MathCity.org Merging man and maths

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1 MathCityorg Mergig ma ad maths Defiitios: Mathematics HSSC-I Textbook of Algebra ad Trigoometry for Class XI Collected by: Muhammad Waqas Sulaima This documet cotais all the defiitios of Mathematics HSSC-I (FSc Part 1) from the Textbook of Algebra ad Trigoometry for Class XI It has bee doe to help the studets ad teachers at o cost This work (pdf) is licesed uder a Creative Commos Attributio-NoCommercial-NoDerivatives 40 Chapter # 1 (Number system) Ratioal umber: A umber which ca be writte i the form of called a ratioal umber Irratioal umber: A real umber which caot be writte i the form of q 0 is called a irratioal umber p q, where p, q Z, q 0 is p q, where p, q Z, Real umber: The field of all ratioal ad irratioal umbers is called the real umbers, or simply the "reals," ad deoted R Termiatig decimal: A decimal which has oly a fiite umber of digits i its decimal part, is called termiatig decimal eg 004, 05, 05 example of termiatig decimal Recurrig decimal: A decimal i which oe or more digits repeats idefiitely is called recurrig decimal or periodic decimal eg , Note: Every termiatig ad recurrig decimal is a ratioal umber because it ca be coverted ito commo fractio No-termiatig, o-recurrig decimal: Decimal which either termiates or it is recurrig it is ot possible to covert it ito a commo fractio Thus o-termiatig, o-recurrig decimals represet irratioal umber eg π = 31415, we do t have exact decimal represetatio of this umber Biary operatios: A biary operatio i a set A is a rule usually deoted by that assigs to ay pair of elemets of A to aother elemet of A eg two importat biary operatios are additio ad multiplicatio i a set of real umbers Complex umber: The umber of the form of z= x + iy, where x, y R, i = 1, is called complex umber Here x is called real part ad y is called imagiary part of z eg, 3+ i, 1 i Real plae or coordiate plae: The geometrical plae o which coordiate system has bee specified is called the real plae or the coordiate plae

2 FSc-I / Defiitios: Mathematics HSSC-I - Argad diagram: The figure represetig oe or more complex umbers o the complex plae is called argad diagram Modulus of complex umber: The modulus of a complex umber is the distace from the origi of the poit represetig the umber It is deoted by x + yi or ( x, y ) Chapter # (Sets, Fuctios ad Groups) Set: A set is geerally described as a well-defied collectio of distict objects or a well-defied object collectio of distict object is called set There are three ways to describe a set, Descriptive method: A method by which a set is described i words Example; N = the set of all ature umbers Tabular method: A set may be described by listig its elemets withi brackets N = 1,,3,4, eg { } Set-builder method: I this form, we use a latter or symbol for a arbitrary elemet of a set ad also statig the property that is commo to all members Example; { x x is ay ature umber} Order of a set: Number of elemets i a set is called its order Α =,4 the order of Α is eg { } Equal set: Two sets A ad B are said to be equal sets if each elemet of set A is a elemet of set B both etries are same so A=B Α =,4,6,8, Β =,8,4,6 Example { } { } Equivalet set: Two sets are said to be equivalet if oe to oe correspodece ca be established betwee them Α =,4,6,8, Β = a, b, c, d Example { } { } Sigleto set: A set havig oe elemet is called sigleto set Α = Example { } Null set: A set havig o elemet is called ull set Example Α = {} or Φ Fiite set: A set havig fiite umber of elemets Α =,4,6,8 100 Example { } Ifiite set: A set havig ifiite umber of elemets Α =, 4,6,8 Example { } Subset: If each elemet of set Α is also a elemet set Β The Α is called sub set of Β writte as Α Β ad i case of Β is called Β super set of Α

3 FSc-I / Defiitios: Mathematics HSSC-I - 3 (i) Empty set is a sub set of every set (ii) Every set is subset of itself Proper subset: if Α is a subset of Β ad cotais at least oe elemet which is ot i Α the Α is called proper subset of Β deoted by Α Β Improper subset: If a set of Β ad Α = Β the Α is improper subset of Β its follow that every set is improper subset of itself Power set: The set of all subset of set Α is called power set of Α, deoted by P( A ) Power set of empty set is ot empty,4, { } P( A) Example A = { } the P( A) { } { } { } = Φ,, 4,4 = Uiversal set: Uiversal set is the set that cotais all the elemets ad objects ivolved i the problem uder cosideratio or the set cotaiig all objects or elemets ad of which all other sets are subsets Complimet of a set: The complimet of a set A, deoted by Α set U is the set of all elemets of U, which do ot belog to Α eg U = N the E = O or c Α relative to the uiversal Deductio: To draw geeral coclusio from well kows facts is called deductio Iductio: To draw geeral coclusio from limited umber of observatio or experiece is called iductio Aristotelia logic: Deductive logic i which every statemet is regarded as true or false is called Aristotelia logic No Aristotelia: Deductive logic i which every statemet is regarded scope of third or fourth is called o-aristotelia logic Truth Table: A table to drives truth values of a give compoud statemet i terms of its compoet parts is called truth table Tautology: A statemet which is true for all possible values of variable ivolved i it is called tautology eg p q (~ q ~ p) is a tautology Cotradictio: A statemet which is always false is called Cotradictio or absurdity eg p ~ p Cotigecy: A statemet which ca be true or false depedig upo the truth values of variable eg ( p q) ( p q) is the cotigecy Fuctio: Let Α ad Β be two o-empty set sets If (1) F is a relatio from Α to Β ie F is a subset of Α Β () Domai of F = A (3) No two ordered pairs of F have same 1 st elemets The F is called a fuctio from Α to Β ad is writte as : y = f x F Α Β deoted by ( )

4 FSc-I / Defiitios: Mathematics HSSC-I - 4 Bijective fuctio: (Rage f = Β ad 1-1) A fuctio f which is both oe to oe ad oto is called bijective fuctio Ijective fuctio : (Rage f Β ad 1-1) A fuctio f which is both oe to oe ad ito is called ijective fuctio Groupoid: A o-empty set which is closed uder give Biary Operatio * is called Groupoid Biary operatio: Ay mappig of G G ito G, where G is o empty set, is called biary operatio Semi group: A o-empty set is called semi group if (i) it is closed uder give Biary operatio (ii) The Biary operatio is associative Mooid: A o-empty set is called Mooid (i) it is closed uder give Biary operatio (ii) The Biary operatio is associative (iii) The set have idetity elemet wrt Biary operatio Group: A o-empty set G id called a group wrt Biary operatio * (i) it is closed uder give Biary operatio (ii) The Biary operatio is associative (iii) The set have idetity elemet wrt Biary operatio (iv) Every elemet of G wrt Biary operatio ie a a = a a = e Abelia group: A group G uder Biary operatio * is called Abelia group if Biary operatio is commutative ie a b = b a if a b b a the this is a No Abelia group uder Biary operatio Liear fuctio: The fuctio ( ) { x, y y mx c} represetatio of liear fuctio is a straight lie Quadratic fuctio: The fuctio ( ) it is defied by secod degree equatio i x, y = + is called a liear fuctio Geometrical { x, y y ax bx c} = + + is called a quadratic fuctio, because Uary Operatio: A mathematical producer that chages oe umber ito aother Or it is a operatio which is applied o a sigle umber to give aother sigle umber eg Chapter # 3 (Matrices ad Determiates) Matrix: A arragemet of differet elemets i the rows ad colums, withi square brackets is called Matrix

5 FSc-I / Defiitios: Mathematics HSSC-I eg Α = Order: Order of Matrix tells us about o of rows ad colums order of a matrix = o of rows o of colum a b c Example Α = d e f order of A = x 3 m Row matrix: A matrix havig sigle row is called Row Matrix A = eg [ ] Colum matrix: A matrix havig sigle colum is called colum Matrix 3 e g A = 1 5 Square matrix: A matrix i which o of rows ad colums are equal is called square matrix 5 eg A = 4 8 Rectagular matrix: A matrix i which o of rows ad colums are ot equal is called square 1 3 matrix eg Α = Diagoal matrix: A square matrix havig each of its elemets excepts priciple diagoal equal to zero ad at least oe elemets i its priciple diagoal matrix eg Α = Scalar matrix: A square matrix havig same elemets i priciple diagoal except 1 is called scalar matrix a 0 eg Α = 0 a, Β = Uit matrix or idetity matrix: Let Α = [ a ij ] be a square matrix of order If a ij = 0 for all i j ad a ij = 1 for all i = j, the the matrix Α is called a uit matrix or idetity matrix of order It is deoted by I

6 FSc-I / Defiitios: Mathematics HSSC-I eg I3 = Null matrix or zero matrix: A square or rectagular matrix whose each elemet is zero, is called a ull or zero matrix It is deoted by O m 0 0 O = 0 0 Equal matrix: Two matrix are said to be equal if they are of same order with the same correspodece elemets eg Α = 4, Β = 4 Upper triagular matrix: If all elemets below the priciple diagoal of square matrix are zero the it is called upper triagular matrix eg Α = Lower triagular matrix: If all elemets above the priciple diagoal of square matrix are zero the it is called lower triagular matrix eg Α = Sigular matrix: A square matrix Α is called sigular if Α = 0 No-Sigular matrix: : A square matrix Α is called o-sigular if Α 0 Adjoit of a matrix: The adjoit of a matrix as d b adjα = c a a Α = c b d is deoted by adj Α ad is defied Symmetric matrix: Let Α be the square matrix if Skew symmetric matrix: Let Α be the square matrix if symmetric matrix Hermitia matrix :Let Α be the square matrix if t Α = Α the Α is called symmetric matrix t Α = Α the Α is called skew t Α = Α the Α is called Hermitia matrix

7 FSc-I / Defiitios: Mathematics HSSC-I - 7 Skew hermitia matrix :Let Α be the square matrix if Hermitia matrix Rak: No zero row i a matrix is called rak of the matrix t Α = Α the Α is called skew Chapter # 4 (Quadratic Equatios) Quadratic Equatio: A equatio of secod degree polyomial i a certai variable is called Quadratic Equatio eg x 4 = 0, 5 x 7 x = 0, x + x + 5 = 0 Equatio of type Equatio + + = 0 where a = b = c 0 is called stadard form of Quadratic ax bx c Solutio of Quadratic Equatio: (i) Factorizatio (ii) Quadratic Formula (iii) Completig Square Expoetial Equatio: Equatios i which variable occur i expoets eg x ad 5 x Reciprocal Equatio: A equatio which remais uchaged whe x is replaced by 1, is called a x reciprocal equatio Radical Equatio: Equatio ivolvig radical expressio of the variable is called radical equatio Example x + + x 3 = 7 Remaider Theorem: If a polyomial f ( x ) of degree 1 is divided by ( x a) exits i the remaider the f ( a ) is remaider till o x term Polyomial fuctio: A polyomial i x is a expressio of the form 1 a x + a x + + a x + a, a 0 where is a o-egative iteger ad the coefficiets a, a 1,, a0 are real umbers It ca be cosidered as a polyomial fuctio of x Factor Theorem: The polyomial ( x a) is a factor of the polyomial f ( ) f ( x ) = 0 Chapter # 5 (Partial Fractios) x if ad oly if Partial fractio: Partial fractio is a expressio of a sigle ratioal fuctio as a sum of two or more sigle ratioal fractio Idetity: It is a equatio which holds good for all values of the variable Ratioal Fractio: The Quotiet of two polyomials factor is called Ratioal Fractio P ( x) Q( x) where Q( x) 0, with o commo

8 FSc-I / Defiitios: Mathematics HSSC-I - 8 Proper Ratioal Fractio: A ratioal Fractio is less degree of polyomial Q( x ) eg 3 x +, x + 5 x + 9 ( ) ( ) P x Q x is called if the degree of polyomial P ( x ) P ( x) Improper Ratioal Fractio: A Improper ratioal Fractio Q( x) polyomial P ( x) is greater tha or equal to the degree of polyomial Q x 3 x 9, x + eg ( ) x x + x is called if the degree of Coditioal Equatio: It is a equatio which is true for particular values of variable eg x = 3 if x = 3 Chapter # 6 (Sequeces ad Series) Sequece: Sequece is a fuctio whose domai is subset of the set of atural umbers Real sequece: If all members of a sequece are real umbers, the it is called a real sequece Fiite Sequece: If the domai of a sequece is a fiite set, the the sequece is called fiite sequece Ifiite Sequece: If the domai of a sequece is a ifiite set, the the sequece is called ifiite sequece Series: The sum of a idicated umber of terms i a sequece is called series eg Arithmetic Sequece: A sequece { a } is a Arithmetic Sequece or Arithmetic progressio if a is the same umber for all Ν ad > 1 a = a + ( ) d a Arithmetic Mea: A umber Α is said to be the Α Μ betwee the two umbers a ad b If a, Α, b are i Α Ρ If d is the commo differece of this Α Ρ, the A a = d ad b A = d a + b Thus A a = b A A = Geometric Progressio: A sequece { a } is geometric sequece or geometric progressio if a a is the same o zero umber of all Ν & > 1 1

9 FSc-I / Defiitios: Mathematics HSSC-I - 9 Geometric Mea: A umber is said to be geometric meas betwee two umbers a ad b If G b a, G, b are i G P Therefore = G = ab G = ± ab a G Harmoic Progressio: A sequece of umbers is called harmoic progressio or harmoic sequece if the reciprocal of its terms are i arithmetic progressio The sequece 1,,, are i harmoic sequece sice there reciprocals 1, 3,5, 7 are i A P Harmoic meas: A umber H is said to be the harmoic meas ( H M ) betwee two umbers a ad b, if a, H, b are i HP Chapter # 7 (Permutatio, Combiatio, Probability) Permutatio: A orderig arragemet of objects is called permutatio Circular Permutatio: The permutatio of thigs which ca be represets by the poits o a circle Probability: Probability is the umerical evaluatio of a chace that a particular evet would occur Sample Space: The set S cosistig of all possible outcome of a give experimet is called sample space Combiatio: Whe a selectio of objects is the made without payig regard to the order of selectio Evet: A evet is a subset of sample space P( Α ) = ( Α ) ( S ) Equally Likely: Two evets Α ad Β are said to be Equally Likely if oe evet is as likely to occur as other Mutually exclusive: Α ad Β are said to be mutually if ad oly if they caot both occur at the same time Chapter # 8 (Mathematical Iductio ad Biomial Theorem) Biomial Theorem: A algebraic expressio cosistig of two terms is called biomial expressio eg a + x, x y, ax + b Chapter # 9,10,11,1,13,14 (Trigoometry) Trigoometry: The word trigoometry has bee derived from three Greek words Trei (three) Goi (agles) ad Metro (measuremet) its mea measuremet of triagle Agle: Two rays with commo startig poit from a agle is called a agle

FSc-I / Defiitios: Mathematics HSSC-I - 10 Degree: If the circumferece of circle is divided ito 360 equal parts i legth, the agle subteded by oe part at the cetre of the circle is called a degree

10 FSc-I / Defiitios: Mathematics HSSC-I - 10 Degree: If the circumferece of circle is divided ito 360 equal parts i legth, the agle subteded by oe part at the cetre of the circle is called a degree Allied agles: The agles associated with basic agles of measure θ to the right agle or its multiple are called allied agles The agles of measure 90 ± θ, 180 ± θ, 70 ± θ, 360 ± θ are kow as allied agles Period: period is the smallest positive umber which, whe added to the origial circular measure of the agle, gives the same value of the fuctio Circular system (Radias): A radia is the measure of the cetral agle of a arc of a circle whose legth is equal to the radius of the circle Sexagesimal system: The system of measuremet i which the agle is measured i degree, ad its sub-uits, miutes ad secods is called the Sexagesimal system Example Period of Trigoometric Fuctio: The smallest positive umber which whe added to the origial circular measuremet of the agle gives same value of fuctio is called period Example π is period of sie fuctio as si( α + π ) = siα Trigoometry equatio: The equatio, cotaiig at least oe trigoometry fuctio are called Trigoometry equatio Example six = / 7, cosx tax = 0 Oblique Triagles: A triagle, which is ot right, is called a oblique triagle Circum-Circle: The circle passes through the three vertices of a triagle is called circum-circle I-Circle: A circle draw iside a triagle touchig its three sides is called its iscribed circle or icircle Escribed Circles: A circle, which touches oe side of the triagle exterally ad the other two produced sides, is called a escribed circle or ex-circle or e-circle Trigoometric fuctio: The equatio, cotaiig at least oe trigoometric fuctio, are called trigoometric fuctio If you foud ay error, please report us at /error Collected by: Muhammad Waqas Sulaima (Prof at Saha Group of Colleges Faisalabad) Edited by MathCityorg These resources are shared uder the licece Attributio- NoCommercial-NoDerivatives 40 Iteratioal Uder this licece if you remix, trasform, or build upo the material, you may ot distribute the modified material

TEACHER CERTIFICATION STUDY GUIDE

TEACHER CERTIFICATION STUDY GUIDE COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra