Ntiol Quli ctios AHEXEMPLAR PAPER ONLY EP/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite workig. Stte the uits for your swer where pproprite. Write your swers clerly i the swer booklet provided. I the swer booklet, you must clerly idetify the questio umber you re ttemptig. Use blue or blck ik. Before levig the emitio room you must give your swer booklet to the Ivigiltor; if you do ot, you my lose ll the mrks for this pper. *EPAH0*
FORMULAE LIST Stdrd derivtives Stdrd itegrls f ( ) f ( ) f ( ) f ( ) d si cos sec ( ) t( ) + c si + c t + + t c + t sec l + c cot cosec e e + c sec sect cosec l coseccot e e Summtios (Arithmetic series) S + ( ) d (Geometric series) S ( ) r r ( + ) ( + )( + ) ( + ) r, r, r 6 r r r Biomil theorem ( + b) r 0 r r b r where Cr r! r!( r)! Mcluri epsio iv f ( ) f ( ) f ( ) f( ) f( ) + f ( ) + + 0 0 0 0 0 + +...!!! Pge 0
FORMULAE LIST (cotiued) De Moivre s theorem Vector product [ (cos si )] r θ + i θ r ( cosθ + isiθ) i j k b bsiθˆ i j + k b b b b b b b b b Mtri trsformtio Ati-clockwise rottio through gle, θ, bout the origi, cosθ siθ si θ cosθ Pge 0
Totl mrks 00 Attempt ALL questios. Use the biomil theorem to epd d simplify.. Give cos y e si () Fid dy d. Give f( ) + (b) Obti f ( ) d simplify your swer.. Use Gussi elimitio o the system of equtios below to give epressio for z i terms of λ. + y+ z + y λz + 6y+ 8z Determie the vlue(s) of λ for which this system does ot hve solutio.. The velocity, v, of prticle, P, t time, t, is give by t t v e e +. () Fid the ccelertio of P t time t. (b) Fid the distce covered by P betwee t 0 d t l.. Give tht z i, write dow the cojugte z d epress z i polr form. 6. The equtio + y + 9 6y defies curve pssig through the poit,. A( ) Fid the equtio of the tget to the curve t A. Pge 0
7. Mtrices A d B re defied by () Fid A. p A d 6 B. (b) Fid the vlue of p for which A is sigulr. (c) Fid the vlues of p d if B A. 8 () Give the first three o-zero terms of the Mcluri series for cos. (b) Write dow the first four terms of the Mcluri series for e. (c) Hece, or otherwise, determie the Mcluri series for icludig, the term i. e cos up to, d 9. Prove by cotrdictio, tht if is irrtiol the is irrtiol. 0. Fid the coordites of the poit of ifleio o the grph of y si + t, where π π < <.. () Write dow the mtri, M, ssocited with reflectio i the y-is. (b) Write dow secod mtri, M, ssocited with ti-clockwise rottio through gle of π rdis bout the origi. (c) Fid the mtri, M, ssocited with ti-clockwise rottio through π rdis bout the origi followed by reflectio i the y-is. (d) Stte the sigle trsformtio ssocited with M.. Prove by iductio tht, for ll positive itegers,. r rr ( + ) + Pge 0
. A semi-circle with cetre ( 0, ) d rdius, lies o the -is s show. Fid the volume of the solid of revolutio formed whe the shded regio is rotted completely bout the -is. y O. Prt of the stright lie grph of fuctio f( ) is show. y (0, c) O (, 0) () Sketch the grph of f ( ) showig poits of itersectio with the es. (b) Stte the vlue of k for which f( ) + k is odd fuctio. (c) Fid the vlue of h for which f ( + h) is eve fuctio.. Use the substitutio tθ to determie the ect vlue of ( + ) 0 d. Pge 06
6. A lie, L, psses through the poit P(,, ) d is prllel to u i+ j k d secod lie, L, psses through Q(,, ) d is prllel to u i+ j+ k. () Determie the vector equtios for L d L. (b) Show tht the lies L d L itersect d fid the poit of itersectio. (c) Determie the equtio of the ple cotiig L d L. 7. () Fid the geerl solutio of the differetil equtio d y dy y e +. d d 7 (b) Fid the prticulr solutio for which y d dy d whe 0. 8. Vegettio c be irrigted by puttig smll hole i the bottom of cylidricl tk, so tht the wter leks out slowly. Torricelli s lw sttes tht the rte of chge of volume, V, of wter i the tk is proportiol to the squre root of the height, h, of the wter bove the hole. This is give by the differetil equtio: dv dt k h, k > 0. () For cylidricl tk with costt cross-sectiol re, A, show tht the rte of chge of the height of the wter i the tk is give by dh k dt A (b) Iitilly, whe the height of the wter is cm, the rte t which the height is chgig is 0 cm/hr. By solvig the differetil equtio i prt (), show tht h. h t. 80 (c) How my dys will it tke for the tk to empty? (d) Give tht the tk hs rdius of 0 cm, fid the rte, i cm /hr, t which the vegettio ws receivig wter from the tk t the ed of the fourth dy. [END OF EXEMPLAR QUESTION PAPER] Pge 07