1. Algebra 1 The properties of powers. iv) y xy y. c c. ii) vi) c c c. 2 Manipulating expressions. a b b a. ab ba. 3 Some useful algebraic identities.

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1 . Alger The properties of powers 0 i) ii) y y y iii) iv) y y y v) vi) y y y y Mipultig epressios. ( ) ( ) ( ) ( ) ( ) ( ) 3 Soe useful lgeri idetities. ( )( ) or, ore geerlly, ( ) 4 Solutio of d order polyoil equtio of the for qudrti forul is 0 0. The, Cople uers re of the for z i where i (or, i ). Alger with ople uers uses the followig idetities: z z ( ) ( ) i z z ( ) i z z z z z z, where z is the ojugte of z, suh tht ( )( ). zz i i. Geoetry Pythgor's theore: B Diste etwee two poits i D spe: A C

2 d ( ) ( y y ) Properties of two-diesiol shpes d three-diesiol solids Squre Digols re equl d iset eh other t right gles. Perieter: 4 Are: Retgle Prllelogr h Trpeziu h Cirle r Ellipse h r Trigle h d Digols re equl d iset eh other Perieter: ( ) Are: Digols iset eh other Perieter: ( ) Are: h Two sides re prllel Perieter: d Are: h All poits o the perieter (iruferee) re the se diste (rdius) fro poit (the etre) iterl to the irle. Perieter: r Are: r A ellipse is the set of poits o ple whose su of distes ( ) fro two give poits is the se. Perieter (pproite): r h 3 h r r h rh rh Are: rh The se re e lulted usig y side of the trigle with its orrespodig height. Perieter: h Are:

re: 4r 4 3 Volue: 3 r Cylider r Surfe re: r( r h) Volue: r h h 3. Trigooetry Nottio.

3 Regulr hego h Cue All the trigles fored y drwig the three digols of the hego re equilterl. Perieter: Are: 3h All sides re ll squres Surfe re: 6 Volue: 3 Cuoid Sphere r All sides re retgles Surfe re: Volue: Surfe re: 4r 4 3 Volue: 3 r Cylider r Surfe re: r( r h) Volue: r h h 3. Trigooetry Nottio. Agles re usully deoted y the Greek letters,, Agle oversios etwee degrees (d) d rdis ( ): 360 / d / useful gle oversios re Degrees Rdis Degrees Rdis Soe

4 3 Defiitios of trigooetri futios os( ) / r, si( ) y / r, t( ) y /, ot( ) / y r y 4 Nueril pproitio of si trigooetri futios: si( ) 3! 5! 7! 4 6 os( )! 4! 6! 5 Covertig polr to Crtesi oordites: (, y) ( r os( ), r si( )) 6 Bsi trigooetri idetities: t si / os os ( ) si ( ) os( ) os si( ) si os( ) os si( ) si os( ) os si( ) si 7 Trigooetri idetities for sus of gles: si( ) si os os si os( ) os os si si t t t( ) t t 8 Trigooetri idetities for doule gles: si( ) si os os( ) os si t t( ) t 9 Idetities for the produt of trigooetri futios: si os si( ) si( ) os si si( ) si( ) os os os( ) os( ) si si os( ) os( ) 0 Idetities for the su of trigooetri futios:

5 si si si os si si os si os os os os os os si si Idetities for the gles d sides of olique trigle (i.e. oe, tht does ot hve right gle). The lw of sies : A B C si si si A B A B C BC os The lw of osies : B A C AC os C A B ABos C Geerl trsfortio of os ito ew periodi futio: f fi f f i f ( t) os ( t t0 ) T where fi d f re the iiu d iu of the ew futio (replig - d ), T is the ew period (replig ) d t 0 the ew pek (replig 0). is the lotio of 3 Superpositio of severl periodi opoets is desried y trigooetri polyoil f ( t) ost os t si t si t 4. Sequees Aritheti progressio. The itertive defiitio is d the orrespodig geerl defiitio is 0. Geoetri progressio. The itertive defiitio is d the orrespodig geerl defiitio is 0.

6 3 A tooy of differee equtios preseted usig geeri eples:, d 3 re ostts d ( t), ( t) d 3 ( t) re futios of tie. The etries desrie the right hd side of the differee equtio, so hoogeeous, utooous, lier, ffie, equtio is t t Lier ffie Autooous t Hoogeeous No-utooous ( t ) t Lier t 0 ( t) t 0 No-lier ffie t t No-lier t t 0 t ( t ) ( t ) ( ) t ( ) t 0 t t t Lier 0 ( t ) No-hoogeeous Autooous No-utooous t t 0 ( t) ( t) No lier t t 0 ( t) t t 0 ( t) ( t) ( t) 5. Logriths Properties of the logrithi futio i) log 0 iv) log ( y) log log y ii) log v) log y log log y iii) log k vi) log k log 3 Reltioship etwee turl d oo logriths: l l0log. This iplies tht the two re proportiol to eh other. 6. Liits Properties of liits. If li f ( ) d li g( ), the i) The liit of the su equls the su of the liits

7 li f ( ) g( ) ii) The liit of the produt equls the produt of the liits li f ( ) g( ) iii) The liit of the rtio equls the rtio of the liits, ssuig tht 0 f ( ) li give 0 ( ) g iv) The liit of the produt etwee ostt d futio equls the produt of the ostt d the futio s liit li f ( ) Liits ivolvig ifiity. i) li ii) li Derivtives Defiitio of the derivtive df f ( ) f ( ) li d or, ltertively df f ( ) f ( ) li d 0 There re five ltertive ottios for the derivtive of futio: df ( ) df d f ( ) f ( ) f ( ) d d d Rules of differetitio Futio Derivtive f ( ) f 0 f ( ) f 3 f ( ) g( ) f g 4 f ( ) g( ) h( ) f g h 5 Produt rule f ( ) g( ) h( ) f gh hg 6 7 Quotiet rule 8 f ( ) g( ) f g g g( ) gh hg f ( ) f h( ) h f ( ) g( ) f g g

8 9 f ( ) l 0 f ( ) e f ( ) si f ( ) os 3 f ( ) t f f e f os f si f se 3 The hi rule For futio f ( ) f ( g( )), df df dg d dg d More geerlly, for f ( ) f ( g( g( g( )))), df df dg dg d dg dg d 4 Tylor epsio ( i) i i i! f ( ) f ( ) f ( )( ) 8. Itegrls Bsi rules of itegrtio Futio ( F( ) ) Idefiite itegrl ( F( ) d ) 3 G( ) G( ) d 4 G( ) H( ) G( ) d H( ) d e ( ) dg G d G( ) dg d G( ) G( ) l e

9 9 si os 0 os si t se Mipultig defiite itegrls i) Give itervl [, ] tht oprises the two suitervls [, ] d [, ] the f ( ) d f ( ) d f ( ) d ii) f ( ) d f ( ) d 9. Mtries Properties of tri ritheti. For y three tries A, B, C d slrs,, the followig re true A B B A ( A B) C A ( B C) 3 ( A B) A B 4 ( ) A A A 5 A( BC) ( AB) C 6 IA A, AI A 7 A( B C) AB AC Properties of the trspose ( A ) T T A ( A B) T A T B T 3 ( AB) T B T A T 3 Properties of the deterit det( AB) det( A)det( B) T det( A ) det( A) 3 det( I ) d, if A is ivertile, 4 det( A ) det( A) 4 Nolier dyil systes. The Joi:

10 f f M f f 0. Desriptive sttistis The verge. For sple of oservtios,,, the verge is: i i. If f ( ) is the reltive frequey with whih vlue ours i the sple, the the verge is lso writte f ( ). The two epressios will e idetil if o iig is used to lulte the reltive frequeies (e.g. s is the se i disrete vriles). Oe rdo vrile Vrie i v ( ) ( ) Stdrd devitio s( ) v( ) Skewess 3 ( ) 3 i s i Kurtosis 4 ( ) 3 4 i s 3 Two rdo vriles i i All Covrie ov(, y ) ( )( y y ) Correltio i ov(, y) r s s y i i. Sets d evets The epty set is writte d the evet spe of eperiet is ovetiolly writte. Coprisos etwee two sets A B The set A is idetil to the set B A B The set A is suset of the set B A B Set A is suset of, or the se s, set B A B Set A is ot suset of B

11 3 Set opertors A Negtio: ot A A B Uio of two sets: A or B A B Itersetio etwee two sets: A d B 4 Iportt reltioships etwee evets B A : Copleetry evets A B : Colletively ehustive evets A B : Mutully elusive evets. Rules of proility For set of utully elusive d olletively ehustive evets E, E, E3,, E, it is lwys true tht i likely, eh with proility p, the opleetry, the P( E) P( E ). P( E ). If these evets re eqully i p. Also, if two evets re The proility of the uio of two utully elusive evets is P( E E ) P( E ) P( E ). The proility of the uio of y two evets is P( E E ) P( E ) P( E ) P( E E ) 3 Rules delig with oditiol proility P( E E ). P( E E ) P( E ). P( E E ) P( E ) 3. P( E E ) P( E ) P( E ) P( E E ) P( E ) (Bye s lw) (Idepedet evets) 4. Totl proility. Give the proilities P( E ),, P( E ) of utully elusive d olletively ehustive evets, d the oditiol proilities P( G E ),, P( G E ) referrig to soe other evet G, the the totl proility of G is P( G) P( G Ei ) P( Ei ) i 3 Byesi proility. P( dt H) P( H) P( H dt) P( dt) 3. Proility distriutios

12 Disrete rdo vriles Proility ss futio (PMF): f ( ) P( X ) Properties of the PMF: i) 0 f ( ) ii) f ( ) All Cuultive distr. futio (CDF): F ( ) P( X ) Properties of the CDF: i) li F( ) 0, ii) li F( ), iii) If, the F( ) F( ) Reltioship etwee the PMF d CDF: f ( ) F( ) F( ) & F( ) f ( i) X X i Cotiuous rdo vriles Defiitio d properties of CDF: se s i disrete se. Proility desity futio (PDF): defied s the derivtive of the CDF. Properties of the PDF: i) f ( ) 0, ii) P( X ) f ( ) d, iii) f ( ) d Reltioship etwee PDF d CDF: df( ) f ( ) d & F ( ) f ( s ) ds 4. Epettio Epettio of rdo vrile: If X is disrete: E( X) f ( ) If X is otiuous: E( X) f ( ) d Moets of distriutio: If X is disrete: ( E X ) f ( ) If X is otiuous: E( X ) f ( ) d

13 3 Epettio of geerl futio g( X) If X is disrete: E( g( X)) g( ) f ( ) If X is otiuous: E( g( X)) g( ) f ( ) d 4 Reltioship etwee oets d desriptive sttistis E( X), Vr( X) E( X ) E( X), ov( X, Y) E( X )( Y ) X Y 5 Mipultig epettios d vries E( X Y) E( X) E( Y) E( X) E( X) Vr X ( ) Vr( X) E( XY) E( X) E( Y) (idepedet vriles) Vr( X Y) Vr( X) Vr( Y) (idepedet vriles) 5. Coitoris The ioil oeffiiet outs he uer of wys of orderig suesses! i strig of trils:!( )! 6. Disrete proility distriutios Ne PMF Me Vrie Uifor U(, k ) k Bioil p q B(, p) ( ) k p pq Poisso e Poisso( )! Geoetri ( is o of f ( ) pq p q p trils) Geoetri( p) Geoetri ( is o of filures) Geoetri( p) Negtive ioil if is o of trils NegB( k, p) Negtive ioil if is o of filures f ( ) pq q p q p f p q k k k k k p q k k p kq p k p q k p q

14 X ~ NegB( k, p) Multioil Multio(, p)! p!! k p k k i p i i p i q i. 7. Cotiuous proility distriutios Ne PMF Me Vrie Uifor U(, ) Epoetil M( ) G G( k, ) Bet Bet(, ) Norl N(, ) t-distriutio ( ) ( ) e k k e ( k) t Logorl LogN(, ) Chi-squre ( ) ( ) ( ) ( ) ( ) y e e e ( ) (l y) k k ( ) ( ) 0 if if e ( e ) e

AIEEE CBSE ENG A function f from the set of natural numbers to integers defined by

AIEEE CBSE ENG A function f from the set of natural numbers to integers defined by AIEEE CBSE ENG. A futio f from the set of turl umers to itegers defied y, whe is odd f (), whe is eve is (A) oe oe ut ot oto (B) oto ut ot oe oe (C) oe oe d oto oth (D) either oe oe or oto. Let z d z e