June Further Pure Mathematics FP2 Mark Scheme
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1 Jne 75 Frher Pre Mheis FP Mrk Shee. e e e e 5 e e 7 M: Siplify o for qri in e ( e )(e 7) e, e 7 M: Solve er qri. ln or ln ln 7 B M A M A A () Mrks. () Using ( e ) or eqiv. o fin e or e: ( = n = ) M A e Using y ( e) y (M reqires vles for n e) M A () () Bf () 5 Mrks. s sin sin M A s k e os e (M rk y e sore y fll ssiion eho) s = = : k = s e sin M Aso () (M rk reqires k, or se of liis) Mrks
2 . y y M: ( ) ( ) ( ). ( ) ( ) y y ( ) y y or = (or ) (or e eqivlen, e.g. 8, ) n M: Qoien or pro rle ep. (Chin rle, if se, s e goo.) r M: Aep wih erivive vles A: Corre erivive vles ( n ) seen or iplie y working. Alernive: (involving iplii iffereniion). y se y [M A] (llow one slip for M) y y y se y se y(se y n y) (or lernive e.g. y os y y y, so os y os y( sin y) ) Then rks s in in shee (n.. n y, se y ). M A M M Aso Aso () Mrks [M]
3 5. () () y seh P y osh osh ln( ) or 8 ln(7 ) or e or e 7 ( or +) ln( ) or ln(7 8 ) (or eqiv.) y ln( ) nh(...) (Ssie for ) seh nh, nh y ln( ) ln (*) () Seon solion, if seen, s e rejee o sore he finl rk. () n M reqires n epression in ers of wiho hyperolis, eponenils n logrihs. B M A A () M M A () 7 Mrks
4 . () y B B, or M, A ln ln ln Lengh (*) M M A (7) () Srfe re = 8 M ) (8 ) (8 5 M M A () Mrks Noe epenene of M s. () n M: Coplee inegrion ep. r M: Ss. orre liis n sr. () n M: Coplee inegrion ep. r M: Ss. orre liis n sr. In (), issing or or hrogho ol sore M M M A. If is here iniilly, hen los, ol sore M M M A.
5 7. rsinh rsinh M A A = 9rsinh rsinh 9ln or B Le M. M M When =, = n when =, =... M A Are = 7 9 9ln rsinh 9 (*) Aso () Mrks Depenen M rks: M: Choose n pproprie ssiion & fin or Se p inegrion y prs. M: Ge ll in ers of or Use inegrion y prs. M: Son inegrion. M: Ssie oh liis (for he orre vrile) n sr.
6 7. Alernive solion: Le sinh osh r sinh sinh osh sinh sinh osh sinh sinh osh rsinh = 9rsinh osh osh osh rsinh osh Are = 7 7 9rsinh 9ln (*) 9 M M M A A B M M A Aso () Mrks
7 7. (i) (ii) A few lernives for:. Le No rks ye nees noher ssiion, or prs, or perhps Liis ( o 9) Le sinh osh sinh osh sinh osh osh Then, s in he lernive solion, osh sinh osh osh Liis ( o rsinh) M M M M M M M M
8 7. (iii) M Le n se n M se n se se se se M se se n se n se Liis se o se M (iv) (By prs s e he righ wy ron, no inegring ) v M,, v M Liis (v) (By prs) v,, v rsinh No progress (vi) ( ) ( ) Liis M M M M M M M
9 8. () () n n osh sinh n n sinh M A n n n sinh n osh n( n ) osh M n n I n sinh n osh n( n ) In (*) A () I sinh osh I M I sinh osh sinh osh I M (This M y lso e sore y fining I y inegrion.) B I osh sinh k I sinh, osh C () sinh osh. A, A (5) = 7sinh 8osh M: = ssie hrogho ( soe M sge) e e e e M 7 8 M: Use of ep. Definiions (n e in ers of ) A () 9e 5e Mrks () Inegrion onsn issing hrogho loses he B rk
10 9. () ) ( M (*) A () () M (*) A () () Fin heigh n se of ringle (perhps in ers of ). M OB n AO = A Are of ringle OAB = M: Fin re n ss. for. M A () () M A (*) A () (e) Roo of qri: (Shol e orre if qoe irely) M Using n : M A () (The n M is epenen on sing he qri eqion). Mrks () Alernive: (sine ) ) ( [M] [A] Conlsion [A] (e) Alernive: Begin wih fll eqn.. In he eqn., se oniions n ) ( [M] Siplify n solve eqn., e.g.
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