Exam Revision 2. Determine whether or not these lines are concurrent. 4. Part Marks Level Calc. Content Answer U1 OC1 4 C NC CGD,G8 1996P1Q14
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1 Exam Revision 1. Threelineshaveequationsx +3y 4 =0,3x y 17 =0andx 3y 10 =0. Determine whether or not these lines are concurrent. 4 Part Marks Level Calc. Content Answer U1 OC1 4 C NC CGD,G8 1996P1Q14. FindtheequationofthemedianADoftriangleABCwherethecoordinatesofA, BandCare (,3), ( 3, 4)and (5,)respectively. 3 Part Marks Level Calc. Content Answer U1 OC1 3 C CN G3,G7 1995P1Q5 hsn.uk.net Page 1
2 3. The diagram shows a sketch of the y functiony =f(x). ( 4, 8) (, 8) (a)copythediagramandonitsketch thegraphofy =f(x). y = f( x) (b)on a separate diagram sketch the graphofy =1 f(x). 3 O x Part Marks Level Calc. Content Answer U1 OC (a) B CN A3 sketch 009 P1 Q3 (b) 3 B CN A3 sketch 1 ic: scalingparalleltox-axis ic: annotategraph 3 ss: correctorderforrefl(x)andtrans 4 ic: starttoannotatefinalsketch 5 ic: completeannotation 1 sketch and one of (0,0), (1,8), (, 8) remainingpoints 3 reflect in x-axis then vertical translation 4 sketch and one of (0,1), (1, 7), (, 7) 5 remainingpoints 4. Functions f andg,definedonsuitabledomains,aregivenby f(x) = xand g(x) =sinx +cosx. Findf ( g(x) ) andg ( f(x) ). 4 Part Marks Level Calc. Content Answer U1 OC 4 C NC A4 1997P1Q3 hsn.uk.net Page
3 5. Showthatx +8x +18canbewrittenintheform (x +a) +b. Henceorotherwisefindthecoordinatesoftheturningpointofthecurvewith equationy =x +8x Part Marks Level Calc. Content Answer U1 OC 3 C NC A5,A6 1994P1Q11 6. Functionsfandgaregivenbyf(x) =3x +1andg(x) =x. (a) (i)findp(x)wherep(x) =f(g(x)). (ii)findq(x)whereq(x) =g(f(x)). 3 (b)solvep (x) =q (x). 3 Part Marks Level Calc. Content Answer U1 OC3 (a) 3 C CN A4 3(x ) +1, (3x +1) 009PQ (b) 3 C CN C1 x = 1 1 ss: substituteforg(x)inf(x) ic: complete 3 ic: sub.andcompleteforq(x) 4 ss: simplify 5 pd: differentiate 6 pd: solve 1 f(x ) 3(x ) +1 3 (3x +1) 4 p(x) =3x 5,q(x) =9x +6x 1 5 p (x) =6x,q (x) =18x +6 6 x = 1 hsn.uk.net Page 3
4 7. ThepointP(,b)liesonthegraphofthefunctionf(x) =3x 3 x 7x +4. (a)findthevalueofb. 1 (b) Prove that this function is increasing at P. 3 Part Marks Level Calc. Content Answer U1 OC3 (a) 1 C NC A P1 Q10 (b) 3 C NC C7 8. Iff(x) =kx 3 +5x 1andf (1) =14,findthevalueofk. 3 Part Marks Level Calc. Content Answer U1 OC3 3 C NC C1,A6 1994P1Q 9. Find the coordinates of the turning points of the curve with equation y =x 3 3x 9x +1anddeterminetheirnature. 8 Part Marks Level Calc. Content Answer U1 OC3 8 C CN C8,C9 max. at ( 1,17)andmin. 009PQ1 at (3, 15) 1 ss: knowtodifferentiate pd: differentiate 3 ss: setderivativetozero 4 pd: factorise 5 pd: solveforx 6 pd: evaluatey-coordinates 7 ss: know to, and justify turning points 8 ic: interpretresult 1 dy dx = (1termcorrect) 3x 6x 9 3 dy dx =0 4 3(x +1)(x 3) 5 x = 1orx =3 6 y = 17ory = 15 7 x 1 3 dy/dx max.at ( 1,17)andmin.at (3, 15) hsn.uk.net Page 4
5 10. Asequenceisdefinedbytherecurrencerelationu n =0 9u n 1 +,u 1 =3. (a)calculatethevalueofu. 1 (b)whatisthesmallestvalueofnforwhichu n >10? 1 (c)findthelimitofthissequenceasn. Part Marks Level Calc. Content Answer U1 OC4 (a) 1 C CR A P1 Q9 (b) 1 C CR A14 (c) C CR A (a)onthesamediagram,sketchthegraphsofy=log 10 xandy= xwhere 0 <x<5. Write down an approximation for the x-coordinate of the point of intersection. 3 (b)findthevalueofthisx-coordinate,correcttodecimalplaces. 3 Part Marks Level Calc. Content Answer U OC1 (a) 3 C CR A 1991 P Q4 (b) 1 C CR A6 (b) A/B CR A6 hsn.uk.net Page 5
6 1. (a)thefunctionfisdefinedbyf(x) =x 3 x 5x +6. Thefunctiongisdefinedbyg(x) =x 1. Showthatf ( g(x) ) =x 3 5x +x (b)factorisefullyf ( g(x) ). 3 (c)thefunctionkissuchthatk(x) = 1 f ( g(x) ). Forwhatvaluesofxisthefunctionknotdefined? 3 Part Marks Level Calc. Content Answer U OC1 (a) 4 C NC A P Q6 (b) 3 C NC A1 (c) C NC A1 hsn.uk.net Page 6
7 13. Thediagramshowspartofthegraphofthe curvewithequationy =x 3 7x +4x +4. (a)findthex-coordinateofthemaximum turning point. 5 y y =f(x) (b)factorisex 3 7x +4x (c)statethecoordinatesofthepointaand A hence find the values of x for which O (,0) x x 3 7x +4x +4 <0. Part Marks Level Calc. Content Answer U OC1 (a) 5 C NC C8 x = PQ3 (b) 3 C NC A1 (x )(x +1)(x ) (c) C NC A6 A( 1,0),x< 1 1 ss: knowtodifferentiate pd: differentiate 3 ss: knowtosetderivativetozero 4 pd: startsolvingprocessofequation 5 pd: completesolvingprocess 6 ss: strategyforcubic, e.g. synth. division 7 ic: extractquadraticfactor 8 pd: completethecubicfactorisation 9 ic: interpretthefactors 10 ic: interpretthediagram 1 f (x) =... 6x 14x x 14x +4 =0 4 (3x 1)(x ) 5 x = x 3x 8 (x )(x +1)(x ) 9 A( 1,0) 10 x < Expressx 3 4x 7x +10initsfullyfactorisedform. 4 Part Marks Level Calc. Content Answer U OC1 4 C NC A1 1998P1Q hsn.uk.net Page 7
8 15. (a) (i)showthat (x 4)isafactorofx 3 5x +x +8. (ii)factorisex 3 5x +x +8fully. (iii)solvex 3 5x +x +8 =0. 6 (b)thediagramshowsthecurvewithequationy =x 3 5x +x +8. y y = x 3 5x + x+ 8 P O Q R x Thecurvecrossesthex-axisatP,QandR. Determine the shaded area. 6 Part Marks Level Calc. Content Answer U OC (a) 6 C CN A1,A 1,,4 01P1Q1 (b) 6 C CN C16, C1 3 3 hsn.uk.net Page 8
9 16. Functions f(x) =sinx,g(x) =cosxandh(x) =x + π 4 aredefinedonasuitable set of real numbers. (a) Find expressions for: (i) f(h(x)); (ii) g(h(x)). (b) (i)showthatf(h(x)) = 1 sinx + 1 cosx. (ii) Find a similar expression for g(h(x)) and hence solve the equation f(h(x)) g(h(x)) =1for0 x π. 5 Part Marks Level Calc. Content Answer U OC3 (a) C NC A4 (i) sin(x + π 4 ), (ii) 001P1Q7 cos(x + π 4 ) (b) 5 C NC T8,T7 (i)proof,(ii)x= π 4,3π 4 1 ic: interpretcompositefunctions ic: interpretcompositefunctions 3 ss: expandsin(x + π 4 ) 4 ic: interpret 5 ic: substitute 6 pd: startsolvingprocess 7 pd: process 1 sin(x + π 4 ) cos(x + π 4 ) 3 sinxcos π 4 + cosxsin π 4 and complete 4 g(h(x)) = 1 cosx 1 sinx 5 ( 1 sinx+ 1 cosx) ( 1 cosx 1 sinx) 6 sinx 7 x = π 4,3π 4 acceptonlyradians 17. Solvetheequationcosx +5cosx =0,0 x< Part Marks Level Calc. Content Answer U OC3 1 C CR T P1 Q15 4 A/B CR T10 hsn.uk.net Page 9
10 18. Findthevaluesoft,where0 <t<π,forwhich4cos ( t π ) 4 hasitsmaximum value. 4 Part Marks Level Calc. Content Answer U OC3 4 C NC T7 1989P1Q (a) Find the equation of AB, the perpendicular bisector of the line y Q(1, 9) joing the points P( 3,1) and Q(1,9). A 4 (b)cisthecentreofacirclepassing throughpandq.giventhatqcis parallel to the y-axis, determine the C equation of the circle. 3 (c)thetangentsatpandqintersectat T. Write down (i)theequationofthetangentatq P( 3, 1) (ii) the coordinates of T. O B x Part Marks Level Calc. Content Answer U OC4 (a) 4 C CN G7 x +y =9 000PQ (b) 3 C CN G10 (x 1) + (y 4) =5 (c) C CN G11,G8 (i)y =9,(ii)T( 9,9) 1 ss: knowtousemidpoint pd: processgradientofpq 3 ss: knowhowtofindperp.gradient 4 ic: stateequ.ofline 5 ic: interpret paralleltoy-axis 6 pd: processradius 7 ic: stateequ.ofcircle 8 ic: interpretdiagram 9 ss: knowtouseequ.ofab 1 midpoint = ( 1,5) m PQ = ( 1) 3 m = 1 4 y 5 = 1 (x ( 1)) 5 y C =4 statedorimpliedby 7 6 radius =5orequiv. statedorimpliedby 7 7 (x 1) + (y 4) =5 8 y =9 9 T= ( 9,9) hsn.uk.net Page 10
11 0. Part Marks Level Calc. Content Answer U3 OC1 (a) C CR G P Q1 (b) 5 C CR G8 (c) C CR CGD hsn.uk.net Page 11
12 1. (a)bywritingsin3xassin(x +x),showthatsin3x =3sinx 4sin 3 x. 4 (b) Hence find sin 3 xdx. 4 Part Marks Level Calc. Content Answer U3 OC (a) C NC T8,T8 1995PQ9 (a) A/B NC T8,T8 (b) 4 A/B NC C3. Givenf(x) =cos x sin x,findf (x). 3 Part Marks Level Calc. Content Answer U3 OC 1 C NC C P1 Q19 A/B NC C1,C0 hsn.uk.net Page 1
13 3. The results of an experiment give rise to the graph shown. P (a)writedowntheequationofthelinein 1 8 termsofpandq. 3O Q ItisgiventhatP =log e pandq =log e q. (b)showthat pandqsatisfyarelationshipoftheform p = aq b,statingthe valuesofaandb. 4 Part Marks Level Calc. Content Answer U3 OC3 (a) A/B CR G3 P =0 6Q PQ11 (b) 4 A/B CR A33 a =6 05,b =0 6 1 ic: interpretgradient ic: stateequ.ofline 3 ic: interpretstraightline 4 ss: know how to deal with x of xlogy 5 ss: knowhowtoexpressnumberas log 6 ic: interpretsumoftwologs 1 m = =0 6 P =0 6Q +1 8 Method 1 3 log e p =0 6log e q log e q log e p =6 05q 0 6 Method lnp =lnaq b 3 lnp =lna+blnq 4 lnp =0 6lnq +1 8 statedorimplied by 5 or 6 5 lna =1 8 6 a =6 05,b =0 6 hsn.uk.net Page 13
14 4. The size of the human population, N, can be modelled using the equation N =N 0 e rt wheren 0 isthepopulationin006,tisthetimeinyearssince006, andristheannualrateofincreaseinthepopulation. (a) In 006 the population of the United Kingdom was approximately 61 million, with an annual rate of increase of 1 6%. Assuming this growth rate remains constant, what would be the population in 00? (b) In 006 the population of Scotland was approximately 5 1 million, with an annual rate of increase of 0 43%. Assuming this growth rate remains constant, how long would it take for Scotland s population to double in size? 3 Part Marks Level Calc. Content Answer U3 OC3 (a) B CR A30, A34 76 million 009 P Q6 (b) 3 A CR A30, A34 t = 161 years 1 ic: substituteintoequation pd: evaluateexponentialexpression 3 ic: interpretinfoandsubstitute 4 ss: convertexpo.equ.tolog.equ. 5 pd: process 1 61e million 3 10 =5 1e t t =ln 5 t =161 years [END OF QUESTIONS] hsn.uk.net Page 14
Prelim practice. Part Marks Level Calc. Content Answer U1 OC1 3 C CR G2 1992P1Q13
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