NATIONAL JUNIOR COLLEGE SENIOR HIGH 1 PROMOTIONAL EXAMINATIONS Higher 2
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1 NATIONAL JUNIOR COLLEGE SENIOR HIGH PROMOTIONAL EXAMINATIONS Higher MATHEMATICS September 06 hours Additioal Materials: Aswer Paper List of Formulae (MF6) Cover Sheet READ THESE INSTRUCTIONS FIRST Write our ame, registratio umber, subject tutorial group, o all the work ou had i. Write i dark blue or black pe o both sides of the paper. You ma use a soft pecil for diagrams or graphs. Do ot use paper clips, highlighters, glue or correctio fluid. Aswer all the questios. Give o-eact umerical aswers correct to sigificat figures, or decimal place i the case of agles i degrees, uless a differet level of accurac is specified i the questio. You are epected to use a approved graphig calculator. Usupported aswers from a graphig calculator are allowed uless a questio specificall states otherwise. Where usupported aswers from a graphig calculator are ot allowed i a questio, ou are required to preset the mathematical steps usig mathematical otatios ad ot calculator commads. You are remided of the eed for clear presetatio i our aswers. At the ed of the eamiatio, faste all our work securel together. The umber of marks is give i the brackets [ ] at the ed of each questio or part questio. This documet cosists of 6 prited pages. Natioal Juior College Page of 6 [Tur_over
2 I basketball, poits are scored b a successful free throw or a field goal. A successful free throw is worth poit while a field goal is worth either or poits. A attempt ca either be a free throw attempt or a field goal attempt. I the recetl cocluded basketball seaso, Stephe had 695 attempts i total. 60% of these attempts were successful ad he scored a total of 97 poits. 40% of his successful free throws ad 5% of his field goal poits were scored durig the secod half of the seaso, where Stephe scored a total of 54 poits. Calculate the umber of -poit field goals durig the seaso. [5] A clider is iscribed i a coe as show i the diagram below. The coe has a height of 8 cm ad a fied radius of R cm. The clider has a radius of cm ad a height of h cm. Show that the volume of the clider is maimised whe kr, where k is a costat to be determied. [5] A curve C has equatio. (i) Sketch the curve C, labellig a poits where C crosses the - ad - aes, the coordiates of the turig poit ad the equatios of the asmptotes. [] (ii) State the rage of values of for which C is cocave dowwards. [] Hece, solve the iequalit e. [] Page of 6 [Tur_over
3 4 f ( ) O, (,0) (7,0) The diagram above shows the curve 4 f. The curve crosses the -ais at (0,0), (,0) ad (7,0). It has a turig poit at, ad the lies, ad 4 are asmptotes to the curve. Sketch, o separate diagrams, the graphs of (i) f, [] (ii) f ( ). [] Label clearl the coordiates of a turig poits, the equatios of a asmptotes ad the poits where the graphs itersect the aes where appropriate. 5 Relative to the origi O, the poits A, B ad C have positio vectors give respectivel b i 4j + k, i +j k ad ( + p)i + ( p)j + pk, where p is real. (a) Determie the agle AOB. [] (b) Fid the values of p for which OC = AB. [] (c) The poit Q is o AB produced such that AB: BQ is :7. Fid the positio vector of the poit Q ad the uit vector i the directio of OQ. Leave our aswers i eact form. [] 6 Prove b the method of differeces that r l l l r rr. [] Hece, fid the least value of iteger k such that r l 0.0 rk rr. [5] Page of 6 [Tur_over
4 7 The diagram below shows a shaded regio which is bouded b the curve C with equatio 4 sec ta ad the lie L with equatio. L C O (i) Verif that the poit, lies o both C ad L. [] 4 (ii) Fid the eact volume of the solid geerated b revolvig the shaded regio about the - ais through. Epress our aswer i the form 4 a b, where a ad b are itegers. [6] 8 I a chemical laborator, the amout of substace X i a chemical reactio is beig measured. The mass of substace X, i g at a time t miutes after the start of the chemical reactio is deoted b ad it satisfies the differetial equatio d d t. The mass of substace X is.5 g at the start of the chemical reactio. (i) Fid the particular solutio, obtaiig a epressio for i terms of t. [5] (ii) What happes to the amout of substace X i the log ru? [] (iii) Sketch a graph to show how the amout of substace X varies with time, labellig the poit where the graph itersects the ais ad the equatio of the asmptote. [] 9 The fuctios f ad g are defied b f :, for, 0, g : cos 4, for,. (i) Sketch the graph of f( ) ad state the eact rage of f. [] (ii) Fid g ( ) ad write dow the eact domai ad rage of g. [4] (iii) Show that the composite fuctio fg eists ad fid the rage of fg. [4] Page 4 of 6 [Tur_over
5 0 (a) Joh owes the bak $0 000 o Jauar 06. The bak offers him two tpes of fiacial schemes to repa his debt. Scheme : At the start of ever moth, iterest is added to the amout he owes at a fied rate of 0.4% of the outstadig amout. At the ed of ever moth, Joh makes a pamet of $500. Scheme : At the start of ever moth, a fied iterest of $75 is charged to the amout he owes. At the ed of ever moth, Joh makes a pamet of $500. (i) (ii) If Joh were to take up Scheme, show that the amout of moe he would owe the bak at the ed of the th moth is [] Joh wats to pa off his debt i the shortest time. Which scheme should he take up? Justif our choice. [4] (b) The sum of the first 0 terms of a arithmetic progressio with a o-zero commo differece is 00. Give that the first, third ad thirteeth terms of the arithmetic progressio are cosecutive terms of a geometric progressio, fid the first term ad commo differece of the arithmetic progressio, ad determie whether the sum to ifiit of the geometric progressio eists. [4] The curve C has parametric equatios t e, t, t for t. (i) Sketch C, labellig a itersectios betwee C ad the aes. [] (ii) Show that the taget to the curve at t, deoted b l, has equatio e dw. w (iii) Fid the eact value of w Hece, fid the eact area bouded b the curve C, the lies l ad. [] e e. [8] Page 5 of 6 [Tur_over
6 A Lorez curve is give b L, where 0 ad 0. represets the proportio of a coutr s total wealth that is owed b the poorest 00% of the coutr s populatio i terms of wealth. L si. C is a Lorez curve with equatio (a) Fid the percetage of total wealth owed b the poorest 0% of the populatio. [] (b) A power series is used to approimate L. (i) Fid the series epasio of L up to ad icludig the term i. [4] (ii) Hece, fid the rage of values of such that the error of approimatig L usig its series epasio i part (b) (i) is less tha or equal to [] (c) The iformatio i the Lorez curve is ofte summarised as a ide to measure the distributio of wealth i a populatio. This ide is called the Gii Ide. C A B O With referece to the diagram above, the Gii Ide for C is defied as follows. Let A be the area of the regio bouded b the lie regio bouded b C, the -ais ad the lie. ad C. Let B be the area of the The Gii Ide for C is evaluated as A. A B (i) Fid the Gii Ide for C, correct to sigificat figures. [] (ii) Eplai whether populatios with the same Gii Ide must have the same Lorez curve. [] --- END OF PAPER --- Page 6 of 6 *
7 Suggested Solutio to 06 SH H Mathematics Term 4 Promotioal Eamiatio with Markig Scheme Let Let Let represet the umber of successful free throws. represet the umber of -poit field goals. z represet the umber of -poit field goals. z 0.6(695) z z 54 Usig GC to solve, 65, 404, z 48. We have 404 successful -poit field goals. B similar triagles, R 8 8 h 8 8 h R 8 h 8 R 8 V 8 8 R R dv d 8 d d R 6 8 R R R R 4 6 R R R 4 6 sice 0 R 6 R 4 R d V d d d R R Whe R,
8 d V 48 dr R 6 R6 0 Statioar poit is a maimum. (i) B log divisio, 6 ( 0.5, 0.) (0, ) = (ii) = = C cocave dowwards or e l Usig GC, the graphs of ad l itersect at.45. From the graph,.4
9 4(i) f 5, 0, O 4 4 4(ii) O (.5,0)
10 5(a) 5(b) cos = ( 4) () () 04.4 or.8 OC = AB p p 4 p p p 6 p p p p 6 6 p p p p p p p or p 5(c) Usig Ratio Theorem, OQ 7OA OB 9 OQ Uit vector parallel to OQ
11 6 6 r r l r rr r r r l l l l l l l l l 4 l l 4 l 5... l l l l l l l l l ll l l l l Alterativel, r l r rr r r l l r r r l l 4 l l... l l l l l l l l r l 0.0 rk rr k r r l l 0.0 r r r r rr As,, l 0.
12 k l l l 0.0 k k l 0.0 k k l 0.0 k k 0.0 e k k e ke k e e e k e k 0 Thus, the least value of iteger k is 0. 7(i) sec ta 4 4 7(ii) 4 4 Volume of solid vase (whe area bouded b sec ta, the -ais ad about the -ais) 4 6 sec ta d ta ta ta 0 rotated completel 4 Volume of coe (whe area bouded b 4 the -ais ad 4 ais), is rotated completel about the Volume of solid whe R is rotated aroud 7 6, where a 7, b 6. 4
13 8(i) d dt d dt 0.5l tc l tc c t e e t Ae where t Ae t Ae t Ae e t A A e c Whe t 0,.5..5 A 5 (0) Ae Hece,. 5e t 8(ii) Whe t, 0, 0,. t t t 5e 5e 5e I the log ru, amout of X decreases to a limit of. 8(iii) (0,.5) 5e t = O t
14 9 (i) (ii) R f [0,] Let g( ) cos 4 cos 4 4 cos 4 g ( ) cos D R g g 0, R D g g, R, (iii) g D 0, f Sice R Df, the composite fuctiofg g eists. 0 (a) (i) D D R fg g g 0, Hece 0,, 0, g f R D g g fg For Scheme, total amout of moe he owes the bak at the ed of the th
15 (a) (ii) moth For Scheme, total amout of moe he owes the bak at the ed of the th moth Joh will take 48 moths to pa off his debt uder Scheme For Scheme, Usig GC, Joh will have to take 44 moths to pa off his debt uder Scheme. Scheme will allow Joh to pa off his debt i the shorter time. (b) Let a ad d be the first term ad commo differece of the arithmetic progressio respectivel a d 5 a9d 00 a9d () Give that the first, third ad thirteeth terms of the arithmetic progressio are cosecutive terms of a geometric progressio, i.e. ad ad a a d ad a ad a 4ad 4d a ad 4d 8ad 0 4d d a 0 d 0 (rejected) or d a Substitute d a ito ():
16 60a 0 a d a a d Commo ratio of the geometric progressio = 5 a Sice the commo ratio is bigger tha, the sum to ifiit of the geometric progressio does ot eist. (i) C O (,0) -. Whe = 0 t 0t e t (ii) Method d t d e, dt dt d t d e At t, e,, d d e Equatio of taget e e e e e Method t te, tt e d d e e d d Whe t, e, d e d e d d e
17 (iii) Equatio of lie: e e e w w e dw w wwe dw w we w w ee e ee ee w e dw e e t l C O + e (t = ) -. Whe e, t. Area of the bouded regio e t d te dt e e e t d t e dt e e e e e e e 4e e e e e 4e e e e 4e e
18 (a) Whe 0., L0. si Percetage owed b the poorest 0% is.8% ( s.f.). (b) (i) si d d d d (b)(ii) d d 0 0; 0 ; 0 0; si si si B G.C., (sice 0 ). (c) (i) (c) (ii) si d Area of B 0 / Gii Ide for C / 0.7 (sf) Sice A B is alwas 0.5, the Gii Ide is ol depedet o the area uder the Lorez curve. Differet curves with the same limits ca have equal area uder it.
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