Aitional Math (07) Prpar b Mr Ang, Nov 07 Fin th valu of th constant k for which is a solution of th quation k. [7] Givn that, Givn that k, Thrfor, k Topic : Papr (00 marks) Tim : hours 0 mins Nam :
Aitional Math (07) (i) Show that tan tan. [] Without using a calculator, fin th valu of ach of th constants a an b for which (i) Lt tan, cos sin cos sin cos tan sc tan a b cos sin cos cos. [] sc tan tan tan tan tan tan tan tan a, b Prpar b Mr Ang, Nov 07
Aitional Math (07) (i) 9 B consiring th gnral trm in th binomial pansion of p, whr p is a constant, plain wh thr ar no vn powrs of in this pansion. [] Givn that th cofficint of in th pansion of p is twic th cofficint of 7. Fin th valu of p. [] 9 9 9r (i) Lt T p r r whr r = 0,,,, 9 r r 9 9r 9r p r 9 9r 7 r p r Sinc r is a non-ngativ intgr, so r is an vn non-ngativ intgr. 7 r is thrfor an o intgr. As a rsult, th powr of cannot b an vn numbr. For th trm, 7 r, r. 9 th cofficint of p 9 For th 7 trm, 7 r 7, r. 9 p 9 p 9 8 p th cofficint of 7 9 9 9 8 p 7 p p p p 0 p 0 7 p p 0 or p p 0, as th cofficint of is twic th cofficint of 7. p p Prpar b Mr Ang, Nov 07
Aitional Math (07) Th curv has a minimum point M. (i) Show that th -coorinat of M satisfis th quation 9. [] Point A(, 7) an B(, 7) li on th curv mt at th point P.. Th tangnts to th curv at A an B Dtrmin, with working, whthr th -coorinat of P is gratr or lss than th -coorinat of M. [] (i) Givn, Lt 0, 0 9 A(, 7) an B(, 7) Whn, Equation of tangnt at A, 7 9 Whn, 8 Equation of tangnt at B, 7 8 8 Solving simultanous quation 9 8 7 Sinc 9 8, M M 7 an 9 8, thrfor P is lss than M. Prpar b Mr Ang, Nov 07
Aitional Math (07) (i) Solv th quation log log log. [] 7 Solv th quation log 00 lg, giving our answr to significant figurs. [] (i) log log log log 7 log log 7 7 7 7 7 log log log log 00 lg lg00 lg lg lg lg lg 0 or 0 or 0. 09 ( s.f.) Prpar b Mr Ang, Nov 07
Aitional Math (07) Th quation of a curv is m c 9, whr m an c ar constants. Th lin m c is tangnt to th curv at th point P. (i) Fin th positiv valu of m. [] Using this valu of m, an givn that th curv passs through (, 9), fin th coorinats of P. [] Th straight lin L mts th curv at on point onl. (iii) Givn that L is not a tangnt to th curv, what can b uc about L? [] (i) Solving simultanous quations m c 9 m 0 Th iscriminant = 0, m 9 0 m m m or m 7 (rjct) 9 an m c Lt m, (, 9), c 9 9 c 9 0 0 P, (iii) L must b a vrtical lin. Prpar b Mr Ang, Nov 07
Aitional Math (07) 7 (a) Th prcntag, P, of carbon- rmaining in a pic of fossilis woo is givn b kt P 00, whr k is a constant an t is masur in ars. It taks 70 ars for th carbon- to b ruc to half of th original amount. Calculat (i) th valu of k, [] th prcntag of carbon- which woul inicat a fossil ag of 8000 ars. [] (b) Th siz, S, an intnsit, I, of a naturall occuring vnt ar connct b th I formula S lg, whr c is a constant, Calculat, to cimal plac, th siz of th c vnt which has intnsit 0 tims that of an vnt of siz.. [] (a) (i) 0 Whn t 0, P 0 00, 0 00 Whn t 70, P 0, k 70 0 00 k 70 0. ln 0. 70k ln 0. k 70 k.0 ( s.f.) Whn t 8000, P P k 0 P P0 P P 0 8000 8000 k 0.80 ( s.f.) prcntag of carbon- is 8% (b) Givn that I S lg, whn S., c. I c0 whn,. I 0c0, S lg S lg 0. S. 0c0 c. Prpar b Mr Ang, Nov 07 7
Aitional Math (07) 8 (a) A particl movs along th curv ln in such a wa that th -coorinat of th particl is incrasing at a constant rat of 0.0 units pr scon. Fin th rat at which th -coorinat of th particl is incrasing at th instant whn 7. [] (b) Th quation of a curv is 8. (i) Eplain wh th curv has onl on stationar point an wh this is a point of inflion. [] Writ own th coorinats of th stationar point. [], 0. 0 t t t (a) Givn that ln 0.0 0 t 0. units pr scon t (b) Givn that 8,, 7, (i) Lt 0, 0, singl stationar point. Whn 0., 0. 0. 0 Whn 0., 0. 0. 0 Th curv rmains crasing aroun th stationar point. Hnc, it is a point of inflion. Whn 0., 8 Th stationar point is ( 0., 8) Prpar b Mr Ang, Nov 07 8
Aitional Math (07) 9 Th iagram shows a trapzium with vrtics A(, ), B(0, p), C(, ) an D. Th sis AB an DC ar paralll an th angl DAB is 90. Angl ABO is qual to angl CBO. (i) Eprss th graints of th lins AB an CB in trms of p an hnc, or othrwis, show that p =. [] Fin th coorinats of th point D. [] (iii) Fin th ara of th trapzium ABCD. [] (i) graint of th lins AB, graint of th lins CB, Hnc p m AB m CB p p p p p p 0 p p 0 Prpar b Mr Ang, Nov 07 9
Aitional Math (07) graint of th lins AB, m AB Sinc AB//DC, m m DC AB Equation of DC, Sinc AB AD, m AD mab Equation of AD, Solving simultanous quation an Substitut into. D, (iii) ara of th trapzium ABCD 9 squar units Prpar b Mr Ang, Nov 07 0
Aitional Math (07) 0 It is givn that sin an cos. (i) Stat th amplitu an th prio, in grs, of (a), (b). [] For th intrval 0 0, solv th quation, [] (iii) sktch, on th sam iagram, th graphs of an, [] (iv) fin th st of valus of for which 0. [] (i) (a) for, amplitu = ; prio = 0 (b) for, amplitu = ; prio = 80 solv th quation, sin cos sin cos sin sin sin sin sin 0 sin sin or sin (no solution) Principal angl, sin 8. 7 80 8. 7 or 0 8. 7 8. or. 8 (.p.) Prpar b Mr Ang, Nov 07
Aitional Math (07) (iii) sktch, on th sam iagram, th graphs of an, sin cos (iv) From th graphs in part (iii), for th st of valus of for which 0. 0 8. or.8 0 Prpar b Mr Ang, Nov 07
Aitional Math (07) Th iagram shows an ara of rough groun borr b a straight roa XY. Th point O is such that OX = km, OY = km an angl XOY = 90. A cross-countr runnr lavs O an rachs P on th roa XY aftr running for km in a straight lin inclin at an angl to OY. (i) Eprss th shortst istanc of P from OX an from OY in trms of. [] Show that 0 cos sin. [] (iii) Eprss 0 cos sin in th form R cos, whr R > 0 an is acut. [] (iv) Fin th valu of. [] (i) P from OX, cos, P from OY, sin. (iii) B similar triangls, cos sin sin 0 cos 0 cos sin R 0 tan 0 cos sin cos 0 Prpar b Mr Ang, Nov 07
Aitional Math (07) (iv) cos cos Principal angl, 9. 97 9. 97 or 0 9. 97 9. 97 0 9. 97 9.97 tan 0 9.97 tan 80. 98 0. 98 Sinc 0 90, rjcts 0. 98 80. 9 (.p.) Prpar b Mr Ang, Nov 07