also + BEC = 5, + ABE =, ad + ADC = 8 Prove that ABCD is a parallelogram (i) Whe is the particle at the origi? (ii) Whe is the particle at rest? (c) The temperature T of a metal is coolig epoetially over t miutes It cools dow from 97 C to 84 C after 5 miutes Fid (i) the temperature after 5 miutes (ii) whe it cools dow to C QUESTION 7 (i) Fid the area bouded by the curve y = ta, the -ais ad the lies = ad =, by usig Simpso s rule 4 with 5 fuctio values (to decimal places) QUESTION 8 (i) Setch the curve y = si for (ii) O the same set of aes, setch y = (iii) How may roots does the equatio si = have i this domai? (b) Solve si - = for 6 (c) If log = p ad log = q, write i terms of p ad q (i) log (ii) log (d) A stac of orages has orage at the top, i the et row dow, the each row has more orages tha the previous oe

SAMPLE EXAMINATION PAPERS 557 (i) How may orages are i the th row? (ii) If there are 89 orages staced altogether, how may rows are there? (iii) Fid the value of if the taget has a -itercept of (e) Fid the area of the sector to decimal place QUESTION 9 The diagram below shows the graph of a fuctio y = f ( ) QUESTION The graph of y = ( + ) is draw below Copy the graph ito your writig boolet (i) Copy this diagram ito your writig boolet (ii) O the same set of aes, draw a setch of the derivative fl ( ) of the fuctio (b) A bag cotais yellow, blue ad white marbles Therefore if I choose oe marble at radom from the bag, the probability that it is blue is Is this statemet true or false? Eplai why i o more tha oe setece (c) Solve the equatio l = l ( + ) (d) The diagram shows the graph of y = e ad the taget to the curve at = (i) Fid the gradiet of the taget at = (ii) Fid the equatio of the taget at = (i) Shade the regio bouded by the curve, the -ais ad the lie = (ii) This area is rotated about the -ais Fid the volume of the solid of revolutio formed (b) The accoutat at Acme Busiess Solutios calculated that the hourly cost of ruig a busiess car is s + 75 cets where s is the average speed of the car The car travels o a m jourey (i) Show that the cost of the jourey 75 is give by C = bs + s l (ii) Fid the speed that miimises the cost of the jourey (iii) Fid the cost of the trip to the earest dollar

558 Maths I Focus Mathematics Etesio HSC Course EXTENSION PAPER Time allowed Two hours ( Plus 5 miutes readig time ) QUESTION Solve for : + 5 (b) Fid the coordiates of the poit that divides the iterval AB with A (-, 8) ad B (, ) i the ratio : d (c) Fid the eact value of # 4 (d) Differetiate (e) Fid # ( 5) 5 d, usig the substitutio u = - 5 QUESTION I the diagram A, B, C ad E are poits o the circle with cetre O AE is produced to D such that BE = DE (i) Show that + BEO = + CDE (ii) Show that + BAO = + BEO (iii) Taget DF is draw to meet the circle at F If BE = 5 cm ad the circle has radius 5 cm, fid the legth of DF i eact form (b) A root of e - = lies ear = -5 Use Newto s method to fid a secod approimatio to the root, correct to decimal places (use = -5 as your first approimatio) (c) Cosider the fuctio f ( ) = si (i) Fid the eact value of f e o (ii) Setch y = f ( ) QUESTION (iii) Fid the equatio of the taget to the curve at the poit where = ABCD is a triagular pyramid with AB = AC = AD = cm, BD = 6 cm, CD = 5 cm ad + BCA = 55 (i) Calculate the legth of BC, to decimal place (ii) Fid + BCD, to the earest miute (b) Fid all solutios of the equatio cos = cos for (c) A group of 6 girls ad 5 boys are to be arraged i a straight lie Fid how may ways they ca be arraged: (i) with o restrictios o the order (ii) if boys ad girls are to alterate (iii) if particular girls are to stad together QUESTION 4 A particle is projected from the origi O with velocity 5 ms -, at a agle of θ

SAMPLE EXAMINATION PAPERS 559 (i) Neglectig air resistace ad assumig acceleratio due to gravity is ms -, show that the equatios for the horizotal ( ) ad vertical ( y ) compoets of the particle s displacemet from O after t secods are give by = 5t cos θ ad y = 5t + 5t siθ (ii) Show that the Cartesia equatio for displacemet is give by y = sec θ + taθ 45 (iii) The particle just clears a object m high stadig out 5 m from the origi Fid possible values of θ for this to happe (b) The probability that a piece of space ju will crash i Australia is estimated at If 8 pieces of space ju are due to crash, fid the probability that of them will crash i Australia Leave your aswer i ide form QUESTION 5 Two poits, P( ap, ap ) ad Q( aq, aq ), lie o the parabola = 4ay (i) Fid the equatio of the taget ( l ) to the parabola at Q (ii) Derive the equatio of chord PQ ad show that pq = - if PQ is a focal chord (iii) Fid the acute agle betwee taget l ad chord PQ if p = ad q = - (b) Use mathematical iductio to prove that for all itegers with ³, + + + + = ( + )( + ) 6 QUESTION 6 The acceleratio of a particle movig i a straight lie is give by 9 =, where metres is the displacemet from the origi after t secods Iitially the particle is m to the right of the origi, with velocity ms - (i) Fid a equatio for the velocity of the particle (ii) Fid the time whe the particle is m from the origi (b) The rate at which a body cools i air is proportioal to the differece betwee the costat air temperature, C, ad its ow temperature, T This ca be epressed by the differetial equatio dt = T ( C), where t is time i dt hours ad is a costat (i) Show that T = C + Ae t is a solutio of the differetial equatio, where A is a costat (ii) A heated piece of metal cools from 9 C to 7 C i hour The air temperature C is 5 C Fid the temperature (to the earest degree) of the body after aother hours QUESTION 7 Assume that for all real umbers ad all positive itegers, ( + ) = Show that: (i) (ii) / = / = / = _ i = _ i _ i = (b) A particle movig i simple harmoic motio has maimum speed 4 ms - ad maimum acceleratio 8 ms - (i) Fid the amplitude ad period of the motio (ii) The particle is at the origi after secods Fid a equatio for the 6 displacemet of the particle (iii) Show that = 4 (iv) Show that the velocity is give by v = 4(4 )

56 Maths I Focus Mathematics Etesio HSC Course EXTENSION PAPER Time allowed Two hours ( Plus 5 miutes readig time ) QUESTION Factorise + 4 (b) Fid the eact value of 4 (i) # d + (ii) # d, usig the + substitutio u = + (c) Fid the umber of differet arragemets possible for the letters i the word CERTIFICATE (d) Fid, to the earest degree, the size of the acute agle betwee the lies - y + 5 = ad + y - 4 = QUESTION AB is a taget to circle ACD, ad AB = 7 cm, BC = 9 cm Fid the legth of chord CD, correct to decimal place (b) Fid the coefficiet of 4 i the 8 epasio of b l (c) By usig the priciple of mathematical iductio, prove 5 / 5 = 5c m 4 = QUESTION (i) Fid the equatio of the ormal to the curve = 4 ay at the poit P ( ap, a p ) (ii) This ormal meets the directri at the poit M Fid the coordiates of M (b) (i) Differetiate y = si + (ii) Hece, or otherwise, evaluate # d QUESTION 4 If α, β ad γ are the roots of - - + =, fid (i) α + β + γ (ii) αβ + βγ + αγ (iii) αβγ (iv) α + + β γ (v) α + β + γ (b) The volume of a epadig balloo is icreasig by a costat rate of cm s - Fid the rate of icrease i its surface area whe the balloo s radius is 8 cm (c) Divide the iterval AB, with A (-, 5) ad B (7, ), i the eteral ratio :4 QUESTION 5 A particle udergoes simple harmoic motio about the origi O Its displacemet metres from O at time t secods is give by = cos bt + l (i) Write the acceleratio as a fuctio of (ii) Write dow the amplitude ad period of the motio (iii) Determie whe the particle is at the origi (b) (i) Divide the polyomial 5 P ( ) = + 5 + 4 by A ( ) = + (ii) Hece write P ( ) = AQ ( ) ( ) + R ( ), where Q ( ) ad R ( ) are polyomials ad R ( ) has degree less tha

SAMPLE EXAMINATION PAPERS 56 (iii) Fid P () ad hece, or otherwise, fid the remaider whe P ( ) is divided by (iv) Apply Newto s method oce to fid a approimate value for a root of P (), begiig with a iitial approimatio of = 5 (c) Show that si bθ l = si θcos θ cos θ + 6 QUESTION 6 Assume ( 5) + = / t = (i) Use the biomial theorem to write a epressio for t, where t + 5 ( ) (ii) Show that = t ( + ) (iii) Hece, or otherwise, fid the largest coefficiet t (you may leave a b your aswer i the form C 5) (b) (i) Fid i how may differet ways 8 people ca be seated aroud a roud table (ii) Two people wish to sit opposite oe aother I how may differet ways ca this be arraged? (iii) If people are allocated seats radomly, fid the probability that these people will ot sit opposite oe aother QUESTION 7 (i) Setch the fuctio f ( ) = cos (ii) Fid the eact value of f c m (iii) Fid the eact area bouded by the curve y = f ( ), the -ais ad the lies = ad = (iv) Fid the volume of the solid formed if the curve f ( ) = cos is rotated about the -ais betwee = ad =, usig Simpso s rule with fuctio values Give your aswer correct to decimal places (b) The diagram shows a cylider of height H ad radius R Poit X is at oe ed of the cylider, o the bottom Poit Y is o the other side, halfway up the cylider Legth XY is D (i) Show that the volume of the cylider is give by Η V = ( D H ) 6 4 (ii) Fid the maimum volume of the cylider i terms of D if D is fied