PREPARATORY MATHEMATICS FOR ENGINEERS

Transcription

1 CIVE 690 This qusti ppr csists f 6 pritd pgs, ch f which is idtifid by th Cd Numbr CIVE690 FORMULA SHEET ATTACHED UNIVERSITY OF LEEDS Jury 008 Emiti fr th dgr f BEg/ MEg Civil Egirig PREPARATORY MATHEMATICS FOR ENGINEERS Tim llwd: hurs Attmpt 4 qustis 4 3 () Fid th qutit d rmidr wh th plymil + is dividd by Hc writ s th sum f plymil d prpr rtil fucti [6 mrks] 3 (b) Lt f ( ) = + + b Supps tht is fctr f f () d tht th rmidr wh f () is dividd by + 3 is Fid th vlus f d b [6 mrks] + 7 (c) Eprss ( )( + ) i prtil frctis [8 mrks] Tur vr

2 CIVE 690 () (i) Fid th quti f th circl with ctr t A = (,) which psss thrugh th pit B = (3,5) (ii) Fid th grdit f th li jiig A d B (iii) Fid th grdit d th quti f th tgt t th circl t B (b) (i) Epd th prssi ( 3 y 3 ) [ mrks] (ii) Hw my rrgmts r thr f th lttrs f th fllwig wrds? ) FOUR, b) TWELVE [8 mrks] 3 () Writ th prssi 4 csθ + siθ i th frm r si( θ + α), fr suitbl cstts r d α, givig bth t 3 sigifict figurs, d givig α i dgrs Hc, r thrwis, fid ll gls 0 θ 360 fr which 4 cs θ + si θ = 3 (b) Fid frmul fr si 3θ i trms fsi θ [8 mrks] [6 mrks] (c) Th vlu V f crti cr is mdlld by th quti V kt = P, whr P d k r psitiv cstts, d t is th tim lpsd i yrs sic th cr ws w (i) (ii) Th vlu f th cr wh w ws 7,000, d wh it ws ctly tw yrs ld, it ws 4,000 Fid th vlus f P d k, givig bth t 4 sigifict figurs Wht ws th vlu, t th rst pud, f th cr wh it ws ctly fur yrs ld? [6 mrks] Tur vr

3 CIVE () Fid i ch f th fllwig css: d (i) (ii) 3 ; y = cs3 si ; (iii) y = l(t ) ; (iv) y = si [ mrks] (b) Lt C b th curv with quti + y + y 7 = 0 Fid th quti f th tgt t th curv C t th pit (,) [6 mrks] (c) Fr th prmtric curv = cst, y = si t, fid prssi fr i trms f d th prmtr t 5 () Usig th substituti u = +, fid th itgrl [ mrks] (b) cs( + ) d Usig itgrti by prts, vlut th dfiit itgrl [6 mrks] (c) / l d Fid th grl sluti t th diffrtil quti [6 mrks] d y = Fid ls th prticulr sluti which stisfis th iitil cditi tht y = 0 wh = 3 [8 mrks] 3 Tur vr

4 CIVE690 PREPARATORY MATHEMATICS FOR ENGINEERS: FORMULA SHEET CIVE 690 Idics p q p+ q =, p q pq ( ) =, ), p p p b = ( b, 0 = =, / =, / p = p Qudrtics If + b + c = 0 th b ± b 4c = Th discrimit is b 4c Crdit gmtry Th li thrugh (, y ) d (, ) y hs grdit ( y y) /( ) Th li y = m + c hs grdit m Th li thrugh (, y ) with grdit m hs quti y y ( = m ) Th li y = m + c is prpdiculr t y = m' + c' if mm ' = Th distc btw, ) d, ) is ( y ( y ( y ) + ( y ) Th circl with ctr (, b) d rdius r is giv by ( ) + ( y b) = r Bimil psis Pscl s rw 0 Trigl: rw rw rw rw ! = 3 K ( ), 0! =, = r! ( ) L( r = = r!( r)! r! C + r ) ( K r r r + ) = K ( + ) ( ) = + +! ( ) L( r + ) r + K + + K r! ( rtil, < ) 4 Tur vr

5 CIVE 690 Trigmtry si( θ ) = siθ, cs( θ ) = csθ siθ t θ =, csθ csc θ =, siθ sc θ =, csθ csθ ct θ = siθ si θ + cs θ =, t θ + = sc θ, ct θ + = csc θ y = si ms = si y d 90 y 90 y = cs ms = cs y d 0 80 y = t ms = t y d y 90 < y < 90 Altrtiv tti: rcsi = si, rccs = cs, rct = t Nt tht si (si ), cs (cs ), t (t ) Hwvr si = (si ), cs = (cs ), t = (t ) si( A ± B) = si Acs B ± cs Asi B si( θ ) = siθ csθ cs( A ± B) = cs Acs B m si Asi B cs( θ ) = cs θ si θ t A ± t B tθ t( A ± B) = t(θ ) = m t At B t θ csθ + bsiθ = r si( θ + α) whr circl with crdits ( b / r, / r) r + b = d α is th gl f th pit th uit Lgrithms p lg = p ms =, lg = 0, lg = k lg + lg y = lg ( y), lg lg y = lg ( ), k lg = lg ( ) y l = lg, l = Diffrtiti If If y = u ± v, th y = uv, th d d du dv = ± If y = u with cstt, th du = d d d d du dv v u du dv u = v + u If y =, th = d d d d v d v du If y is fucti f u, whr u is fucti f, th = d d du 5 Tur vr

6 CIVE 690 Itgrti If y = u ± v, th y d = u d ± v d If y = u with cstt, th y d = u d If u is fucti f th d dv du y du u d = uv v d d d d = y du Stdrd drivtivs d itgrls y = f () d y = f () y d + + c + ( ) + c l l +c si cs si cs + c cs si cs si + c t sc t l cs + c si cs t + + si t + c + c 6 END