Advanced Higher Grade

Transcription

1 Prelim Emitio / (Assessig Uits & ) MATHEMATICS Avce Higher Gre Time llowe - hors Re Crefll. Fll creit will be give ol where the soltio cotis pproprite workig.. Clcltors m be se i this pper.. Aswers obtie b reigs from scle rwigs will ot receive creit.. This emitio pper cotis qestios gre t ll levels.

2 All qestios shol be ttempte. Use Gssi elimitio to solve the followig sstem of eqtios (). () Differetite t. () (b) Give tht f l, obti simplif f. (). The coefficiets of i the epsio of b re respectivel. Fi the vle of the costts b. (). Give tht i is root of the eqtio, fi ll the soltios. (). () Epress i prtil frctios. () (b) Hece evlte (). If the prmetric eqtios of crve re cos si cos si, evlte t. (). Use the sbstittio si to obti ()

3 . The fctio f is efie b f where R,,. () Fi the eqtios of the smptotes of the grph of f. () (b) Prove tht the grph of f hs trig poit etermie its tre coorites. () (c) Sketch the grph of f iictig ll the importt fetres. (). A sphere hs ris of cm. A sphericl cp of epth cm, is remove from the sphere, s show. cm cm Usig itegrtio methos, fi the volme of the cp hece, or otherwise, fi the rtio of the volme of the cp to the volme of the whole sphere. [ volme of sphere, V ] r (). (),, re the first three terms of rithmetic seqece. For wht vle of oes S, the sm of the first terms, first ecee? () (b) The sm of the first three terms of positive geometric seqece is the sm of the th, th th terms is. Ietif the seqece. () [ END OF QUESTION PAPER ]

4 Mrkig Scheme - Avce Higher Prelim / Give oe mrk for ech. s:,, sig gmete mtri Illstrtios for wrig ech mrk first moifie sstem seco moifie sstem (echgig rows & elimitig frctios) fiig fiig ; () t s: mrks kows how to se chi rle kows how to ifferetite t - t t simplifies swer t

5 Mrkig Scheme - Avce Higher Prelim Mthemtics & (b) Give oe mrk for ech s: kows how to ifferetite logs chi rle fctor mrks ses qotiet rle correctl for chi rle fctor swer Illstrtios for wrig ech mrk This c lso be oe sig properties of logs: i.e. l l l etc.. s:, b fis correct terms for correct biomil coefficiets sbstittes oe epressio ito the other fis b, b, b b b fis b b. s: i, i mrks sttig cojgte root fiig qrtic fctor kowig to ivie b qrtic fctor iviig correctl i i i i sig qrtic forml to fi other roots ll roots correct

6 i, i Mrkig Scheme - Avce Higher Prelim Mthemtics & () Give oe mrk for ech s: mrks Illstrtios for wrig ech mrk fctorises eomitor correctl metho fis coefficiets A, B, C A B C (b) s: l sbstittes prtil frctios ito itegrl kows how to itegrte log term itegrtes trig fctio correctl sbstittes limits correctl swer l t l l t l t. s: fis erivtive of term correctl fis erivtive of term correctl kows how to fi fis evltes si cos si cos si cos si cos

7 Mrkig Scheme - Avce Higher Prelim Mthemtics &. Give oe mrk for ech s: mrks Illstrtios for wrig ech mrk ses give sbstittio correctl re-writes itegrl correctl chges limits kows how to itegrte cos si si cos cos, cos cos cos cos itegrtes correctl sbstittes limits correctl swer () s:,, mrks kows how to fi eqtios of verticl smptotes fis eqtios of verticl smptotes fis eqtio of horiotl smptote (b) s: m trig poit t, cos si,, As, i.e. = fis f ttempts to solve fis vle for fis coorite f = ses tre tble or otherwise to etermie tre f f mimm trig poit, (c) s: see grph t e of mrkig scheme mrks ll importt poits show correct behvior t smptotes

8 Mrkig Scheme - Avce Higher Prelim Mthemtics & Give oe mrk for ech. s: : mrks sets p sphere o pproprite coorite es (see igrm o et pge) sttes eqtio of circle Illstrtios for wrig ech mrk circle, cetre (, ), ris cm (for emple) kows how to fi volme of revoltio ses correct limits of itegrtio sbstittes ito itegrl correctl itegrtes correctl sbstittes limits correct volme of cp fis volme of whole sphere sttes rtio () s: = mrks its (c be fo sig iteg.) : kows how to fi seqece (b) fis seqece sbstittes correctl ito sm to terms solves ieqlit s:,,,,, ttempts to se geerl term correct epressio for sm of first three terms epressio for th, th th terms kows how to solve eqtios fis vles for r ietifies seqece mrks seqece is,,,... S r r. r r r r r r, r r iviig r r,,,,,, etc r Mrks

9 Grph for Qestio - (, -/) = - = Digrm for Qestio. cm