ALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions

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1 Qudrtic Equtios ALGEBRA Remider theorem: If f() is divided b( ), the remider is f(). Fctor theorem: If ( ) is fctor of f(), the f() = 0. Ivolutio d Evlutio ( + b) = + b + b ( b) = + b b ( + b) 3 = 3 + b 3 + 3b ( + b) ( b) 3 = 3 b 3 3b ( b) b = ( b) ( + b) ( + b + c) = + b + c + (b + bc + c) 3 + b 3 = ( + b) ( b + b ) 3 b 3 = ( b) ( + b + b ) 3 + b 3 + c 3 3bc = ( + b + c) ( + b + c b bc c) if + b + c = b 3 + c 3 = 3bc Lier Equtios A pir of lier equtios i two vribles, s d, is sid to form sstem of simulteous lier equtios i two vribles. The geerl form of sstem of lier equtios i two vribles d is + b + c = 0 + b + c = 0 Set of Equtios Cosistet Sstem mes hve solutio Icosistet Sstem mes hve o solutio b c = b c Depedet sstem hs ifiitel m solutios b c Coditio: = = b c Idepedet sstem: Ol oe uique solutio Coditio: b b For free olie MOCK CATs, logi to Pge :

2 Qudrtic Equtios Geerl form of qudrtic equtio is + b + c = 0 Roots re b m b 4c Sum of roots = b Product of roots = c Nture of roots If b 4c = 0 rel d equl b 4c > 0 b 4c < 0 rel d distict imgir Formig Equtio from roots: If α d β re the roots of qudrtic equtio the tht equtio c be writte i the form (α + β) + αβ = 0 i.e. (sum of the roots) + Product of the roots = 0. Cubic Equtios Cubic equtio 3 + b + c + d = 0 will hve three roots (s,, 3 ). The b Sum = = Sum of Products (tke t time) = = c d Product = 3 = Importt: If the grph of give equtio cuts -is times the it will hve rel roots. E. Wht is the vlue of Sol. = + = + = 0 = For free olie MOCK CATs, logi to Pge :

3 E. = Wht is the vlue of? () () 3 3 (3) (4) 5 3 (5) Noe of these Aswer: (4) E.3 Oe root of + k 8 = 0 is squre of the other, the, the vlue of k is () () 8 (3) 8 (4) (5) 0 Hit: Let the roots be α, α. Iequlities, Mim d Miim Itervls (, b) mes < < b (, b] mes < b [, b] mes b For iequlities questios lws go with the optios d elimite the wrog optios b tkig vlues for the vrible i the give rge. Some Importt Poits If > b d b > c, the > c. If > b d c is rel umber, the + c > b + c d c > b c. If > b > 0, the <. b If > > 0, the log > log, if > d log < log, if 0 < <. A.M G.M H.M + b + c b + bc + c (!) > for >. + 3 for iteger. b + b c + c 3bc. b c d b c d 4 + b 4 + c 4 + d 4 4bcd. If ( ) ( b) 0, < b, the or b. For free olie MOCK CATs, logi to Pge : 3

4 If ( ) ( b) 0, < b, the b. If > b the > + b or < b If < b, the b < < + b. + b + b b b E.4 Lrgest vlue of mi ( +, 6 3), whe > 0, is Sol. For this tpe of questios, Tke, +, = 0 = 4,. (Sice it is give > 0 = ) Put = i either + or 6 3 So, the swer is + = 3. Mimum /Miimum vlue of Qudrtic Equtio. If the sum of two qutities is costt, the the product will be mimum, if both re equl.. If the product is costt, the the sum will be miimum, if both re equl. 3. The equtio + b + c = 0, will hve mimum vlue whe < 0 d miimum vlue whe > 0. The mimum or miimum vlues re give b 4c b 4, d will occur t = b. Fuctios If f() = 3 3 +, the f() = fog() = f{g()} gof() = g{f()} If f() = f( ), the f() is eve fuctio. E.g. If f() = f( ), the f() is odd fuctio. E.g. 3 If f() = = f () Fidig the iverse of give fuctio. E.5 Fid the iverse of = d determie whether the iverse is lso fuctio. 5 Sol. Sice the vrible is i the deomitor, this is rtiol fuctio. Here's the lgebr: Step : Write the origil fuctio = 5 Step : Represet i terms of i the equtio. = 5 Step 3: Replce the 's d 's with ech other. = 5 Thus, the iverse fuctio is =. 5 For free olie MOCK CATs, logi to Pge : 4

5 Importt: If f d g re two fuctios defied from set A i to set B, the. Sum / differece of two fuctios is ( f + g ) () = f() + g (). Product of two fuctios is ( f g) () = f() g () f 3. Divisio of two fuctios is g ( ) = f (). g() Poits o grph If the grph is smmetricl bout -is the it is eve fuctio. If the grph is smmetricl bout origi the it is odd fuctio. For grph questios it is lws better to tke vlues d check the optios. Some Importt grphs Modulus Fuctio f() = or f() = whe < 0 Domi: R Rge: R + whe 0 f() = k o Gretest Iteger Fuctio or Step Fuctio f() = [] Domi: R Rge: Iteger Reciprocl fuctio f() = Domi: R {0} Rge: R {0} f() = / For free olie MOCK CATs, logi to Pge : 5

6 Logrithms Properties of Logrithms log = 0 log = log m = m log log b = m = m logb log m + log = log m log m log = log m/ log log b = = log b m log = m log If log N =, the N =. b For free olie MOCK CATs, logi to Pge : 6

Content: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.

Content: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B. Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe