remainder theorm Part Marks Level Calc. Content Answer U2 OC1 (a) 3 C CN A21 k = P2Q1 (b) 2 C CN A22 x = 2, 1 2,1
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1 remainder theorm [SQA] 1. (a) Giventhatx +2isafactorof2x 3 +x 2 +kx +2,findthevalueofk. 3 (b)hencesolvetheequation2x 3 +x 2 +kx +2 =0whenktakesthisvalue. 2 (a) 3 C CN A21 k = P2Q1 (b) 2 C CN A22 x = 2, 1 2,1 1 ss: use synth division or f(evaluation) 2 pd: process 3 pd: process 4 ss: findaquadraticfactor 5 pd: process 1 f( 2) =2( 2) ( 2) 3 + ( 2) 2 2k +2 3 k = 5 4 2x 2 3x +1 or 2x 2 +3x 2 or x 2 +x 2 5 (2x 1)(x 1)or (2x 1)(x +2)or (x +2)(x 1) andx= 2, 1 2,1
2 2. Thediagramshowspartofthegraphofthe curvewithequationy =2x 3 7x 2 +4x +4. (a)findthex-coordinateofthemaximum turning point. 5 y y =f(x) (b)factorise2x 3 7x 2 +4x (c)statethecoordinatesofthepointaand A hence find the values of x for which O (2,0) x 2x 3 7x 2 +4x +4 <0. 2 (a) 5 C NC C8 x = P2Q3 (b) 3 C NC A21 (x 2)(2x +1)(x 2) (c) 2 C NC A6 A( 1 2,0),x< ss: knowtodifferentiate 2 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) = x 2 14x x 2 14x +4 =0 4 (3x 1)(x 2) 5 x = x 2 3x 2 8 (x 2)(2x +1)(x 2) 9 A( 1 2,0) 10 x < 1 2
3 3.Functionsf,gandharedefinedonthesetofrealnumbersby f(x) =x 3 1 g(x) =3x +1 h(x) =4x 5. (a)findg(f(x)). 2 (b)showthatg(f(x)) +xh(x) =3x 3 +4x 2 5x 2. 1 (c) (i)showthat (x 1)isafactorof3x 3 +4x 2 5x 2. (ii)factorise3x 3 +4x 2 5x 2fully. 5 (d)hencesolveg(f(x)) +xh(x) =0. 1 (a) 2 C CN A4 3(x 3 1) P2Q2 (b) 1 C CN A6 proof (c) 5 C CN A21 (x 1)(3x +1)(x +2) (d) 1 C CN A22 2, 1 3,1 1 ic: interpretnotation 2 ic: completeprocess 3 ic: substituteandcomplete 4 ss: knowtousex=1 5 pd: completeevaluation 6 ic: stateconclusion 7 ic: findquadraticfactor 8 pd: factorisecompletely 9 ic: interpretandsolveequationin (d) 1 g(x 3 1) 2 3(x 3 1) (x 3 1) +1+x(4x 5) =3x 3 +4x 2 5x 2 4 evaluatingatx= =0 6 (x 1)isafactor 7 (x 1)(2x 2 +7x +2) 8 (x 1)(3x +1)(x +2) 9 2, 1 3,1
4 4. (a) (i)showthat (x 1)isafactoroff(x) =2x 3 +x 2 8x +5. (ii) Hence factorise f(x) fully. 5 (b)solve2x 3 +x 2 8x +5 =0. 1 (c)thelinewithequationy =2x 3isatangenttothecurvewithequation y =2x 3 +x 2 6x +2atthepointG. Find the coordinates of G. 5 (d)thistangentmeetsthecurveagainatthepointh. Write down the coordinates of H. 1 (a) 5 C CN A21 (x 1)(x 1)(2x +5) 2010P1Q22 (b) 1 C CN A22 x =1, 5 2 (c) 5 C CN A23 (1, 1) (d) 1 C CN A23 ( 5 2, 8) 1 ss: knowtousex=1 2 ic: completeevaluation 3 ic: stateconclusion 4 pd: findquadraticfactor 5 pd: factorisecompletely 6 ic: statesolutions 7 ss: sety curve =y line 8 ic: expressinstandardform 9 ss: comparewith(a)orfactorise 10 ic: identifyx G 11 pd: evaluatey G 12 pd: statesolution 1 evaluatingatx= =0 3 (x 1)isafactor 4 (x 1)(2x 2 +3x 5) 5 (x 1)(x 1)(2x +5) 6 x =1andx= x 3 +x 2 6x +2 =2x 3 8 2x 3 +x 2 8x +5 =0 9 (x 1)(x 1)(2x +5) =0 10 x =1 11 y = 1 12 ( 5 2, 8) [SQA] 5. Factorisefully2x 3 +5x 2 4x C NC A P1Q2
5 6. (a) 1 C NC A P1 Q1 (b) 3 C NC A21 [SQA] 7. Findpif (x +3)isafactorofx 3 x 2 +px C CN A P1Q1 [SQA] 8. Whenf(x) =2x 4 x 3 +px 2 +qx +12isdividedby (x 2),theremainderis114. Onefactoroff(x)is (x +1). Findthevaluesofpandq. 5 5 C CN A P1Q6
6 9. Findkifx 2isafactorofx 3 +kx 2 4x C CN A P1Q3 [SQA] 10. Onerootoftheequation2x 3 3x 2 +px +30 =0is 3. Findthevalueofpandtheotherroots. 4 4 C NC A P1Q7 [SQA] 11. (a) 2 C NC A P1 Q2 (b) 2 C NC A21
7 12. Expressx 4 xinitsfullyfactorisedform. 4 4 C NC A P1Q7 [SQA] 13. (a) 2 C NC A P1 Q5 (b) 3 C NC A21 [SQA] 14. Expressx 3 4x 2 7x +10initsfullyfactorisedform. 4 4 C NC A P1Q2
8 15. (a)thefunctionfisdefinedbyf(x) =x 3 2x 2 5x +6. Thefunctiongisdefinedbyg(x) =x 1. Showthatf ( g(x) ) =x 3 5x 2 +2x (b)factorisefullyf ( g(x) ). 3 (c)thefunctionkissuchthatk(x) = 1 f ( g(x) ). Forwhatvaluesofxisthefunctionknotdefined? 3 (a) 4 C NC A P2 Q6 (b) 3 C NC A21 (c) 2 C NC A1
9 16. (a) 3 C NC A P2 Q1 (b) 2 C NC A6 (c) 6 C NC C8
10 17. (a) 3 C NC A P2 Q1 (b) 4 C NC A21
11 18. 8 C CN A21,C8 1993P2Q1
12 19. (a) 7 C NC A23, A P2 Q7 (b) 1 A/B NC A24
13 20. The diagram shows a sketch of the graphofy =x 3 3x 2 +2x. (a)find the equation of the tangent to this curve at the pointwherex =1. O 5 (b)thetangentatthepoint (2,0) x hasequationy=2x 4.Find the coordinates of the point where this tangent meets the curve again. 5 y y =x 3 3x 2 +2x (a) 5 C CN C5 x +y =1 2000P2Q1 (b) 5 C CN A23,A22,A21 ( 1, 6) 1 ss: knowtodifferentiate 2 pd: differentiatecorrectly 3 ss: knowthatgradient =f (1) 4 ss: knowthaty-coord =f(1) 5 ic: stateequ.ofline 6 ss: equateequations 7 pd: arrangeinstandardform 8 ss: knowhowtosolvecubic 9 pd: process 10 ic: interpret 1 y = x 2 6x +2 3 y (1) = 1 4 y(1) =0 5 y 0 = 1(x 1) 6 2x 4 =x 3 3x 2 +2x 7 x 3 3x 2 +4 = identifyx = 1fromworking 10 ( 1, 6)
14 21. (a) (i)showthatx =1isarootofx 3 +8x 2 +11x 20 =0. (ii)hencefactorisex 3 +8x 2 +11x 20fully. 4 (b)solvelog 2 (x +3) +log 2 (x 2 +5x 4) =3. 5 Part Marks Level Calc. Content Answer U3 OC3 (a) 4 C CN A21 (x 1)(x +4)(x +5) 2009P2Q3 (b) 5 B CN A32 x =1 1 ss: know and use f(a) =0 aisaroot 2 ic: starttofindquadraticfactor 3 ic: completequadraticfactor 4 pd: factorisefully 5 ss: useloglaws 6 ss: know to & convert to exponential form 7 ic: writecubicinstandardform 8 pd: solvecubic 9 ic: interpretvalidsolution 1 f(1) = =0sox =1 isaroot 2 (x 1)(x 2 3 (x 1)(x 2 +9x +20) 4 (x 1)(x +4)(x +5) 5 log 2 ( (x +3)(x 2 +5x 4) ) 6 (x +3)(x 2 +5x 4) =2 3 7 x 3 +8x 2 +11x 20 =0 8 x =1orx = 4orx = 5 9 x =1only [END OF QUESTIONS]
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