Notes for SPM SPM Candidates
IMPORTANT NOTES ON SPM MATHEMATICS PAPER 2 No 1
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HIGHLIGHTED NOTES FOR THE STUDENTS SETS Shade the region Intersetion ! Union ! Co"#$i"ent o% A ! A& GRAPHS OF FUNCTIONS- Inequalities Shade the region that satis%ies the ine'(a$ities )no* ho* to DRA+ a straight $ine , e-g- x ! ! 2 is a .ertia$ $ine and y ! ! / is a hori0onta$ $ine Understand the ine'(a$it sign , or 3 is re#resented 4 a 4ro5en $ine 6 7 VOLUME / SURFACE AREA OF A SOLID Deter"ine the t*o so$ids in.o$.ed Choose the Mathe"atia$ o#eration ,8 addition or s(4tration Use the orret gi.en %or"($ae S(4stit(te *ith the orret .a$(es o% r, d, h, l and and w LINES AND PLANES IN 3D S5eth the right:ang$ed triang$e Identi% the ang$e and na"e it Ca$($ate the ang$e (sing (sing trigono"etr; trigono"etr; e-g- tan MATHEMATICAL MATHEMATICAL REASONING REASONI NG =A state"ent> or =Not a state"ent> state"ent> , Ans*ers s(h as ?ES ; NO on$ on$ are not ae#ted Use o% AND ; OR ; ALL ; SOME Sentene , @ i% and on$ i% @ , *rite *rite do*n t*o i"#$iations i"#$iations I"#$iation , I% I% @--; then @- , e-ge-g- , I% x is is greater than 0ero; then x is is a #ositi.e n("4erCon.erse i"#$iation i"#$iation , e-g- I% x is is a #ositi.e n("4er; n("4er; then x is greater than 0ero Arg("ents , e-g- 1 ,Pre"ise ,Pre"ise 1, A$$ heagons ha.e si sidessidesPre"ise2 ,
PQRSTU is a hexagon
Con$(sion , PBRSTU has si sidese-g- 2 , Pre"ise 1 , I% x is is greater than 0ero; then x is is a #ositi.e n("4erPre"ise 2 ,
6 is greater than zero
Con$(sion , is a #ositi.e #ositi.e n("4ern("4ere-g- / , Pre"ise 1 , I% 9 x ! 1 ; then x ! ! 9Pre"ise 2 ,
x 4
Con$(sion , 4 x 16 A$$ the ans*ers ans*ers an 4e deri.ed %ro" %ro" the '(estion itse$%itse$%- Dots "(st 4e at $east threethreeSIMULTANEOUS SIMULTANEOUS LINEAR EUATIONS EUATIONS i7 +hen there is a %ration ; "($ti#$ e.er ter" o% the e'(ation 4 the deno"inator to reate a ne* e'(ation So$.e 4 e$i"ination or s(4stit(tion ii7 Use in.erse "atri to so$.e use a calculator to check the answers
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Notes for SPM SPM Candidates
STRAIGHT LINE
y1 y 2
Gradient ; m
x1 x2
E'(ation o% a straight $ine , y ! ! " x Para$$e$ $ines ; sa"e gradient x :intere#t :intere#t ; y ! ! y :intere#t :intere#t ! ; *hen x ! ! x :intere#t :intere#t ! 8 12 ; Do Do not *rite *rite x ! ! 8 12 or 6 8 12 ; 7
UADRATIC UADRATIC EUATION Rearrange to genera$ %or" o% '(adrati e'(ation ; ax 2 bx c 0 Fatorise ; 6 76 7 ! - E'(ation "(st 4e *ritten in this %or"e-g- , 3 x 4 x 2 0 4 x x 2 0 3
annot 4e ae#ted
State the .a$(es o% x.
MATRICES +hen the "atri has no in.erse ; a! " #$ % & Con.ert si"($taneo(s e'(ations into the "atri %or" e-g- , 2 8 / ! 8 19 2 3 x 14 8 / ! 1/ 1 3 y 13 For"($a o% the in.erse "atri 8 *rite the in.erse "atri #e'()e#e'()e e-g- , 2 3 x 14 x
1
1 3 3 1 y
3 y 13
3 14
2 13
inverse matrix on the left
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Gi.e the .a$(e o% x and and the .a$(e o% y 6not 6not in "atri %or"7 CIRCLES III 2 Use the orret gi.en %or"($ae ; 2 r and r Stress on , arclength
11
360
and
area of sec tor
360
r 2
S(4stit(te *ith the orret .a$(es o% r and d. Ans*ers an 4e in in %rations or 9 signi%iant %ig(res %ig(res or 2 dei"a$ #$aes e-g- /2-/2 PRO*A*ILIT+ II M(st 5no* *hen to (se P ( A)
12
2 r
n ( A) and n ( S )
P ( A
'
)
1
P ( A)
GRADIENT AND AREA UNDER THE GRAPH D(ration or $ength $ength o% ti"e is the tota$ ti"e ta5en Area o% tra#e0i(" !
1 2
( a b) h ; Area o% triang$e and Area o% retang$e or s'(are
Distane:ti"e Distane:ti"e gra#h; gradient ! s#eed 6the rate o% hange o% distane7 : gra#h sho*s a hori0onta$ $ine .ehi$e is at rest S#eed:ti"e gra#h ; gradient ! ae$eration 6 the rate o% hange o% s#eed7 : gra#h sho*s a hori0onta$ $ine (ni%or" s#eedonstant s#eed
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SPM 2012- Quadratics Function Notes for SPM SPM Candidates 2011- Cubic Function 2010- Quadratics Function 200- Cubic Function GRAPHS OF FUNCTIONS E"#hasise on the the (se o% the a$($ator to to a$($ate the (n5no*ns (n5no*ns 200!- Reci"roca# Dra* the gra#h , e"#hasise on , Function T,e S$ale an! an! t,e )ane )ane (' (' . ien 200$- Cubic Function Pl(t t,e 0(ints a$$u)atel1 usin a $)(ss s12#(l 4 2006- Reci"roca# Can use 'le.i#le $u)e Function T,e $u)e 2ust #e s2((t,5 0assin t,)(u, ee)1 0(int6 Mar5s ded(ted *hen , 200%- Quadratic Function the sa$e is *rong The sa$e is not (ni%or" thi5ness o% (r.e is not (ni%or" 64o$d thin $ines7 The (r.e does not #ass thro(gh a$$ the #oints
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x is is not in the gi.en range the (r.e$ine is do(4$e $ine roo5ed (r.e is dra*n
r($er is (sed to Join the #oints
Dra* the straight $ine re'(ired eli2inate a)ia#les t,at ,ae in!i$es " to get e'(ation o% straight line straight line i% there are 3 inte)se$ti(ns inte)se$ti(ns ; gi.e 3 ans7e)s %or ans7e)s %or the .a$(es o% A$$ ans*ers are o4tained %ro" the gra#h- Cal$ulate! alues n(t n(t a$$e0te!a$$e0te!TRANSFORMATIONS III Co"4ined trans%or"ation RS "eans trans%or"ation S %o$$o*ed 4 trans%or"ation RP$ease (se the )i,t te)2in(l(16 Al7a1s sta)t t,e ans7e) 7it, t,e t)ans'()2ati(n 'i)st6 S#e$$ing o% the trans%or"ationDes$)i#e in Des$)i#e in %($$ the trans%or"ationT)anslati(n Re'le$ti(n In the $ine ! " x 4 or ! 5 or ! y R(tati(n o 6o; 1o; 2o; o7 entre 6 x ; y 7 7 diretion 8 $o5*iseanti$o5*ise $o5*iseanti$o5*ise
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Enla)e2ent entre 6 x ; y 7 7 sa$e %ator ! k
SPM
STATISTICS 2012- Fre&uenc' Moda$ $ass Po#'gon Mean 2011- (istogra) : the (se o% %or"($ae Frequency Polygon 2010 - *gi+e ta4$e "(st ha.e o$("n %or "id#oint 200 - (istogra) $a4e$ing at the x :ais :ais "(st (se "id#oint 200! - Fre&uenc' the gra#h "(st 4e $osed at 4oth ends Po#'gon A$$ $ines "(st "(st 4e dra*n 4 (sing (sing r($err($er200$ - *gi+e Histogram : ta4$e "(st ha.e o$("ns %or "id#oint or (##er 4o(ndar 2006 - Fre&uenc' : $a4e$ing at the x :ais :ais an 4e "id#oint; (##er 4o(ndar and $ass inter.a$ Po#'gon : there sho($d 4e a ga# 4et*een the y :ais :ais and the %irst 4ar 200% - (istogra) Ogive : ta4$e "(st ha.e o$("n %or ("($ati.e %re'(en and (##er 4o(ndar : etra one $ass 4e%ore the %irst gi.en $ass inter.a$ *ith ("($ati.e %re'(en ! : $a4e$ing at the x :ais :ais "(st 4e (##er 4o(ndar : the gra#h "(st interset at the x :ais :ais : "edian : '(arti$es and inter'(arti$e range
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Notes for SPM SPM Candidates
In%or"ation o% the gra#h , Median is @ Inte Inter' r'(ar (arti ti$e $e rang range e is is @
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@ '(arti$e is @ Stat State" e"en entt , 9< st(d st(den ents ts ha.e ha.e a hei heigh ghtt o% o% 1<< 1<< " "
PLANS AND ELEVATIONS Corret sha#e Satis%ies onditions gi.en Meas(re"ent "(st 4e a(rate 8 .ie* d iret$ %ro"@ Hidden $ine 8 dashed $ine M(st (se the geo"etria$ geo"etria$ instr("ents instr("ents to dra*dra*- e-g- , r($er ; o"#asses; set s'(are DO NOT S)ETCH--- MUST DRA8 'ull s$ale Less "ar5s *i$$ 4e gi.en i% a $ine is a do(4$e $ine ga#s in the dra*ing o% the $ines thi5ness o% $ine is not an etension is (ni%or" 64o$d thin $ines7 seen re%$eted i"age is dra*n it is not a right ang$e 4igger si0e s"a$$er si0e is it is not a straight dra*n $ine EARTH AS A SPHERE - l(nitu!e an! latitu!e G)eat Ci)$le S9et$, the S9et$, the earth and the ang$es i- the %or"($a o% a.erage s#eed; s#eed; Speed
Dis tan ce time
ii- 5not ! na(tia$ na(tia$ "i$es #er ho(r iii- distane a$ong the #ara$$e$ #ara$$e$ $atit(de ! 6di%%erene 6di%%erene in $ongit(de in degrees7 os ang$e o% $atit(de $atit(de
SOME
s"a$$ ir$e
REMINDERS Ro(nd o%% on$ at the $ast ans*er A$$ ste#s "(st 4e $ear$ sho*n Read the instr(tions and '(estion .er are%($$are%($$ The ans*er "(st 4e in the $o*est %or"; to 9 signi%iant %ig(res and to 2 dei"a$ #$aes Cal$ulate 5 'in! 5 s(le 8 a$$ ste#s are $ear$ sho*n'ull 666 Des$)i#e in 'ull 8 a$$ the #ro#erties Master the sienti%i a$($atora$($ator State - ie (nl1 (nl1 t,e ans7e) La#elin 8 Kerte ; #oint or "eas(re"ent "(st 4e $a4e$ed orret$ La#elin 8
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