National 5 Mathematics

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1 St Anrw s A Mttis Drtnt Ntionl Mttis PRACTICE EXAM REVISION 0-

Ntionl Mttis Sintii Nottion Stnr For Rvision P. Writ o t ollowin nurs in sintii nottion. 00 000 000 000 000 0 000 000 000. For o t ollowin nurs i writ it out in iurs ; ii writ it in sintii nottion.

Clult o t ollowin rssin our nswr in stnr or: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i 0 0. Answr o t ollowin ustions lvin our nswrs in stnr or n orrt to s.. wr nssr. Lit trvls t 0 ils r son.

2 Ntionl Mttis Sintii Nottion Stnr For Rvision P. Writ o t ollowin nurs in sintii nottion For o t ollowin nurs i writ it out in iurs ; ii writ it in sintii nottion. illion tousn illion tousn. Writ o t ollowin nurs in sintii nottion Writ o t ollowin nurs out in ull i 00 0 j 0 0 l Clult o t ollowin rssin our nswr in stnr or: i 0 0. Answr o t ollowin ustions lvin our nswrs in stnr or n orrt to s.. wr nssr. Lit trvls t 0 ils r son. How r will it trvl in n our? T rius o t rt is 0 trs. Wt is its irurn in? I rt ts 0 tis inut, ow n tis will it t in liti o 0 rs? 00 rs o wtr ontins 000 ros. How n ros woul tr in tn ontinin tonn o wtr? In r o ron tr r tr in o ur ron? 0 tos. How n ron tos r St Anrw s A Mttis Dt 0-

3 Ntionl Mttis Siilrit Rvision P. E ir low sows ir o siilr ss or ojts. For ir. i stt t sl tor ro lt to rit ii lult t lnt r E ir o ss low is ttill siilr. Clult t r o rit-n s E ir o ontinrs low is ttill siilr. Clult t volu o rit-n ontinr. litrs 0 0 0l? 0l St Anrw s A Mttis Dt 0-??

4 Ntionl Mttis Siilrit Rvision P. Clult t lnt o t si r in ir low In t ir ABC CED, AB =, AC = n ED =. A D Elin w t trinls ABC n CDE r siilr. Clult t lnt o CD. C Givn tt t r o trinl ABC is sur ntitrs, lult t r o trinl CDE.. Clult t lnt o t si r in ir low. B E 0 R. T ir oosit sows n luiniu i r. T ross rs QS n PT r rlll. RS =, QS = n PT = s sown. Clult t lnt o ST. In t ir lr is li inst two wlls s sown. P Q S T T ir wll is trs i, n t lowr wll is 0 trs. T istn twn t two lt n s o t wlls is 0 trs. Clult t istn twn t oot o t lr n t lowr wll. St Anrw s A Mttis Dt

5 Ntionl Mttis Rvision P Ptors You n lultor or tis worst. Roun our nswrs to on il l wr nssr.. Clult t lnt o t si r in trinl low 0 i 0. Consir t rwor oosit. B Clult t lnt o BD. Hn lult t lnt o BC. Clult t r o trinl ABC. A D C. A rous s sis o 0 n its lonst ionl surin. 0 Clult t lnt o t sortr ionl. Clult t r o t rous. Clult t istn twn ir o oints low. A,, B, 0 P,, Q, E,, F, R,, F, St Anrw s A Mttis Dt 0-

6 Ntionl Mttis Rvision P Ptors You n lultor or tis worst. Roun our nswrs to on il l wr nssr.. Consir t uoi oosit. F G Clult t lnt o t ionl AC. Hn lult t lnt o t s ionl AG. E A B H D C. T ri oosit s rtnulr s. T Clult t lnt o t s ionl PR. Givn tt TR =, lult t vrtil it o t ri. Q R P S. Wi o t ollowin trinls r rit-nl? i ii iii. Consir t ir oosit. All lnts r ntitrs. A B Clult t lnt o AC. Clult t lnt o ED. E C Prov tt trinl ACD is rit-nl t C. Hn lult t lnt o BC n t r o trinl ABC orrt to t nrst wol nur. 0 St Anrw s A Mttis Dt 0- D

Ntionl Mttis Rvision P Ptors - Prols You n lultor or tis worst. Roun our nswrs to on il l wr nssr. Bin ustions to rwin lr n wll lll ir. Lt t lnt to oun.. A si sils u Nort n tn urtr u Est.

. A si sils u Nort n tn urtr istn u Wst. T si is now ro its strtin oint. Clult.. How lon is t ionl o sur o si?. A rtnl surs. Clult t lnt o its ionls.

7 Ntionl Mttis Rvision P Ptors - Prols You n lultor or tis worst. Roun our nswrs to on il l wr nssr. Bin ustions to rwin lr n wll lll ir. Lt t lnt to oun.. A si sils u Nort n tn urtr u Est. How r is t si ro its strtin oint?. An irrt lis 00 u Wst n tn urtr 0 u Sout. How r is t irrt ro its strtin oint?. A si sil u Est ollow u Sout. How r woul it v sil i it ollow irt ours?. A si sils u Nort n tn urtr istn u Wst. T si is now ro its strtin oint. Clult.. How lon is t ionl o sur o si?. A rtnl surs. Clult t lnt o its ionls.. A lr o lnt trs lns inst vrtil wll wit t oot o t lr trs ro t s o t wll. How i u t wll os t lr r?. A lr is l inst vrtil wll. I t istn twn t oot o t lr n t wll is trs, n t lr rs trs u t wll, lult t lnt o t lr.. A irl s itr o 0. A or is rwn wi is ro t ntr o t irl. Clult t lnt o t or. 0. A irl s itr o. A or is rwn wi is ro t ntr o t irl. Clult t lnt o t or. or rius Tis ir l ou wit ustions n 0. T roo sown oosit s two rlll sis. Usin t ivn insions lult t ritr o t roo.. Clult t lnt o t nistr ril sown in t ir i tr r stirs, n i tr surs n risr 0. Giv our nswr in trs. St Anrw s A Mttis Dt 0-

8 Ntionl Mttis Rvision P Brts An Eutions. En n sili : i j + + l + n o + r. En o t ollowin : s + s i + + j l n o s s r. En o t ollowin : + + t t i + j z + z + l + + n. + o r t t + s + t + u + v + w En o t ollowin : + + t t i j z + z + l + + n. + o r t t + s t + u v + + w + +. En : + w i j 0 l 0 n o + r + s + t u v + w + St Anrw s A Mttis Dt 0-

9 Ntionl Mttis Rvision P St Anrw s A Mttis Dt 0- Brts An Eutions - Etnsion. En, sili n solv o t ollowin utions :. En n sili o t ollowin rssions :. Solv o t ollowin utions : n l j i 0 0 l j i 0 0 j i

10 Ntionl Mttis Brts An Eutions - Alitions Rvision P. For t rtnl oosit writ own, in its silst, n rssion or: i its ritr P; ii its r A. Clult P wn =.. For t rtnl oosit writ own, in its silst, n rssion or: i its ritr P; ii its r A. Clult A wn =.. Consir t trinl oosit. Writ own n rssion or in its silst or. Hn lult wn.. Consir t trinl oosit. Writ own n rssion or in its silst or. Hn lult wn.. For t trinl oosit, writ own silii rssions or: P its r A; t lnt o PQ. R Q. For t rtnl oosit, writ own silii rssions or: P, its ritr. A, its r. T lnt,, o its ionl.. Rt ustion. or rtnls surin: St Anrw s A Mttis Dt 0-

11 Ntionl Mttis Rvision P St Anrw s A Mttis Dt 0- Siultnous Eutions. Solv t ollowin ssts o utions "liintion". 0 0 i v u v u j 0 l 0 0. Solv t ollowin ssts o utions "sustitution". 0 0 i 0 v u v u j 0 l Solv t ollowin siultnous utions "rill". Drw s wit n ro 0 to Drw s wit n ro 0 to Drw s wit ro 0 to n ro to 0 Drw s wit n ro 0 to Drw s wit n ro 0 to Drw s wit ro - to n ro 0 to 0 0

12 Ntionl Mttis Siultnous Eutions Rvision P Prols Lin to Siultnous Eutions Not: ustion ust in wit t onstrution o two srt utions wit two stt unnowns.. Fin two nurs wos su is n wos irn is.. Fin two nurs wos su is n wr twi t i on inus tr tis t sll on is.. Two nurs r su tt twi t sllr lus t lrr is ul to n t irn twn twi t lrr n t sllr is. Fin t two nurs.. Two nurs r su tt tr tis t lrr lus twi t sllr is ul to n t su o twi t sllr lus t lrr is. Fin t two nurs.. Consir t two rtnls oosit. T sllr on s ritr o 0. T lrr on s ritr o twi t sllr. For two utions n solv t siultnousl to in t vlus o n. Hn lult t r o t sllr rtnl.. A vn is rrin it intil os n iv intil rls. I os n rls wi totl o n os n rls wi 0, in t wit o n iniviul o n sinl rl. Wt is t totl wit rri t vn?. ouns o uttr n ints o il osts.. ouns o uttr n ints o il osts.. Fin t ost o oun o uttr n sinl int o il.. In rtin tor, t si rt o is.0 r our, wit ovrti t.0. His totl w or rtin w ws.0. I wor totl o ours in ll, ow n ours i wor t t sis rt?. At onrt 00 tits wr sol. C tits ost wrs or nsiv ons ost. I t totl rits wr 0, ow n tits wr sol? 0. Jon svs on uttin vr 0 n vr 0 oin rivs in o. Atr wil isovr's tt s oins ountin to.0. How n 0 oins os v? St Anrw s A Mttis Dt 0-

13 Ntionl Mttis Volu Rvision P Iortnt orul: Cuoi : Pris : Clir: V V V l l r r r insion A l Sr: Con: Pri V V V r r s r it A. Clult t volu o linr wit ivn rius n it. r r =, = r =, = r =, = 0 r =, =. A linr s itr o 0 n it o. Clult its volu.. A linr s itr o n it o 0. Clult its volu to t nrst ui tr.. Clult t volu o sr sri low, rounin our nswr to il l. r r = r = r = r =. A sr s itr o. Clult its volu ivin our nswr orrt to siniint iurs.. Clult t volu o on sri low, rounin our nswrs to il l. r =, = r r =, = r =, = r =, =. A on s s itr o n it o. Clult t volu o tis on. St Anrw s A Mttis Dt 0-

14 Ntionl Mttis. A on s s itr o 0 n slnt it o. Clult t volu o t on.. A on s s rius o n slnt it o. Clult t volu o t on. r Rvision P s 0. A ri s sur s o si n vrtil it o. Clult t volu o t ri orrt to siniint iurs.. A ri s rtnulr s surin n vrtil it o 0. Clult t volu o t ri. St Anrw s A Mttis Dt 0-

15 Ntionl Mttis Rvision P St Anrw s A Mttis Dt 0- Ftoristion. Ftoris o t ollowin irst rovin oon tor. 0 o n l r j i t. Ftoris o t ollowin irst rovin t ist oon tor. t t Ftoris rssion ontins irn o surs: i j w l n w o r s t 00. Ftoris o t ollowin urti rssions: o s s n l j i s s. Ftoris: o n l z z j i t t

16 Ntionl Mttis Forul - Cnin t Sujt o Forul Rvision P. Cn t sujt o orul to.. = +. =. = +. =. =. = 0. =. = i. = + j. = t. = + l. =. = + n. = + o. = + 0. M t sujt o orul low.. = +. = +. t = s r. = + r. = n. = +. =. = + i. =. Cn t sujt o orul to t lttr sown in rts.. P = l l. V = IR I. S = DT T. A = l. C =. G = UT U. v = u + t t. P = l + l i. H = +. M t sujt o orul.. =. s =. =. =. w =. = z. =. = i. = j. = +. z = l. = +. Cn t sujt o orul to t lttr sown in rts.. v = u + s s. v = u + s u. V = r. V = r r. r = A A. L = +. =. z = t i. r = j. st = A A. R = A l. n = n n t n. = n n. R n R r r St Anrw s A Mttis Dt 0-

17 Ntionl Mttis Grint & Eution o Lin Rvision P. Clult t rint o lin in t ir oosit. Co n olt sttnt low : i ii iii T rint o n orizontl lin is. iv v T rint o n vrtil lin is.. Clult t rint o t lin joinin ir o oints low. A,, B, C,, D, E-,, F, G,-, H,- I0,, J, K-,-, L,- M-,0, N-, P,, Q-, i R,-, S-,0. Drw o t ollowin lins on oorint ir ou soul n lins on ir. 0 i. Stt t rint n t -intrt oint or lin in ustion.. Stt t rint n t -intrt oint or lin low. 0 0 i 0. Writ own t ution o t lin: wit rint, ssin trou t oint 0, wit rint -, ssin trou t oint 0, wit rint, ssin trou t oint 0,-. Estlis t ution o t lin ssin trou ir o oints low. A,, B, C,, D,- E-,-, F, G,-, H-,- I,, J0, K-,, L,-. tnsion & tr rti Fin t ution o lin in ustion usin. St Anrw s A Mttis Dt 0-

18 Ntionl Mttis T Cirl - Ars & Stors Rvision P. Clult t lnt o t r in ir low, ivin our nswr orrt to.. A E D O 0 o 0 o o F B C O O. Clult t ritr o stor in ustion.. Clult t r o stor in ustion.. A sool sll il is in t s o stor o irl s sown. Givn tt O is t ntr o t irl, lult: t ritr o t lin il; O 0 o t r o t lin il. 0. In t ir oosit, O is t ntr o two onntri irls wit rii n 0 s sown. o Anl AOB 0. A Clult: T ritr o t s s. T s r. 0 0 o O B. A Jns r n is ull on wn nl o PQR 0 s sown. Usin t insions sown in ir, lult t roit r o r tril in t n. Dortiv sil ns r l lon t s s sown in ir, lult t roit totl lnt o tis sil in stri. ir P 0 o Q R St Anrw s A Mttis Dt 0- ir

19 Ntionl Mttis T Cirl - Stors, Snts & Cors Rvision P. Clult t r o s snt in t irs low. A O 0 o O 0 o O B AOB 0 o. Clult t lnt o t or in ir low. C A O o O O o Q B D P. Clult t ritr o snt in ustion.. T loo or sll i-r on is sown oosit. It is sil stor o irl wit its ntr t O. Clult t ritr o t loo. P Q Clult t r o t i-r rt. O. A sinr tl-to is s s si-irl wit snt rov s sown in t ir. Fro t inortion suli lult: t ritr o t tl-to; t r o t tl-to. o St Anrw s A Mttis Dt 0-

Ntionl Mttis T Cirl - Etn Qustions Q Rvision P. A wr isuit onsists o stor o irl wit trinulr rt rov s sown in iur. S T rius o t irl PQ is n PS =. Anl QPR = 00 o. Clult t r o t isuit.

A rntr lo s nulu wi trvls lon n r o irl, ntr O. O T r lnt o t nulu is 0. T nulu swins ro osition OA to OB. T lnt o t r AB is. Clult t siz o nl AOB to t nrst r. A B. A irl, ntr O, s n r PQ o lnt 0.

20 Ntionl Mttis T Cirl - Etn Qustions Q Rvision P. A wr isuit onsists o stor o irl wit trinulr rt rov s sown in iur. S T rius o t irl PQ is n PS =. Anl QPR = 00 o. Clult t r o t isuit. iur P T Wr Q A sllr vrsion o t wr is rou. R In tis sllr isuit PQ =. Givn tt t r lnt QR =, lult t siz o nl QPR, orrt to t nrst r, or tis vrsion o t isuit. P Wr iur R. A rntr lo s nulu wi trvls lon n r o irl, ntr O. O T r lnt o t nulu is 0. T nulu swins ro osition OA to OB. T lnt o t r AB is. Clult t siz o nl AOB to t nrst r. A B. A irl, ntr O, s n r PQ o lnt 0. I t itr o t irl is 0, lult t siz o nl POQ orrt to.. A irl, ntr O, s stor EOF wit n r o 0. I t rius o t irl is, lult t siz o nl EOF orrt to.. An r AB on irl, ntr O, s lnt o. I nl AOB = o, lult t rius o tis irl. A stor o irl s n r o. I t nl t t ntr is 0 o, lult t itr o t irl orrt to -il ls. Q. T s oosit is t stor o irl, ntr P, rius 0. 0 T r o t stor is sur trs. P Fin t lnt o t r QR. 0 St Anrw s A Mttis Dt 0- R

21 Ntionl Mttis T Cirl - Anls in Cirls Rvision P Co n lrn t ollowin.... Anls in si-irl ul 0 o.. T nl twn tnnt n rius is 0 o.. Loo or isosls trinls in ll irs.. T nls in n trinl u to 0 o.. Two nls in strit lin u to 0 o.. B wr o t tnnt it. Eris : Fin t siz o t s nls in ir. O is t ntr o irl, PS n PT r tnnts... o o O o O O... o O O o O 0 o P S S P.. T o O o O o P S Eris : Clult t siz o nl in ir low..... O O O 0 o 0 o O o 0 o St Anrw s A Mttis Dt 0- o 0

22 Ntionl Mttis Inutions Rvision P. Solv o t ollowin inutions wr n onl t vlus ro t st o nurs..,, 0,,,,,. El nswr {,, }. Solv o t ollowin inutions. 0 El. Solv o t ollowin inutions. 0 i w w w w j El. I tin o wol nur, trl it n sutrt. T nswr ust lss tn or ul to. For n inution n solv it to in t ossil strtin wol nurs.. I sutrt wol nur ro n oul t nswr. T rsult ust rtr tn 0. For n inution n solv it to in t ossil strtin wol nurs.. Fr n Jn r rotr n sistr. Fr is rs olr tn twi Jn's. T su o tir s is lss tn rs. Tin Jn's to rs or n inution. Wt n ou s out Jn's? St Anrw s A Mttis Dt 0-

23 Ntionl Mttis Trionotr - T Sin Rul Rvision P. Us t sin rul to lult t si r in trinl low. o o o o o o 0 o o o 0 o o o o o o 0 o. Us t sin rul to lult t siz o t nl r in trinl low. o 0 o 0 o 0 0 rul! o 0 o o St Anrw s A Mttis Dt 0-

24 Ntionl Mttis Trionotr - T Cosin Rul Rvision P. Us t osin rul to lult t si r in trinl low. o o o 0 0 o o 0 o 0 0 o 0 0 o. Us t n or o t osin rul to lult t siz o t nl r in trinl low St Anrw s A Mttis Dt 0-

25 Ntionl Mttis Trionotr - T Ar o Trinl Rvision P. Us trionotr to lult t r o trinl low.. Clult t r o rlllor low.. A isin ot sts its nt in t or o trinl s sown oosit. T totl lnt o its nt is trs. Clult t sur r o t wtr nlos witin t nt. St Anrw s A Mttis Dt 0-

Ntionl Mttis Rvision P Trionotr - Mi Eris & Frwors.

26 Ntionl Mttis Rvision P Trionotr - Mi Eris & Frwors. Clult t vlu o in trinl low.. Clult t r o t trinl wit sis surin, n 0.. Clult t lnt o BD. Clult t lnt o AD. Clult t r o trinl ABC. Fro t rwor oosit: Clult t lnt o AC. Clult t siz o BAC. A o B o C Writ own t siz o ACD. Clult t lnt o AD. Clult t r o t uriltrl ABCD. o St Anrw s A Mttis Dt 0- D

27 Ntionl Mttis Trionotr - Brins Rvision P N. Co t rin ir oosit n ill in s n nls s ou n. Now nswr t ollowin ustions.. A o o N Wt is t rin o i B ro A ii A ro B iii C ro B iv A ro C v C ro A vi B ro C N C N 0 o B N B 0 o. Rt ustion. or tis rin ir. A 0 o o N C N. A si sils ro rour H on rin o 0 o or 0 until it rs oint P. It tn sils on rin o 0 o or 0 until it rs oint Q. Clult t istn twn oint Q n t rour. On wt rin ust t si sil to rturn irtl to t rour ro Q? H N o 0 P 0 o N 0 Q. A n B rrsnt two orst loo-out towrs. A is n on rin o 0 o ro B. A orst ir is sit t F, on rin o 00 o ro A n 0 o ro B. B A ir-itin liotr lvs A or F. Wt istn os tis liotr v to trvl to r t ir? A F. A survor is wlin u wst wn os to rs. To voi t rs turns t P n wls or 0 trs on rin o o n tn or trs on rin o 0 o. H tn lults t istn PR, t irt istn ross t rs. Wt nswr soul t?. Two sis lv Livrool t t s ti. On o t trvls nort-wst t n vr s o 0 / wil t otr trvls t n vr s o / on rin o 0 o. How r rt r ts sis tr ours? St Anrw s A Mttis Dt 0-

28 Ntionl Mttis Rvision P St Anrw s A Mttis Dt 0- Eutions Involvin Frtions. Solv t ollowin utions. 0 0 v o n l j i w Solv o t ollowin utions. 0 u t s r o n w l j i. Solv o t ollowin utions. w w w i

29 Ntionl Mttis Eutions Involvin Frtions Rvision P. Solv o t ollowin utions. i. Solv t ollowin utions. 0 i j. I tin o nur. I I ivi t nur n tn 0, t rsult uls l o t oriinl nur. For n ution n solv it to in. I tin o nur. I I in two-tirs o t nur n sutrt, t rsult is t s s lvin t oriinl nur. For n ution n solv it to in. I tin o nur n. I oul tis nur n on svn. I tis totl is tn ivi tr t rsult is on lss tn t oriinl nur. For n ution n solv it to in n. A nur is oul n sutrt ro 0. Wn tis rsult is lv t nswr is oun to on-n--l tis t oriinl nur. For n ution n solv it to in.. M trol tn ol litrs o trol wn ull. I son o litrs to ut in lwn-owr n us l o wt is lt on journ. I tr r tn litrs lt, ow n litrs wr in t ull tn?. A roll o wir ws trs lon. A tr lnt is ut-o n us to rwir juntion o. Hl o wt is lt is us on notr jo. I tr is on tir o t roll still unus stlis t vlu o.. A sll us, wn lnt, n oriinl it o ntitrs. Atr ont it rown urtr l o its oriinl it. T it o t us ws tn ru ntitrs. Atr tis runin t us ws oun to tr urtrs o its oriinl it. Fin t oriinl it o t us. St Anrw s A Mttis Dt 0-

30 Ntionl Mttis Rvision P St Anrw s A Mttis Dt 0- Frtionl Erssions. Sili t ollowin vt t v. Sili t ollowin n r t t r l j i n n. Writ o t ollowin s sinl rtion in its silst or. l z w j i 0. Sili t ollowin. i v u v u

31 Ntionl Mttis Surs Rvision P. Erss o t ollowin in its silst or i 00 j l n o 0 r s t u. Sili: i 0 - j l Sili: i j l n o. En n sili: i - j + + l En n sili: i + + j + + l - - n -. Erss o t ollowin wit rtionl nointor. o 0 i j 0 r 0 l s 0 t n u. tnsion Rtionlis t nointor, in rtion, usin t rorit onjut sur. St Anrw s A Mttis Dt 0-0

32 Ntionl Mttis Surs - Prols Rvision P. A rit nl trinl s sis, n s sown. For s low lult t lnt o t tir si, rssin our nswr s sur in its silst or. Fin i = n =. Fin i = n =. Fin i = n = Fin i = n =.. Givn tt = + n = -, sili: Givn tt = + n = -, sili: - -. A rtnl s sis surin + n -. Clult t t vlu o i its r; ii t lnt o ionl.. A urv s s its ution. I t oint P, lis on tis urv in t t vlu o. Fin t t lnt o OP wr O is t oriin.. In ABC, AB = AC = n BC =. Erss t lnt o t ltitu ro A to BC s sur in its silst or.. An uiltrl trinl s o its sis surin trs. Fin t t lnt o n ltitu o t trinl in trs o. Hn in t t r o t trinl in trs o.. T t r o rtnl is + sur ntitrs. Givn tt t rt o t rtnl is, sow tt t lnt is ul to +.. tnsion Erss in t or tnsion Givn tt tn, sow tt tn +. St Anrw s A Mttis Dt 0-

33 Ntionl Mttis Rvision P St Anrw s A Mttis Dt 0- Inis You ust wr o t lws o inis n ll t otr inortion sown in tis o to sussull olt tis worst. Sili t ollowin rssin our nswrs wit ositiv inis. 0 0 r w w w w o n l r r j i. Sili t ollowin rssin our nswrs wit ositiv inis. 0 o n l j i. Erss witout root sins. r. Erss wit root sins writ wit ositiv inis irst wr nssr. w. Evlut o t ollowin witout t us o lultor r o n l j i. Sili o t ollowin. i nin root sins to rtionl owrs; ii ovin 's onto t nurtors; iii nin rts wr nssr. l j i,,,,, 0 n n n n n

34 Ntionl Mttis Rvision P Ssts o Eutions Prols. Solv t ollowin ssts o utions A n is i.0 r our si n 0.0 r our ovrti. In rtiulr w wor totl o ours. I is totl w or t w ws, ow n ours ovrti i wor?. A so slls two ts o otition il, rin n ross-ountr. On rtiulr Stur t so sol totl o ls wit till rits totllin. Givn tt rin i rtils or n ross-ountr i or, stlis ow n o t o i ws sol.. A so slls r- oos or.0 n r-s or.0. On rtiulr oos wr sol risin.0. How n r- oos wr sol?. A orul or vloit v is ivn s v u t t, wr u is t initil vloit, is lrtion n t is t ti in sons. It is nown tt wn t =, v = n wn t =, v =. For sst o utions n solv to in u n. Hn in v wn t = 0.. A son orul or vloit v is v u s, wr u is t initil vloit, is lrtion n s is islnt It is nown tt wn s = 0, v = 0 n wn s = 0, v =. For sst o utions n solv to in u n. Hn in v wn s = 00. St Anrw s A Mttis Dt 0-

35 Ntionl Mttis Rvision P N Mts - Mi Eris. Solv o t ollowin utions.. A irl ntr O s rius o s sown. Cor AB s lnt o. Clult t siz o nl AOB to t nrst r. Hn lult t lnt o r AB. O B Clult t r o t s snt. A. Sili.. Clult t totl volu o tor unit wi is 0 trs lon n wos si viw insions r sown oosit 0. Erss s sinl rtion. Solv, orrt to two il ls, t ution 0.. M t sujt o t orul. St Anrw s A Mttis Dt 0-

Ntionl Mttis N Mts - Mi Eris Rvision P. Solv t inution Ftoris t rssion. Hn, i solv t ution = 0; ii sili t rtion. Solv t sst o utions Solv t ution. A orul onnt wit lns sis is ivn s u.

36 Ntionl Mttis N Mts - Mi Eris Rvision P. Solv t inution Ftoris t rssion. Hn, i solv t ution = 0; ii sili t rtion. Solv t sst o utions Solv t ution. A orul onnt wit lns sis is ivn s u. v M v t sujt o t orul. Hn in v wn u = n =.. Consir t ir oosit. Q Clult t lnt o QR. Hn lult t siz o nl RQS. Clult t r o uriltrl PQSR. P o R S. Two intil soli srs r in t sllst o ossil wi is uoi in s. Clult t ount o unoui s lt in t o ivn tt t rius o t sr is 0. [ Volu o sr : V r ] St Anrw s A Mttis Dt 0-

37 Ntionl Mttis N Mts - Mi Eris Rvision P. Solv t ution.. A sur o si s t s ritr s rtnl o rt n lnt l. Fin t lnt o t rtnl in trs o. + + Writ own n rssions, in trs o, or t rs' o t two ss. l Givn tt t r o t sur is or tn t r o t rtnl, or n ution n solv it to in. +. T ir oosit rrsnts rn or litin wits. A AB = trs, AC = trs n CB = trs. Clult t siz o nl ABC. Clult t vrtil it o oint A ov t roun. C B. Givn tt 0 n 0 Writ our nswr in stnr or., in t vlu o E wr E.. T ost C o nuturin n rtil is ivn C, wr n r onstnts. Wn =, t ost o nutur is 0 n wn =, t ost o nutur is 0. Writ own two utions involvin n n solv t siultnousl to in n. Hn lult t ost o nutur wn = 0.. A soli on s s rius o n vrtil it o. Clult t on's i slnt it; ii totl sur r; iii volu. Giv ll nswrs orrt to siniint iurs. [ Sur Ar = r rs ; V r ] St Anrw s A Mttis Dt 0-

38 Ntionl Mttis Crit Mts - Mi Eris Rvision P. Solv i ii. A s sr os in uoi s o surin 0. I t o is to n to linril on o t wt soul t itr o t nw o i t volu o t ontnts is to rin t s?. T ir rrsnts vrtil tlr ol wi is suort two strit ros AB n CD o lnt trs. CP is trs n nl PAB = o. Clult t istn twn t oints B n D ivin our nswr in ntitrs to t nrst ntitr. B D C P o A. Trinl PQR s sis PQ =, QR = 0 n RP = s sown. PS is n r o irl wit ntr Q. P Clult t s r.. T lowr vss oosit r ttill siilr. To ill t sll vs wit wtr ts 00 illilitrs. How u wtr, in litrs, is n to ill t lr vs? R S 0 Q. Sili: St Anrw s A Mttis Dt 0-

39 Ntionl Mttis Answrs to N Mttis P Sintii Nottion Rvision P , 0 000, , 0 000, i j l i 0. 0 ils tis 0 ros 0 0 tos Siilrit. i or ii i 0.. or ii 0 i or ii i 0 or ii.. 00 l 0 l litrs St Anrw s A Mttis Dt 0-

40 Ntionl Mttis Siilrit Rvision P. = 0 =. Bus t r uinulr CD =. = =. ST =. istn = 0 Ptors. 0 i. BD = 0 BC = Ar = 0. or Ar. 0 Ptors.. 0. i Ys ii No iii Ys. Proo BC = 0, Ar = St Anrw s A Mttis Dt 0-

41 Ntionl Mttis Rvision P St Anrw s A Mttis Dt 0- Ptors - Prols P =. Brts & Eutions r o n l j i r s s o n l j i s s w v u t s t t r o n l z z j i t t... 0

42 Ntionl Mttis Rvision P St Anrw s A Mttis Dt w v u t s t t r o n l z z j i t t z w v u t s r o n l j i w w Brts & Eutions - Etnsion. 0 0 n l j i l j i. j i Brts & Eutions - Alitions. i P ii A. i P ii A..

43 Ntionl Mttis Rvision P... A PQ. P. P, 0 P, A A, A, 0 0 Siultnous Eutions. i u v j l. i u v j l.,,,,,, St Anrw s A Mttis Dt 0-

44 Ntionl Mttis Siultnous Eutions Rvision P. n 0. n. n. n. =, =. o =, rl =. uttr =, il =. ours. 0 r tits 0. it n oins Volu Ftoristion t + + i + j r l + + n o t t + +. i j w w l n w w o r s t s + s i j l n s s o St Anrw s A Mttis Dt 0-

45 Ntionl Mttis Rvision P. + + t t i + j z + z + l + + n + o + 0 Forul.. =. = +. =. = i. = j. = + t. l.. n. o n... t r s.. r i.... P l. G U T. V I R. v u t. S T. D P l i. A. l H C.... z w. i. = j.... s.. = + z l. v u V.. s. u v s. r V. r. A = r. L. = t. i. z r j.. st R A A. l. n A t r r n n. R t r r n St Anrw s A Mttis Dt 0-

46 Ntionl Mttis Answrs to N Mttis Rvision P Grint & Eution o Lin Rvision P. i ii iii iv v zro 0, unin. i. Stunts own rwins.., 0,, 0,-, 0,0, 0,, 0,, 0,-, 0,, 0, i, 0,-., 0,-, 0,, 0,-, 0,0, 0,, 0,, 0,, 0, i, 0, i St Anrw s A Mttis Dt 0-

47 Ntionl Mttis T Cirl Rvision P T Cirl..... T Cirl. 0 o. 0 o. o o. T Cirl Eris. o. o. o. o. o, o`. 0 o. o. o, o St Anrw s A Mttis Dt 0-

48 Ntionl Mttis Rvision P Eris. 0 o. o. 0 o. o Inutions. {-, -, 0, } {,,, } {} {-, -} {-, -, 0} {,,,, } {-, -, 0,, } {,,,, }. > < - > < >. < > - 0 < - - > 0 < i w < - - j > 0. {0,,,,, }. {0,,}. Jn ust ounr tn rs. St Anrw s A Mttis Dt 0-

49 Ntionl Mttis Rvision P Trionotr. 0. o o o o o o Trionotr.. o o 0 o o o o Trionotr.... Trionotr!. o o.. 0. o 00 o St Anrw s A Mttis Dt 0-

50 Ntionl Mttis Rvision P Trionotr N. i o 0 o N N o o 0 o ii o iii 0 o iv o v o vi 00 o o N. i 00 o N 0 o 0 o ii 0 o 0 iii 0 o iv o v 0 o vi 0 o o 0 o o 0 o N Trionotr. ont. o.. 0. St Anrw s A Mttis Dt 0-

51 Ntionl Mttis Eutions Involvin Frtions Rvision P. i j l n o 0. 0 i 0 0 j l 0 0 n o - r s - t u. - - i Eutions Involvin Frtions. 0 i. 0 - i j. 0. litrs... St Anrw s A Mttis Dt 0-0

52 Ntionl Mttis Frtionl Erssions Rvision P v. t. n i j l 0 n r. 0 i w z z j 0 w z l. 0 u v i Surs. i 0 j l n o 0 00 r s t u. i j 0 l. i 0 j l 0 n o St Anrw s A Mttis Dt 0-

53 Ntionl Mttis Surs.ont Rvision P i - j + + l i + j + + l n i j l 0 0 n o r s t u Surs -Prols i ii.... Proo Proo. St Anrw s A Mttis Dt 0-

54 Ntionl Mttis Inis Rvision P. 0 - i j r l n o w r. 0 i j 0 l n o. r. w. i j - l 000 n o r i - - j l + + Ssts o Eutions Prols. =, = =, =. ours. rin n ross ountr.. u + =, u + =, 0, 00. u + 0 = 00, u + 0 =,, St Anrw s A Mttis Dt 0-

55 Ntionl Mttis Mi Eris Rvision P.. 0 o. t sur o v n invrsl s t. 00, ,. = Mi Eris. < + -,- - i -0, ii.,. u v = u 0. o.. 0. o, o Mi Eris. -, , o = 0, + = 0,, 0. i 0 ii. / St Anrw s A Mttis Dt 0-

56 Ntionl Mttis Rvision P Mi Eris. i ii, , 0. or n.. litrs. St Anrw s A Mttis Dt 0-

d e c b a d c b a d e c b a a c a d c c e b

d e c b a d c b a d e c b a a c a d c c e b FLAT PEYOTE STITCH Bin y mkin stoppr -- sw trou n pull it lon t tr until it is out 6 rom t n. Sw trou t in witout splittin t tr. You soul l to sli it up n own t tr ut it will sty in pl wn lt lon. Evn-Count