Mthemtics Lst Miutes Review 60606 Form 5 Fil Emitio Dte: 6 Jue 06 (Thursdy) Time: 09:00-:5 (Pper ) :45-3:00 (Pper ) Veue: School Hll Chpters i Form 5 Chpter : Bsic Properties of Circles Chpter : Tgets to Circles Chpter 3: Iequlities Chpter 4: Lier Progrmmig Chpter 5: Applictios of Trigoometry i -D Prolems Chpter 6: Applictios of Trigoometry i 3-D Prolems Chpter 7: Equtios of Circles Chpter 8: Locus Chpter 9: Mesures of Dispersio Chpter 0: Permuttio d Comitio Chpter : More out Proility Chpters i Form 4 Chpter : Qudrtic Equtios i Oe Ukow (I) Chpter : Qudrtic Equtios i Oe Ukow (II) Chpter 3: Fuctios d Grphs Chpter 4: Equtios of Stright Lies Chpter 5: More out Polyomils Chpter 6: Epoetil Fuctios Chpter 7: Logrithmic Fuctios Chpter 8: More out Equtios Chpter 9: Vritios Chpter 0: More out Trigoometry Fil Remider Step y step (Do ot jump!) Drw the grph d look! Get the keywords! Check swers (try to put them ck!) Check uits! Brig Clcultor!!! Try your est! Never give up! Lst Miutes Review []
Mthemtics Lst Miutes Review (Form 5 Chpters d ) 5008 Revisio Circle properties Lst Miutes Review (F5 Chpters d ) []
5008 Revisio Tget properties Lst Miutes Review (F5 Chpters d ) []
Mthemtics Lst Miutes Review (Form 5 Chpters 3 d 4) 56 Revisio Iequlities c c c c (if c 0 ) c c (if c 0, chge sig!) (if c 0 ) (if c 0, chge sig!) c c c c (reciprocl, chge sig!) Ay umer 0 Lst Miutes Review (F5 Chpters 3 d 4) []
56 If 0 d, we hve c 0 0 or c 0 0 or c 0 0 c 0 0 No -itercepts, ll ove -is: 4c 0 (No roots) 4c Verte, 4 Steps of Lier progrmmig:. Let d y e the mout of those items Note: 0, y 0, iteger / umer? Set up equtios ccordig to the questios. DRAW lies! (Get 3 poits solid lie vs. dotted lie) SHADE re! (Try poit (0,0) shde / plot itegrl poits) 3. Write dow the Cost fuctio of d y Set Cost = 0 Plot the lie move i prllel to the m/mi 4. Get m/mi poit d fid Cost (or get the poit from ll etremes) Itersectio of lies 0 y c 0 p qy r () Su () ito () () Lst Miutes Review (F5 Chpters 3 d 4) []
Mthemtics Lst Miutes Review (Form 5 Chpters 5 d 6) 60 Revisio Trigoometry fuctios For right-gled trigle si c cos c t c Pythgors theorem: c Idetities: si cos si t cos For y trigle: Are: Are: Are h Are si C B Are: Are ss s s c c where s (Hero s formul, FMLA03) c Sie lw: si A si B c si C A C Cosie lw: c cosc (FMLA0) c cosc (No FMLA) Lst Miutes Review (F5 Chpters 5 d 6) []
60 Revisio 3-D prolem Directio: True erig: 045 00 (Alwys from N, clockwise) Compss erig: N45E S70W (From N or S to E or W) Drw trigles (or other side) from top view frot view side view Projectio of poit to ple K is verticlly elow H OHK is right-gled trigle Agles defiitios: Lst Miutes Review (F5 Chpters 5 d 6) []
Mthemtics Lst Miutes Review (Form 5 Chpters 7 d 8 ) 60407 Revisio Crtesi coordites For y two poits A, y d, y Mid-poit formul: Sectio formul (r : s): y y, B, s r sy ry, r s r s Distce formul: AB Distce of, y Distce of y y A to lie y k : y k A, y to lie h : h For y -itercept, put 0. For -itercept, put y 0. Horizotl lie: y k Verticl lie: h y Slope of lie ( poits): m Agle of iclitio: m t Two-poit form: y y y y y y y Poit-slope form: m Itercept-slope form: y m c k y h or or Reltioship etwee two lies: prllel ( m m ) perpediculr ( m m ) overlppig (sme slope m m & y-itercept c c ) itersectig ( y y fid the poit) Check whether poit, y is o lie: Put, y ito the lie equtio d see whether LHS = RHS Lst Miutes Review (F5 Chpters 7 d 8) []
60407 Revisio Equtio of circle Give Cetre h, k d rdius r : h y k r Give y D Ey F 0 : D E Cetre =, Rdius = D E F Give 3 poits: Let y D Ey F 0 Su 3 poits 3 equtios 3 ukows Reltioship etwee circle (with cetre O, rdius r ) d poit A : Iside circle: OA r O the circle: OA r Outside circle: OA r Reltioship etwee circle C d stright lie L : C : y y L : y m c D Ey F () () Su () ito () Qudrtic equtio i 0 itersectios 4c 0 itersectio (lie is tget to circle) 0 0 itersectios Fidig locus:. Let y P, e the movle poit of the locus. Use distce formul, slope properties, to set up reltioship 3. Epress the equtio s A By D Ey F 0 Descriptio of locus: Fied distce to poit A circle Fied distce to lie prllel lies Fied distce to lie segmet semi-circles & prllel lies Equl distce to prllel lies A lie i the midwy Equl distce to poits A perpediculr isector Equl distce from crossig lies A pir of gle isectors Equl distce to poit d lie A prol Lst Miutes Review (F5 Chpters 7 d 8) []
Mthemtics Lst Miutes Review (Form 5 Chpters 9, 0 d ) 6050 Revisio Sttistics Me: or f f f f f f (Clcultor: SD Shift ) Medi: Mode: Middle of the sorted dt (odd vs. eve) Numer with highest frequecy Upper clss oudry = (upper limit + lower limit of et higher clss) Lower clss oudry = (lower limit + upper limit of et lower clss) Clss size (Clss width, Clss legth) = upper clss oudry lower clss oudry Clss mid-poit (Clss mrk) = (lower clss limit + upper clss limit) Rge = Highest vlue Lowest vlue (for sigle dt) = Highest clss oudry Lower clss oudry (for grouped dt) Upper Qurtile (UQ, Q3) = medi of upper hlf dt (odd vs. eve) Lower Qurtile (LQ, Q) = medi of lower hlf dt (odd vs. eve) Iter-Qurtile Rge (IQR) = UQ LQ Bo d Whisker Digrm: Mi, LQ, medi, UQ, M (Note: every prt is 5%) Stdrd Devitio (Clcultor SD Shift ) Vrice = Stdrd Score z Ectly 50% of dt more th (or less th) Aout 68% of dt lyig etwee d ( z ) Aout 95% of dt lyig etwee d ( z ) Aout 99.7% of dt lyig etwee 3 d 3 ( 3 z ) Chge of dt: X X c Me: c S.D.: X kx Me: k S.D.: k Lst Miutes Review (F5 Chpters 9, 0 d ) []
6050 Revisio Permuttio d Comitio Coutig: Use whe the cses re hppeed together (AND cse) Use whe the cses re hppeed either wy (OR cse) Fctoril:! With positios: Select group: P r C r! ( r)!! ( r)! r! Revisio Proility outcome _ for _ E Proility: P( E), 0 P ( E) possile _ outcome Complemetry: P ( E) P( E ') where E ' is the complemetry of E Additio: A d B re mutully eclusive, P( A or B) P( A) P( B) A d B re ot mutully eclusive, P( A or B) P( A) P( B) P( A d B) Multiplictio: A d B re idepedet, P( A d B) P( A) P( B) Coditiol: P( A d B) P( B A) (Note: still proility of B!) P( A) Set lguge: AB is the evet tht oth A d B occurs AB is the evet tht A or B (or oth) occurs AB mes A is suset of B {}B mes is elemet of B Lst Miutes Review (F5 Chpters 9, 0 d ) []
Mthemtics Lst Miutes Review (Form 4 Term ) 50609 Revisio Qudrtic epressios c 0 4c c 0 k lm 0 k l 0 or m 0 l or k m, (Not just 3 3 3 3 Specil cses: 0 0!!!) 0 or 0 (Not ccellig!) 4 (Not just, Not 4!!!) For prolems, Let e set equtio The swer is (with uits) Give c 0 Discrimit ( 4c ): 4c 0 rel roots 4c 0 repeted rel root 4c 0 0 rel roots ( comple roots) Give d re the roots ( - itercepts), we hve 0 Give c 0, if d re the roots, we hve (sum of roots), c (product of roots) Lst Miutes Review (Term Emitio) []
50609 Specil cses: 4 If SUM d PRODUCT, the the equtio is ( SUM ) ( PRODUCT) 0 Revisio Comple umers 6 6 4i (Puttig i for simplicity) 3 i i, i, i i, 4 5 i, i i Note: 4 4 i i (Not 4i, Not just i ) i c dii i c di c d i 4 4, (so i, i 4 4 i, i, i 3 i ) ic di c di ci di c di ci d c d d ci i c di i c di c di c di i c di c di c di c di ci di c cdi cdi d i c d d c c d i Lst Miutes Review (Term Emitio) []
50609 Revisio Fuctios d grphs Domi: f, the 0 f, the Fuctio: f c The, f c Grph of y c : y c itercept (Put 0 the y c ) Roots itercept (Put y 0, fid, FMLA) 0, 0 4c : 0 roots ( itercepts); 0 root ( itercept, just touch); 0 0 roots (0 itercepts, ot touch) 4c Verte, 4 Ais of symmetry: 4c Miimum / Mimum vlue: 4 For, verte o right verte o left, differet sig, sme sig Grph of y h k : Verte h, k Completig squre: y 5 (Not h!!!) Ais of symmetry: h (Not just h!!!) Miimum / Mimum vlue: k 6 5 6 9 9 5 6 9 8 5 y Note: Tke 3 y Note: / / y Note: Oly 3 terms, lst term out! 3 3 y Note: Just / Lst Miutes Review (Term Emitio) [3]
50609 Revisio Stright lies Horizotl lie: y k Verticl lie: h For y two poits A, y d, y B, Distce formul: AB y y y y Mid-poit formul:, s r sy ry Sectio formul (r : s):, r s r s y y Slope of lie: m ( t ) Two-poit form: y y y y y y Poit-slope form: m y Itercept-slope form: y m c (Itercept form: ) itercept y itercept Geerl form: A By C 0 -itercept = y-itercept = A C y Slope = B B A B C (Put y = 0) A C (Put = 0) B Reltioship etwee two lies: overlppig (sme slope m m & y-itercept c c ) prllel ( m m ) perpediculr ( m m ) itersectig ( y y fid the poit, see elow) Itersectio of lies 0 y c () 0 p qy r () Su () ito () Get oth d y for the coordites! Lst Miutes Review (Term Emitio) [4]
50609 Revisio Polyomils 3 For polyomil f c d For f is divided y k, Remider is k For f is divided y, Degree = 3, Coefficiet of f. m, Remider is f. m f k 0 k is fctor of 0 0 m f. f m is fctor of 0 f. Log Divisio: Rtiol fuctios: is, Costt Term is d 3 3 3 4 6 0 0 0 8 8 60 78 78 0 0 0 78 34 4 3 6 3 6 5 3 5 ( 5) 3( 5) 3 5 5 3( 5) ( 5) ( 5)( 5) ( 5)( 5) 3 5 0 ( 5)( 5) 5 ( 5)( 5) Divided f ( Divisor ) ( Quotiet ) + ( Remider ) 3 4 8 0 H.C.F. d L.C.M.: f ( ) ( )( ) g( ) 3( )( )( ) H.C.F. ( )( ) 3 0 78 4 L.C.M. 3 ( ) ( ) ( ) 6( )( ) ( ) Lst Miutes Review (Term Emitio) [5]
50609 Revisio Ide d logrithms m m, m m m, m m 0 m m, m m m m m m m m m m,,, 3 3,,, m, o clcultio! (commo fctor) log0, log, log 0, log 0 udefied! log MN log M log N M log log M log N N k M k log M log M N log M log M, log y, log y y y log log y y m log o clcultio! y k log y log k log (By tkig log o oth sides) Grphs: Lst Miutes Review (Term Emitio) [6]
50609 Revisio More out Equtios Specil cses: 0 0 0 or 0 (Not ccellig!) For 4 or 6, let u 4 u ; 3 u u 6 For or 3, let u ; u 4 u 3 u 3 9 For, sme deomitor dd ( ) epd crefully rek ( ) For, put o left oly the squre. MUST CHECK swers For log, comie log the ccel. MUST CHECK swers d d For prolem, Let e set equtio ( s or t ) The swer is t s 0 y Reltioship etwee circle d stright lie: y m c D Ey F () () Su () ito () Qudrtic equtio i 0 itersectios 0 itersectio (lie is tget to circle) 0 0 itersectios Revisio iequlities c c Ay umer 0 c c (if c 0 ) c c (if c 0, chge sig!) (if c 0 ) (if c 0, chge sig!) c c c c (reciprocl, chge sig!) Revisio Rte d Rtio : w : : : c wy : y : z : c y : z Direct vries: y k, where k is o-zero costt k Iversely vries: y, where k is o-zero costt Joitly vries: z ky, where k is o-zero costt Prtly vries: z k k, where k d k re o-zero costts z k k y, where k d k re o-zero costts Check uit! Lst Miutes Review (Term Emitio) [7]
50609 Revisio Estimtio & Percetge is icresed y r%, ew vlue r% ; is decresed y r%, ew vlue r% y y is icresed to y, % icrese 00% ; is decresed to y, % icrese 00% ew old % chge 00% old profit % profit 00% Cost loss % loss 00% Cost Discout % discout 00% Mrked _ price ; Sellig price Cost % profit ; Sellig price Cost %loss Amout for Simple Iterest P P r% Amout for Compoud Iterest P r% ; Sellig price Mrked _ price % discout Revisio Trigoometry fuctios For right-gled trigle si, cos, t c c For P, y o rectgulr ple y y si, cos, t, r r r y c Idetities: si cos, si t cos 80 o 360 o 90 70 o f / f, f / cof o * f f f si, cof cos, cof t, cof cos si t Revisio Grph of the fuctios 0 90 80 70 360 si 0 0 0 cos 0 0 t 0 udef. 0 udef. 0 Properties: si, cos t hs o m/mi si d cos re periodic of 360 t re periodic of 80 Lst Miutes Review (Term Emitio) [8]
Mthemtics Lst Miutes Review (Etr) 60606 Revisio Estimtio & Percetge Asolute error = differece etwee Actul d Mesured vlues Mimum error = Hlf of uit Upper limit = Mesured vlue + Mimum error Lower limit = Mesured vlue Mimum error mimum _ error solute _ error Reltive error = or mesured _ vlue ctul _ vlue % error = Reltive error 00% Revisio Ares d Volumes Circle Are Circumferece r r Sector Are r o 360 Arc legth r o 360 Lst Miutes Review (Etr) []