Exam Revision. y B A E. Find the coordinates of E, the point of intersection of the diagonals. 3

Transcription

1 Eam Revision 1. quadrilateral has vertices ( 1, 8), B(7, 12), C(8, 5) and D(2, 3) as shown in the diagram. y B E C O D (a) Find the equation of diagonal BD. 2 (b)theequationofdiagonalcis +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 B. (ii) Show that this line passes through E (a)showthatthefunction f() = canbewrittenintheform f() =a( +b) 2 +cwherea,bandcareconstants. 3 (b) Hence, or otherwise, find the coordinates of the turning point of the function f Calculate, to the nearest degree, the angle between the -ais and the tangent to thecurvewithequationy = 3 4 5atthepointwhere =2. 4 hsn.uk.net Page 1 Questions marked c SQ ll others c Higher Still Notes

2 4. curvehasequationy = 16, >0. Findtheequationofthetangentatthepointwhere = (a)sequenceisdefinedbyu n+1 = 1 2 u nwithu 0 = 16. Writedownthevaluesofu 1 andu 2. 1 (b)secondsequenceisgivenby4,5,7,11,... Itisgeneratedbytherecurrencerelationv n+1 =pv n +qwithv 1 =4. Findthevaluesofpandq. 3 (c)eitherthesequencein(a)orthesequencein(b)hasalimit. (i) Calculate this limit. (ii)whydoestheothersequencenothavealimit? 3 6. Giventhatkisarealnumber,showthattherootsoftheequationk =k are always real numbers Whenf() = p 2 +q +12isdividedby ( 2),theremainderis114. Onefactoroff()is ( +1). Findthevaluesofpandq. 5 hsn.uk.net Page 2 Questions marked c SQ ll others c Higher Still Notes

3 8. The diagram shows a sketch of the graphs of y = and y = ThetwocurvesintersectatandtouchatB,i.e.atBthecurveshaveacommon tangent. y y = O B y = (a) (i)findthe-coordinatesofthepointofthecurveswherethegradientsare equal. 4 (ii)by considering the corresponding y-coordinates, or otherwise, distinguish geometrically between the two cases found in part(i). 1 (b)thepointis ( 1,12)andBis (3, 8). Find the area enclosed between the two curves curveforwhich dy d =32 +1passesthroughthepoint ( 1,2). Epressyintermsof Solvetheequationcos2 +cos =0,0 < Solvetheequationsin2 +sin =0,0 < Giventhattan α = 11 3,0 < α < π 2,findtheeactvalueofsin2α. 3 hsn.uk.net Page 3 Questions marked c SQ ll others c Higher Still Notes

4 13. (a)usingthefactthat 7π 12 = π 3 + π 4,findtheeactvalueofsin ( 7π 12 ). 3 (b)showthatsin( +B) +sin( B) =2sincosB. 2 (c) (i)epress 12 π intermsof π 3 and π 4. (ii)henceorotherwisefindtheeactvalueofsin ( ) ( ) 7π 12 +sin π CirclePhasequation 2 +y y +9 =0. CircleQhascentre ( 2, 1) andradius2 2. (a) (i)showthattheradiusofcirclepis4 2. (ii)henceshowthatcirclespandqtouch. 4 (b)findtheequationofthetangenttothecircleqatthepoint ( 4,1). 3 (c)thetangentin(b)intersectscirclepintwopoints.findthe-coordinatesof thepointsofintersection,epressingyouanswersintheforma ±b bo 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 centresofthecirclesare(6,0,7), B(0,5,6)andC(4,5,0). B FindthesizeofangleBC. 7 O y C hsn.uk.net Page 4 Questions marked c SQ ll others c Higher Still Notes

5 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 suitableaesthetopleftcornersofthe B T-rods are the points ( 8, 10, 2), B( 2, 1,1)andC(6,11,5). Determine whether or not the section of roadbchasbeenbuiltinastraightline. 3 C (b)furthert-rodisplacedsuchthatdhas coordinates (1, 4, 4). Show that DB is perpendicular to B. B 3 D 17. (a)showthat (cos +sin) 2 =1 +sin2. 1 (b) Hence find (cos+sin) 2 d (a) (i)showthat =1isarootof =0. (ii)hencefactorise fully. 4 (b)solvelog 2 ( +3) +log 2 ( ) =3. 5 hsn.uk.net Page 5 Questions marked c SQ ll others c Higher Still Notes

6 Differentiatesin2 + 2 withrespectto. 4 [END OF QUESTIONS] hsn.uk.net Page 6 Questions marked c SQ ll others c Higher Still Notes