TEMASEK JUNIOR COLLEGE, SINGAPORE JC One Promotion Examination 2014 Higher 2

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TEMASEK JUNIOR COLLEGE, SINGAPORE JC Oe Promotio Eamiatio 04 Higher MATHEMATICS 9740 9 Septemer 04 Additioal Materials: Aswer paper 3 hours List of Formulae (MF5) READ THESE INSTRUCTIONS FIRST Write

1 TEMASEK JUNIOR COLLEGE, SINGAPORE JC Oe Promotio Eamiatio 04 Higher MATHEMATICS Septemer 04 Additioal Materials: Aswer paper 3 hours List of Formulae (MF5) READ THESE INSTRUCTIONS FIRST Write your Civics Group ad Name o all the work that you had i. Write i dark lue or lack pe o oth 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-eact umerical aswers correct to 3 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 epected to use a graphic calculator. Usupported aswers from a graphic calculator are allowed uless a questio specifically states otherwise. Where usupported aswers from a graphic 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. The umer of marks is give i rackets [ ] at the ed of each questio or part questio. At the ed of the eamiatio, faste all your work securely together. This documet cosists of 5 prited pages. TJC 04 [ Tur over TJC/MA9740/JCPromo04

2 Solve the iequality, givig your aswers i eact form. [4] (i) A graph with equatio y = f() udergoes trasformatio A followed y trasformatio B where A ad B are descried as follows: A: a traslatio of 5 uits i the egative directio of the -ais, B: a scalig parallel to the y-ais y factor. The equatio of the resultig graph is y l 5. Fid f(), showig your workigs clearly. [] (ii) The diagram elow shows a sketch of the graph y l 5. Eplai clearly why y is udefied whe 4 < < 0. [] (iii) If there are 3 distict solutios for the followig simultaeous equatios y l 5, y 3 5, where, fid the possile values of. [] 3 A sequece u 0, u, u, is such that 0 u ad u u 4 for =,, 3,. Prove y iductio that u for all o-egative itegers. [5] 4 The populatio of a city is P at time t years from a certai date. There is a 0% populatio growth ad five thousad people leave the coutry every year. Write dow a differetial equatio to relate P ad t. Give that the populatio was 5 millio at the start of year 000, epress P i terms of t. Fid the year i which the coutry will have a populatio of 0 millio. [6] TJC/MA9740/JCPromo04

3 5 Epress r 3 rr ( ) Deotig S 3 r r r 3 i partial fractios. [] rr ( ) 3, fid S i terms of. [3] Hece determie whether the series coverges. [] 6 (a) Usig a algeraic method, fid the eact value of 4 d. [3] () Sketch ad shade the fiite regio ouded y the curve y = +, the lies y = ad =, ad the y-ais. Fid the eact volume of the solid formed whe the regio is rotated radias aout the y-ais. [4] 7 The diagram shows the graph of is deoted y. 5 y 8. The root of the equatio 5 + 8= 0 (i) Fid the value of correct to 3 decimal places. [] The real umers satisfy the recurrece relatio (8 ) 5 for. (ii) Usig the result i (i), show that if the sequece coverges, it will coverge to. [] (iii) By cosiderig the graphs of y = ad y (8 ) 5 o the same diagram, or otherwise, prove that if 4 the (a) 0, (). [3] (iv) It is give that = 3. Use the results i part (iii) to otai a iequality relatig 0,,, ad 3. With the help of a graphic calculator, descrie the ehaviour of the sequece. [] TJC/MA9740/JCPromo04

4 4 8 Give that y si, show that d y dy y y 0 d d. [] By further differetiatio, fid the Maclauri s series of y i ascedig powers of up to ad icludig the term i 3. [4] Deduce the Maclauri s series of cos up to ad icludig the term i. [3] p q 9 The curve C has equatio y 5, 5 where p ad q are positive costats. If C has o turig poits, fid the coditio satisfied y p ad q. [4] It is give that q < p. (i) Show that C has a egative gradiet at all poits o the graph. [] (ii) Write dow the equatios of the asymptotes of C. [] (iii) Sketch C, givig the coordiates of the poits where the graph crosses the aes.[] 0 A rectagle is iscried i a ellipse y, with its four vertices eig i cotact 4 with the ellipse. Give that the legth of the rectagle is, show that the area of the rectagle, A, is 4. [] (a) Usig differetiatio, fid the maimum value of A. [5] () Give that at a particular istat, is icreasig at the rate of uits per secod ad the rate of chage of A is 4 uits per secod, fid the value of at this istat. [3] The fuctios f ad g are defied y 3 for 4 4 f( ) 4 for 4 g( ) e,, 0. (i) Sketch the graph of f. [] (ii) Defie f ( ) i a similar form. [4] (iii) Fid the set of values of for which ff ( ) f f( ). [] (iv) Eplai why the composite fuctio gf does ot eist. [] TJC/MA9740/JCPromo04

5 A curve has parametric equatios give y 3 5 acos t, y si t, where 0 t ad a ad are positive costats. (i) Fid d y i terms of a, ad t. [] d (ii) The taget to the curve at the poit a, cuts the y-ais at T. Fid the eact coordiates of T i terms of. [3] (iii) A sketch of the curve is show elow ad regio R is the fiite regio eclosed etwee the curve ad the aes. y O a Show that the area of R ca e writte i the form t f dt where ad ad f are to e determied. By usig the sustitutio u = si t or otherwise, fid the eact area of R. [5] 3 (a) The sum of the first terms of a series is give y the epressio 5 5. Show that the series is a geometric series. [3] Hece, fid the least value of k such that the sum of the series from the k th term owards is less tha 50. [4] () Nicholas ad his father start a race at the same time. Nicholas hops at a costat distace of 0.4 m. His father makes a first hop of m ad each susequet hop is 0.05 m less tha that of the previous hop. Assume that at the start of the race, Nicholas is d m i frot of his father ad that they start each hop at the same time. Fid the miimum value of d such that Nicholas s father will ot e ale to catch up with him. Leave your aswer correct to decimal place. [4] TJC/MA9740/JCPromo04

MATHEMATICS 9740 (HIGHER 2)

MATHEMATICS 9740 (HIGHER 2) VICTORIA JUNIOR COLLEGE PROMOTIONAL EXAMINATION MATHEMATICS 970 (HIGHER ) Frida 6 Sept 0 8am -am hours Additioal materials: Aswer Paper List of Formulae (MF5) READ THESE INSTRUCTIONS FIRST Write our ame