Higher Mathematics. Exam Revision. Questions marked [SQA] c SQA All others c Higher Still Notes. hsn.uk.net Page 1

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1 Exam Revision hsn.uk.net Page 1

2 1. A quadrilateral has vertices A( 1, 8), B(7, 12), C(8, 5) and D(2, 3) as shown in the diagram. y B A E C O x D (a) Find the equation of diagonal BD. 2 (b)theequationofdiagonalacisx +3y =23. Find the coordinates of E, the point of intersection of the diagonals. 3 (c) (i) Find the equation of the perpendicular bisector of AB. (ii) Show that this line passes through E. 5 Part Marks Level Calc. Content Answer U1 OC1 (a) 2 C CN G3,G2 y 12 =3(x 7) 2011P1Q21 (b) 3 C CN G8 E(5, 6) (ci) 4 C CN G7 y 10 = 2(x 3) (cii) 1 C CN A6 proof 1 pd: findgradientofbd 2 ic: stateequationofbd 3 ss: start solution of simultaneous eqs 4 pd: solveforonevariable 5 pd: solveforsecondvariable 6 ss: knowandfindmidpointofab 7 pd: findgradientofab 8 ic: interpretperpendiculargradient 9 ic: stateequationofperp.bisector 10 ic: justificationofpointonline hsn.uk.net Page orequiv. 2 y ( 3) =3(x 2) 3 3x y =9andx +3y =23 4 x =5ory =6 5 y =6orx=5 6 (3,10) orequiv orequiv 9 y 10 = 2(x 3) 10 whenx =5,y = =6

3 2. (a)showthatthefunction f(x) = 2x 2 +8x 3 canbewrittenintheform f(x) =a(x +b) 2 +cwherea,bandcareconstants. 3 (b) Hence, or otherwise, find the coordinates of the turning point of the function f. 1 Part Marks Level Calc. Content Answer U1 OC2 (a) 3 C NC A P1 Q9 (b) 1 C NC A6 3. Calculate, to the nearest degree, the angle between the x-axis and the tangent to thecurvewithequationy =x 3 4x 5atthepointwherex =2. 4 Part Marks Level Calc. Content Answer U1 OC3 4 C NC C4,G2 1989P1Q13 4. Acurvehasequationy =x 16,x >0. x Findtheequationofthetangentatthepointwherex =4. 6 Part Marks Level Calc. Content Answer U1 OC3 6 C CN C4,C5 y =2x P2Q2 1 ic: findcorrespondingy-coord. 2 ss: expressinstandardform 3 ss: starttodifferentiate 4 pd: diff.fractionalnegativepower 5 ss: findgradientoftangent 6 ic: writedownequ.oftangent 1 (4, 4) statedorimpliedby x dy dx = x m x=4 =2 6 y ( 4) =2(x 4) hsn.uk.net Page 3

4 5. (a)asequenceisdefinedbyu n+1 = 1 2 u nwithu 0 = 16. Writedownthevaluesofu 1 andu 2. 1 (b)asecondsequenceisgivenby4,5,7,11,... Itisgeneratedbytherecurrencerelationv n+1 =pv n +qwithv 1 =4. Findthevaluesofpandq. 3 (c)eitherthesequencein(a)orthesequencein(b)hasalimit. (i) Calculate this limit. (ii)whydoestheothersequencenothavealimit? 3 Part Marks Level Calc. Content Answer U1 OC4 (a) 1 C CN A11 u 1 =8,u 2 = P2Q3 (b) 3 C CN A11,A10 p =2,q = 3 (c) 3 C CN A12,A13 (i)l =0 (ii)outside 1 <p <1 1 pd: findtermsofasequence 2 ic: interpretsequence 3 ss: solveforonevariable 4 pd: statesecondvariable 5 ss: knowhowtofindvalidlimit 6 pd: calculateavalidlimitonly 7 ic: statereason 1 u 1 =8andu 2 = 4 2 e.g.4p +q =5and5p +q =7 3 p =2orq = 3 4 q = 3orp=2 5 l = 1 2 lorl = 0 1 ( 1 2 ) 6 l =0 7 outsideinterval 1 <p <1 6. Giventhatkisarealnumber,showthattherootsoftheequationkx 2 +3x +3 =k are always real numbers. 5 Part Marks Level Calc. Content Answer U2 OC1 1 C NC A P1 Q18 4 A/B NC A17 hsn.uk.net Page 4

5 7. Whenf(x) =2x 4 x 3 +px 2 +qx +12isdividedby (x 2),theremainderis114. Onefactoroff(x)is (x +1). Findthevaluesofpandq. 5 Part Marks Level Calc. Content Answer U2 OC1 5 C CN A P1Q6 hsn.uk.net Page 5

6 8. The diagram shows a sketch of the graphs of y = 5x 2 15x 8 and y =x 3 12x +1. ThetwocurvesintersectatAandtouchatB,i.e.atBthecurveshaveacommon tangent. y y =x 3 12x +1 A O x B y =5x 2 15x 8 (a) (i)findthex-coordinatesofthepointofthecurveswherethegradientsare equal. 4 (ii)by considering the corresponding y-coordinates, or otherwise, distinguish geometrically between the two cases found in part(i). 1 (b)thepointais ( 1,12)andBis (3, 8). Find the area enclosed between the two curves. 5 Part Marks Level Calc. Content Answer U2 OC2 (ai) 4 C NC C4 x = 1 3andx=3 2000P1Q4 (aii) 1 C NC CGD parallel and coincident (b) 5 C NC C ss: knowtodiff.andequate 2 pd: differentiate 3 pd: formequation 4 ic: interpretsolution 5 ic: interpretdiagram 6 ss: knowhowtofindareabetween curves 7 ic: interpretlimits 8 pd: formintegral 9 pd: processintegration 10 pd: processlimits hsn.uk.net Page 6 1 findderivativesandequate 2 3x 2 12and10x x 2 10x +3 =0 4 x =3,x= tangentsat x = 1 3 are parallel, at x =3coincident 6 (cubic parabola) or (cubic) (parabola) dx 8 (x 3 5x 2 +3x +9)dxorequiv. 9 [ 1 4 x4 5 3 x x2 +9x ] 3 1 orequiv

7 9. Acurveforwhich dy dx =3x2 +1passesthroughthepoint ( 1,2). Expressyintermsofx. 4 Part Marks Level Calc. Content Answer U2 OC2 4 C NC C P1Q4 10. Solvetheequationcos2x +cosx =0,0 x< Part Marks Level Calc. Content Answer U2 OC3 5 A/B CR T P1Q Solvetheequationsin2x +sinx =0,0 x< Part Marks Level Calc. Content Answer U2 OC3 5 C NC T P1 Q10 hsn.uk.net Page 7

8 12. Giventhattan α = 11 3,0 < α < π 2,findtheexactvalueofsin2α. 3 Part Marks Level Calc. Content Answer U2 OC3 3 C NC T8 1995P1Q (a)usingthefactthat 7π 12 = π 3 + π 4,findtheexactvalueofsin ( 7π 12 ). 3 (b)showthatsin(a +B) +sin(a B) =2sinAcosB. 2 (c) (i)express 12 π intermsof π 3 and π 4. (ii)henceorotherwisefindtheexactvalueofsin ( ) ( ) 7π 12 +sin π12. 4 Part Marks Level Calc. Content Answer U2 OC (a) 3 C NC T8, T P1Q24 2 (b) 2 C CN T8 proof π (c) 3 B NC T11 12 = π 3 π 4 (c) 1 C NC T or ss: expandcompoundangle 2 ic: substituteexactvalues 3 pd: processtoasinglefraction 4 ic: startproof 5 ic: completeproof 6 ss: identifysteps 7 ic: startprocess(identify A & B ) 8 ic: substitute 9 pd: process 1 sin π 3 cos π 4 +cos π 3 sin π orequivalent 4 sinacosb+cosasinb+ 5 +sinacosb cosasinb and complete 6 π 12 = π 3 π or hsn.uk.net Page 8

9 14. CirclePhasequationx 2 +y 2 8x 10y +9 =0. CircleQhascentre ( 2, 1) andradius2 2. (a) (i)showthattheradiusofcirclepis4 2. (ii)henceshowthatcirclespandqtouch. 4 (b)findtheequationofthetangenttothecircleqatthepoint ( 4,1). 3 (c)thetangentin(b)intersectscirclepintwopoints.findthex-coordinatesof thepointsofintersection,expressingyouanswersintheforma ±b 3. 3 Part Marks Level Calc. Content Answer U2 OC4 (a) 2 C CN G9 proof 2001 P1 Q11 (a) 2 A/B CN G14 (b) 3 C CN G11 y =x +5 (c) 3 C CN G12 x =2 ±2 3 1 ic: interpretcentreofcircle(p) 2 ss: findradiusofcircle(p) 3 ss: findsumofradii 4 pd: comparewithdistancebetween centres 5 ss: findgradientofradius 6 ss: usem 1 m 2 = 1 7 ic: stateequationoftangent 8 ss: substitutelinearintocircle 9 pd: expressinstandardform 10 pd: solve(quadratic)equation 1 C P = (4,5) 2 r P = = 32 =4 2 3 r P +r Q = =6 2 4 C P C Q = = 6 2and so touch 5 m r = 1 6 m tgt = +1 7 y 1 =1(x +4) 8 x 2 +(x+5) 2 8x 10(x+5)+9 =0 9 2x 2 8x 16 =0 10 x =2 ±2 3 hsn.uk.net Page 9

10 15. A box in the shape of a cuboid is designed with circles of different sizes on each face. The diagram shows three of the circles, where the origin represents oneofthecornersofthecuboid.the z centresofthecirclesarea(6,0,7), B(0,5,6)andC(4,5,0). B FindthesizeofangleABC. 7 A O y x C Part Marks Level Calc. Content Answer U3 OC1 5 C CR G17,G16,G P2Q4 2 A/B CR G26,G ss: use BA. BC BA BC 2 ic: statevectore.g. BA 3 ic: stateaconsistentvectore.g. BC 4 pd: process BA 5 pd: process BC 6 pd: processscalarproduct 7 pd: findangle 1 use BA. BC BA BC statedorimpliedby 7 2 BA 6 = BC 4 = BA = 62 5 BC = 52 6 BA. BC =18 7 A BC =71 5 hsn.uk.net Page 10

11 16. (a) Roadmakers look along the tops of a set C of T-rods to ensure that straight sections of road are being created. Relative to suitableaxesthetopleftcornersofthe B T-rods are the points A( 8, 10, 2), B( 2, 1,1)andC(6,11,5). A Determine whether or not the section of roadabchasbeenbuiltinastraightline. 3 C (b)afurthert-rodisplacedsuchthatdhas coordinates (1, 4, 4). Show that DB is perpendicular to AB. B 3 A D Part Marks Level Calc. Content Answer U3 OC1 (a) 3 C CN G23 the road ABC is straight 2001 P1 Q3 (b) 3 C CN G27, G17 proof 1 ic: interpretvector(e.g. AB) 2 ic: interpretmultipleofvector 3 ic: completeproof 4 ic: interpretvector(i.e. BD) 5 ss: staterequirementforperpend. 6 ic: completeproof hsn.uk.net Page 11 or 1 e.g. AB 6 = e.g. BC = 12 = 4 3AB or 4 2 AB =3 3 andbc 2 = a common direction exists and a common point exists, so A, B, C collinear 4 BD 3 = AB. BD =0 6 AB. BD = =0 5 AB. BD = AB. BD =0soABisatrightanglesto BD

12 17. (a)showthat (cosx +sinx) 2 =1 +sin2x. 1 (b) Hence find (cosx+sinx) 2 dx. 3 Part Marks Level Calc. Content Answer U3 OC2 (a) 1 C NC T P1 Q19 (b) 3 A/B NC C (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 hsn.uk.net Page 12

13 19. Part Marks Level Calc. Content Answer U3 OC3 (a) 2 C CR A30, A P2 Q7 (a) 3 A/B CR A30, A34 (b) 1 C CR A30 (b) 3 A/B CR A34 hsn.uk.net Page 13

14 20. Part Marks Level Calc. Content Answer U3 OC4 (a) 4 C CR T P2 Q7 (b) 3 C CR T Differentiatesin2x + 2 withrespecttox. 4 x Part Marks Level Calc. Content Answer U3 OC2 2 C NC C3 1989P1Q10 2 A/B NC C20 [END OF QUESTIONS] hsn.uk.net Page 14