CATHOLIC JUNIOR COLLEGE Geeral Certificate of Educatio Advaced Level Higher JC Prelimiary Examiatio MATHEMATICS 9740/0 Paper 4 Aug 06 hours Additioal Materials: List of Formulae (MF5) Name: Class: READ THESE INSTRUCTIONS FIRST Write your ame ad class o all the work you had i. Write i dark blue or black pe o both sides of the paper. You may use a soft pecil for ay diagrams or graphs. Do ot use staples, paper clips, highlighters, glue or correctio fluid. Aswer all the questios. Give o-exact umerical aswers correct to sigificat figures, or decimal place i the case of agles i degrees, uless a differet level of accuracy is specified i the questio. You are expected to use a graphig calculator. Usupported aswers from a graphig calculator are allowed uless a questio specifically states otherwise. Where usupported aswers from a graphig calculator are ot allowed i a questio, you are required to preset the mathematical steps usig mathematical otatios ad ot calculator commads. You are remided of the eed for clear presetatio i your aswers. At the ed of the examiatio, arrage your aswers i NUMERICAL ORDER. The umber of marks is give i brackets [ ] at the ed of each questio or part questio. Questio 4 5 6 7 8 9 0 Total Marks Total 5 5 5 7 7 8 9 9 5 7 00 This documet cosists of 5 prited pages, icludig the cover page. 9740/0/Prelims/06
[I this questio, sketches of the give graphs are ot draw to scale] The graphs of y f x y f x are give below. ad y f x y f x O separate diagrams, draw sketches of the graphs of (a) y f x (b) y f x,, [] statig the equatios of ay asymptotes ad the coordiates of ay poits of itersectio with the axes. The vectors a ad b are give by a 4i 6pj 8k ad b i j 4pk, where p 0. It is give that a b. Fid p. [] Give a geometrical iterpretatio of ba. [] b (iii) Usig the value of p foud i part, fid the exact value of b ba. [] The cubic equatio x ax bx c 0, where a, b ad c are costats, has roots + i ad. Oe JC studet remarked that the third root is i. State a ecessary assumptio the studet made i order that the remark is true. [] Give that the assumptio i part holds, fid the values of a, b ad c. 9740/0/Prelims/06
4 A closed cotaier is made up of a cylider of base radius r cm ad height h cm, ad a hemispherical top with the same radius r. It is iscribed withi a fixed right circular coe of base radius 5 cm ad height cm, as show i the diagram below. By usig similar triagles, show that h r. 5 Determie the exact rage of possible values of legth r. Fid the total volume V of the closed cotaier i terms of r. By differetiatio, fid the exact value of r that produces the maximum cotaier volume V, as r varies. [Volume of a sphere with radius R is 4 π R.] 5 A sequece,,, u u u satisfies the recurrece relatio u u ( ) (a) Give that u u, for. ( )! (b) Give that u, for., use the method of mathematical iductio to prove that a, where a is ay costat. Write dow u, u ad u 4 i terms of a. Hece or otherwise, fid u i terms of a. 9740/0/Prelims/06
4 6 Give that r A( r ) Br, fid the costats A ad B. [] Use the method of differeces to fid (iii) Hece fid the value of r r. r r r. r 7 Give that y l( x), fid the exact rage of values of x for y to be well defied. Show that d y y e d x y. (iii) Hece, fid the Maclauri s series for l( x), up to ad icludig the term i x. (iv) Verify that the same result is obtaied usig the stadard series expasios give i the List of Formulae (MF5). [] [] 8 Do ot use a calculator i aswerig this questio. It is give that complex umbers z ad z are the roots of the equatio such that arg(z ) > arg(z ). Fid exact expressios of z ad z i the form r 0 ad π π. Fid the complex umber z 6z6 0 i re, where 4 z iz i exact polar form. (iii) Fid the smallest positive iteger such that z is a positive real umber. [] 9 By usig the substitutio u x, fid The regio R is bouded by the curve Fid (a) (b) y x d x. [5] x x ad the lies x = 8 ad y =. x B the exact area of R, simplifyig your aswer i the form A l B [5] where A ad B are itegers to be determied, the volume of the solid geerated whe R is rotated π radias about the x-axis, givig your aswer correct to decimal places. 9740/0/Prelims/06
0 The plae p passes through the poits A, B ad C with coordiates 0,,,,,4,,0 respectively. 5 ad Show that a cartesia equatio of the plae p is x y z =. The lie l has equatio r 0,. 4 Fid the acute agle betwee l ad p. [] The poit Q has positio vector 5i jk. (iii) Show that Q lies o the lie l. [] (iv) It is give that a variable poit R lies o the plae p ad is at a distace of 45 from the poit Q. Fid the foot of perpedicular from the poit Q to the plae p ad hece describe geometrically the locus of R. [6] (v) Fid a vector equatio of the lie which is a reflectio of the lie l i plae p. The fuctio f is defied by where k is a positive costat. x k f : x, x, x, x Sketch the graph of y f x, statig the equatios of ay asymptotes ad the coordiates of ay poits where the curve crosses the x ad y axes. Describe fully a sequece of trasformatios which would trasform the curve oto (iii) Fid x y f. f (iv) Hece or otherwise, fid Fid the value of The fuctio g is defied by where a is a real costat. i a similar form ad write dow the rage of f 07 f. y x f., leavig your aswer i terms of k. g : x a x, x, x, (v) Give that fg exists, write dow a iequality for a ad explai why gf does ot exist. THE END 9740/0/Prelims/06