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PREFACE
‘‘THE ALGEBRAIC SUM OF ALL THE TRANSFORMATIONS OCCURRING IN A CYCLICAL PROCESS CAN ONLY BE POSITIVE, OR, AS AN EXTREME CASE EQUAL TO NOTHING’’ MEANS IF YOU CONTINUOUSLY PUT YOUR EFFORTS ON AN ASPECT YOU HAVE VERY GOOD CHANCE OF POSITIVE OUTCOME i.e. SUCCESS
It is a matter of great pride and honour for me to have received such an overwhelming response to the previous editions of this book from the readers. In a way, this has inspired me to revise this book thoroughly as per the changed pattern of JEE Main & Advanced. I have tried to make the contents more relevant as per the needs of students, many topics have been re-written, a lot of new problems of new types have been added in etcetc. All possible efforts are made to remove all the printing errors that had crept in previous editions. The book is now in such a shape that the students would feel at ease while going through the problems, which will in turn clear their concepts too
A Summary of changes that have been made in Revised & Enlarged Edition
— Theory has been completely updated so as to accommodate all the changes made in JEE Syllabus & Pattern in recent years.
— The most important point about this new edition is, now the whole text matter of each chapter has been divided into small sessions with exercise in each session. In this way the reader will be able to go through the whole chapter in a systematic way.
— Just after completion of theory, Solved Examples of all JEE types have been given, providing the students a complete understanding of all the formats of JEE questions & the level of difficulty of questions generally asked in JEE.
— Along with exercises given with each session, a complete cumulative exercises have been given at the end of each chapter so as to give the students complete practice for JEE along with the assessment of knowledge that they have gained with the study of the chapter
— Last 13 Years questions asked in JEE Main & Adv, IIT-JEE & AIEEE have been covered in all the chapters.
However I have made the best efforts and put my all Algebra teaching experience in revising this book Still I am looking forward to get the valuable suggestions and criticism from my own fraternity i.e. the fraternity of JEE teachers.
I would also like to motivate the students to send their suggestions or the changes that they want to be incorporated in this book.
All the suggestions given by you all will be kept in prime focus at the time of next revision of the book.
Dr. SK Goyal
CONTENTS
COMPLEX NUMBERS 1.
LEARNING PART
Session 1
— Integral Powers of Iota (i)
— Switch System Theory
Session 2
— Definition of Complex Number
— Conjugate Complex Numbers
— Representation of a Complex Number in Various Forms
Session 3
— amp (z)– amp (–z)=± p, According as amp (z) is Positive or Negative
— Square Root of a Complex Number
— Solution of Complex Equations
— De-Moivre’s Theorem
— Cube Roots of Unity
THEORY OF EQUATIONS 2.
LEARNING PART
Session 1
— Polynomial in One Variable
— Identity
— Linear Equation
— Quadratic Equations
— Standard Quadratic Equation
Session 2
— Transformation of Quadratic Equations
— Condition for Common Roots
Session 3
— Quadratic Expression
— Wavy Curve Method
— Condition for Resolution into Linear Factors
— Location of Roots (Interval in which Roots Lie)
Session 4
— nth Root of Unity
1-102
— Vector Representation of Complex Numbers
— Geometrical Representation of Algebraic Operation on Complex Numbers
— Rotation Theorem (Coni Method)
— Shifting the Origin in Case of Complex Numbers
— Inverse Points
— Dot and Cross Product
— Use of Complex Numbers in Coordinate Geometry
PRACTICE PART
— JEE Type Examples
— Chapter Exercises
103-206
Session 4
— Equations of Higher Degree
— Rational Algebraic Inequalities
— Roots of Equation with the Help of Graphs
Session 5
— Irrational Equations
— Irrational Inequations
— Exponential Equations
— Exponential Inequations
— Logarithmic Equations
— Logarithmic Inequations
PRACTICE PART
— JEE Type Examples
— Chapter Exercises
3.
SEQUENCES AND SERIES
LEARNING PART
Session 1
— Sequence
— Series
— Progression
Session 2
— Arithmetic Progression
Session 3
— Geometric Sequence or Geometric Progression
Session 4
— Harmonic Sequence or Harmonic Progression
4.
LOGARITHMS AND THEIR PROPERTIES
LEARNING PART
Session 1
— Definition
— Characteristic and Mantissa
Session 2
— Principle Properties of Logarithm
5.
PERMUTATIONS AND COMBINATIONS
LEARNING PART
Session 1
— Fundamental Principle of Counting
— Factorial Notation
Session 2
— Divisibility Test
— Principle of Inclusion and Exclusion
— Permutation
Session 3
— Number of Permutations Under Certain Conditions
— Circular Permutations
— Restricted Circular Permutations
Session 4
— Combination
— Restricted Combinations
207-312
Session 5
— Mean
Session 6
— Arithmetico-Geometric Series (AGS)
— Sigma (S) Notation
— Natural Numbers
Session 7
— Application to Problems of Maxima and Minima
PRACTICE PART
— JEE Type Examples
— Chapter Exercises
313-358
Session 3
— Properties of Monotonocity of Logarithm
— Graphs of Logarithmic Functions
PRACTICE PART
— JEE Type Examples
— Chapter Exercises
359-436
Session 5
— Combinations from Identical Objects
Session 6
— Arrangement in Groups
— Multinomial Theorem
— Multiplying Synthetically
Session 7
— Rank in a Dictionary
— Gap Method [when particular objects are never together]
PRACTICE PART
— JEE Type Examples
— Chapter Exercises
6.
BINOMIAL THEOREM
LEARNING PART
Session 1
— Binomial Theorem for Positive Integral Index
— Pascal’s Triangle
Session 2
— General Term
— Middle Terms
— Greatest Term
— Trinomial Expansion
Session 3
— Two Important Theorems
— Divisibility Problems
7.
DETERMINANTS
LEARNING PART
Session 1
— Definition of Determinants
— Expansion of Determinant
— Sarrus Rule for Expansion
— Window Rule for Expansion
Session 2
— Minors and Cofactors
— Use of Determinants in Coordinate Geometry
— Properties of Determinants
Session 3
— Examples on Largest Value of a Third Order Determinant
— Multiplication of Two Determinants of the Same Order
8.
MATRICES
LEARNING PART
Session 1
— Definition
— Types of Matrices
— Difference Between aMatrix and a Determinant
— Equal Matrices
— Operations of Matrices
— Various Kinds of Matrices
437-518
Session 4
— Use of Complex Numbers in Binomial Theorem
— Multinomial Theorem
— Use of Differentiation
— Use of Integration
— When Each Term is Summation Contains the Product of Two Binomial Coefficients or Square of Binomial Coefficients
— Binomial Inside Binomial
— Sum of the Series
PRACTICE PART
— JEE Type Examples
— Chapter Exercises
519-604
— System of Linear Equations
— Cramer’s Rule
— Nature of Solutions of System of Linear Equations
— System of Homogeneous Linear Equations
Session 4
— Differentiation of Determinant
— Integration of a Determinant
— Walli’s Formula
— Use of S in Determinant
PRACTICE PART
— JEE Type Examples
— Chapter Exercises
605-690
Session 2
— Transpose of a Matrix
— Symmetric Matrix
— Orthogonal Matrix
— Complex Conjugate (or Conjugate) of a Matrix
— Hermitian Matrix
— Unitary Matrix
— Determinant of a Matrix
— Singular and Non-Singular Matrices
Session 3
— Adjoint of a Matrix
— Inverse of a Matrix
— Elementary Row Operations
— Equivalent Matrices
— Matrix Polynomial
— Use of Mathematical Induction
9.
PROBABILITY
LEARNING PART
Session 1
— Some Basic Definitions
— Mathematical or Priori or Classical Definition of Probability
— Odds in Favours and Odds Against the Event
Session 2
— Some Important Symbols
— Conditional Probability
Session 3
— Total Probability Theorem
— Baye’s Theorem or Inverse Probability
10.
MATHEMATICAL INDUCTION
LEARNING PART
— Introduction
— Statement
— Mathematical Statement
11.
SETS, RELATIONS AND FUNCTIONS
LEARNING PART
Session 1
— Definition of Sets
— Representation of a Set
— Different Types of Sets
— Laws and Theorems
— Venn Diagrams (Euler-Venn Diagrams)
Session 2
— Ordered Pair
— Definition of Relation
— Ordered Relation
— Composition of Two Relations
Session 4
— Solutions of Linear Simultaneous Equations Using Matrix Method
PRACTICE PART
— JEE Type Examples
— Chapter Exercises
691-760
Session 4
— Binomial Theorem on Probability
— Poisson Distribution
— Expectation
— Multinomial Theorem
— Uncountable Uniform Spaces
PRACTICE PART
— JEE Type Examples
— Chapter Exercises
761-784
PRACTICE PART
— JEE Type Examples
— Chapter Exercises
785-836
Session 3
— Definition of Function
— Domain, Codomain and Range
— Composition of Mapping
— Equivalence Classes
— Partition of Set
— Congruences
PRACTICE PART
— JEE Type Examples
— Chapter Exercises
SYLLABUS
JEE MAIN
Unit I Sets, Relations and Functions
Sets and their representation, Union, intersection and complement of sets and their algebraic properties, Power set, Relation, Types of relations, equivalence relations, functions, one-one, into and onto functions, composition of functions
Unit II Complex Numbers
Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality
Unit III Matrices and Determinants
Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three. Properties of determinants, evaluation of deter-minants, area of triangles using determinants. Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations, Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices
Unit IV Permutations and Combinations
Fundamental principle of counting, permutation as an arrangement and combination as selection, Meaning of P(n,r) and C (n,r), simple applications
Unit V Mathematical Induction
Principle of Mathematical Induction and its simple applications.
Unit VI Binomial Theorem and its Simple Applications
Binomial theorem for a positive integral index, general term and middle term, properties of Binomial coefficients and simple applications.
Unit VII Sequences and Series
Arithmetic and Geometric progressions, insertion of arithmetic, geometric means between two given
numbers. Relation between AM and GM Sum upto n 2 3 terms of special series: ∑ n, ∑ n , ∑n . ArithmeticoGeometric progression.
Unit VIII Probability
Probability of an event, addition and multiplication theorems of probability, Baye’s theorem, probability distribution of a random variate, Bernoulli and Binomial distribution.
JEE ADVANCED
Algebra
Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations.
Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots.
Arithmetic, geometric and harmonic progressions, arithmetic, geometric and harmonic means, sums of finite arithmetic and geometric progressions, infinite geometric series, sums of squares and cubes of the first n natural numbers
Logarithms and their Properties
Permutations and combinations, Binomial theorem for a positive integral index, properties of binomial coefficients
Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables.
Addition and multiplication rules of probability, conditional probability, independence of events, computation of probability of events using permutations and combinations
01 CHAPTER ComplexNumbers
LearningPart
Session1
● IntegralPowersofIota(i)
● SwitchSystemTheory
Session2
● DefinitionofComplexNumber
● ConjugateComplexNumbers
● RepresentationofaComplexNumberinVariousForms
Session3
● amp () z amp (), z Accordingasamp () z isPositiveorNegative
Thesquareofanyrealnumber,whetherpositive,negative orzero,isalwaysnon-negativei.e. x 2 0 forall xR. Therefore,therewillbenorealvalueof x , whichwhen squared,willgiveanegativenumber.
Thus,theequation x 2 10 isnotsatisfiedforanyreal valueof x .‘Euler’wasthefirstMathematicianto introducethesymbol i (read‘Iota’)forthesquarerootof 1withtheproperty i 2 1 Thetheoryofcomplex numberwaslaterondevelopedbyGaussandHamilton. AccordingtoHamilton,‘‘Imaginarynumberisthat numberwhosesquareisanegativenumber’’.Hence,the equation x 2 10
x 2 1 or x 1 (in the sense of arithmetic, 1 has no meaning). Symbolically, 1 isdenotedby i (thefirstletterofthe word‘Imaginary’).
Solutions of x 2 10 are xi
Also, i istheunitofcomplexnumber,since i ispresentin everycomplexnumber.Generally,if a ispositivequantity, then
i
IntegralPowersofIota () i
(i)If the index of i is whole number, then i 0 1, ii 1 , i 2211 (), iiii 32 , ii4222()()11
To find the value of i n () n 4 First divide n by 4.
Let q be the quotient and r be the remainder. i.e. 4) ( nq 4q r
Remark
aia, where a is positive quantity. Keeping this result in mind, the following computation is correct abiaibiabab 2 where, a and b are positive real numbers. But the computation, ababab()()||||is wrong. Because the property, abab is valid only when atleast one of a and b is non-negative. If a and b are both negative, then abab
(iii)43 n leavesremainder3,whenitisdividedby4. i.e.,443 ) ( nn 4 3 n iii n 433
Now,()()() 1 434343 nnn ii () i i Aliter ()() 1 434343 nnn ii ()ii n 43 ()() 1 n i i
❙ Example5. Findthevalueof 1 2462 iiii n ..., where inN 1and.
Sol. Q 111111 2462 iiii nn ......()
Case I If n is odd, then 1 2462 iiii n ...1111110 ...
Case II If n is even, then 1111111 2462 iiii n
❙ Example6. If a i 1 2 , where i 1, thenfindthe valueof a1929
Sol. Q a iii 2 2 2 1 2 12 2
112 2 i i aaaaaai 192919282964964 ()() ai()4241 ai()4241 a
❙ Example7. Dividing fz() by zi, where i 1, we obtaintheremainder i anddividingitby zi, weget theremainder 1 i Findtheremainderuponthe divisionof fz() by z 2 1
Sol. zi 0 zi
Remainder, when fz() is divided by ()zii i.e. fii()(i) and remainder, when fz() is divided by ()zi 11 i.e. fii ()1[] Qzizi0(ii)
Since, z 2 1 is a quadratic expression, therefore remainder when fz() is divided by z 2 1, will be in general a linear expression.Let gz() be the quotient and azb (where a and b are complex numbers) be the remainder, when fz() is divided by z 2 1
Then, fzzgzazb ()()() 2 1 K (iii) fiigiaibaib ()()() 2 1 or aibi [from Eq. (i)] K (iv) and fiigiaib ()()() 2 1 aib or aibi1[from Eq. (ii)] …(v) FromEqs. (iv) and (v), we get bi 1 2 and a i 2
Hence, required remainder azb 1 2 1 2 izi
TheSumofFourConsecutive Powersof i (Iota) isZero
If nI and i 1,then iiiiiiii nnnnn12323 1() iii n () 11 0
Remark 1. frfrp rp m r mp ()() 1 1 1 2. frfrp rp m r mp ()() 1 1 1
4 TextbookofAlgebra
❙ Example8. Findthevalueof n nnii 1 1 13 () () where ,i 1
Sol. Q n nn n nn n iiii 1 13 1 1 13 1 1 13 ()()() ii00 2 i 1 Q n n n nii 2 13 2 13 1 00 and (threesetsoffourconsecutivepowersof) i
❙ Example9. Findthevalueof n n i 0 100 ! (, where i 1).
Sol. Let fxxxxx pqrs () 4414243 and xxxxx 322111 ()() ()()() xixix 1 , where i 1
Now, fiiiii pqrs () 4414243 1 23iii 0
[sum of four consecutive powers of i is zero] fiiiii pqrs ()()()()()4414243 1 123 ()()() iii 110 ii and f pqrs ()()()()() 111114414243
11110
Hence, by division theorem, fx() is divisible by xxx 32 1
SwitchSystemTheory (FindingDigitintheUnit’sPlace)
Wecandeterminethedigitintheunit’splacein a b ,where abN,.Iflastdigitof a are015,, and6,then digitsintheunit’splaceof a b are0,1,5and6 respectively,forall bN .
Powersof2
222222222 123456789 ,,,,,,,,,...thedigitsinunit’splace ofdifferentpowersof2areasfollows: 248624862 ,,,,,,,,,... (period being 4)
123012301...(switch number)
(Theremainderwhen b isdividedby4,canbe1or2or3or0). Then,presstheswitchnumberandthenwegetthedigit inunit’splaceof a b (justabovetheswitchnumber)i.e. ‘pressthenumberandgettheanswer’.
Anumberoftheform aib,where abR , and i 1,is calleda complexnumber. Itisdenotedby z i.e. zaib
Acomplexnumbermayalsobedefinedasanorderedpair ofrealnumbers;andmaybedenotedbythesymbol(a, b). Ifwewrite zab(,),then a iscalledtherealpartand b is theimaginarypartofthecomplexnumber z andmaybe denotedbyRe() z andIm(z),respectivelyi.e., azRe() and bzIm()
Twocomplexnumbersaresaidtobeequal,ifandonlyif theirrealpartsandimaginarypartsareseparatelyequal. Thus, aibcid ac and bd where, abcdR ,,, and i 1 i.e. zz12
ReRe()() zz12 and ImIm()() zz12
ImportantPropertiesofComplexNumbers
1.The complex numbers do not possess the property of order, i.e., () aib or () cid is not defined. For example, 9632 ii makes no sense.
2.A real number a can be written as ai 0.Therefore, every real number can be considered as a complex number, whose imaginary part is zero. Thus, the set of real numbers (R) is a proper subset of the complex numbers() C i.e. RC Hence, the complex number system is NWIQRC
3.A complex number z is said to be purelyreal,ifIm() z 0; and is said to be purelyimaginary, ifRe() z 0. The complex number000 i is both purely real and purely imaginary.
4.In real number system, ab22 0 ab 0.
But if z 1 and z 2 are complex numbers, then zz 1 2 2 2 0 does not imply zz12 0
[multiplying numerator and denominator by cid whereatleastone of c and d is non-zero] aciadibcibd cid 2 22()() acibcadbd cid () 222 ()() acbdibcad cd22 () () () () acbd cd i bcad cd 2222
Remark 1 1 i i i and 1 1 i i i, where i 1.
PropertiesofAlgebraicOperations onComplexNumbers
Let zz12 , and z 3 beanythreecomplexnumbers. Then,theiralgebraicoperationssatisfythefollowing properties:
PropertiesofAdditionofComplexNumbers
(i) Closure law zz12 is a complex number.
(ii) Commutative law zzzz 1221
(iii) Associative law ()() zzzzzz 123123
8 TextbookofAlgebra
(iv) Additive identity zzz 00 , then0 is called the additive identity.
(v) Additive inverse z is called the additive inverse of z, i.e. zz()0.
(v) Multiplicative inverse If z is a non-zero complex number, then 1 z is called the multiplicative inverse of z i.e. z. 1 1 1 zz z
(vi) Multiplication is distributive with respect to addition zzzzzzz 1231213 ()
ConjugateComplexNumbers
Thecomplexnumbers zabaib (,) and zabaib (,) ,where a and b arerealnumbers, i 1 and b 0,aresaidtobecomplexconjugateofeach other(here,thecomplexconjugateisobtainedbyjust changingthesignof i).
Notethat,sum ()() aibaib 2a,whichisreal.
Andproduct ()() aibaib aib22 ()
aib222 ab22 1()
ab22 , which is real.
Geometrically, z is the mirror image of z along real axis on argandplane.
Remark
Let zaibab,,00 (,) ab (III quadrant )
Imaginaryaxis
Then, zaib (,) ab (II quadrant). Now,
(i)If z lies in I quadrant, then z lies in IV quadrant and vice-versa.
(ii)If z lies in II quadrant, then z lies in III quadrant and vice-versa.
PropertiesofConjugate ComplexNumbers
Let z, z 1 and z 2 becomplexnumbers.Then,
(i) ()zz
(ii) zzz2Re()
(iii) zzz2Im()
(iv) zz 0 zz z is purely imaginary.
(v) zz 0 zz z is purely real.
(vi) zzzz 1212 Ingeneral, zzzzzzzz nn123123
(vii) zzzz 1212
In general, zzzzzzzz nn123123
(viii) z z z z z 1 2 1 2 2 0 ,
(ix) zz nn ()
(x) zzzzzzzz 12121212 22Re()Re()
(xi)If zfzz(,), 12 then zfzz(,) 12
❙ Example 18. If x i y i i 3 3 3 3 , where xyR , and i 1,findthevaluesof x and y
On comparing real and imaginary parts, we get 33180 xy xy 6…(i) and yx 10...(ii)
On solvingEqs. (i) and (ii), we get x 2, y 8
❙ Example19. If (), aibpiq 5 where i 1, provethat () biaqip 5 .
Sol. Q ()aibpiq 5 ()aibpiq 5 ()() aibpiq 5
()() iaibipiq 252 [] Qi 2 1
()()()() ibiaiqip 55
()()()() ibiaiqip 5 ()() biaqip 5
❙ Example20. Findtheleastpositiveintegralvalueof n,forwhich 1 1 i i n , where i 1,ispurely imaginarywithpositiveimaginarypart.
Sol. 1 1 1 1 1 1 i i i i i i nn 12 2 2 ii n 112 2 i n
() i n Imaginary n 135,,,... for positive imaginary part n 3.
❙ Example21. Ifthemultiplicativeinverseofa complexnumberis ()/3419 i ,where i 1,find complexnumber.
Sol. Let z bethecomplexnumber.
Then, z i34 19 1 or z i i i 19 34 34 ()34 () ()
1934 19 () i () 34i
❙ Example22. Findreal ,suchthat 32 12 i i sin sin , where i 1,is
(i)purelyreal.(ii)purelyimaginary.
Sol. Let z i i 32 12 sin sin
On multiplying numerator and denominator by conjugate of denominator, z ii ii (sin)(sin) (sin)(sin) 3212 1212 (sin)sin (sin) 348 14 2 2 i (sin) (sin) (sin) (sin) 34 14 8 14 2 22 i
(i)Forpurelyreal, Im() z 0 8 14 0 2 sin sin orsin0 n , nI
(ii)Forpurelyimaginary, Re() z 0 (sin) (sin 34 14 0 2 2 or340 2 sin
orsinsin 2 2 2 3 4 3 23 nnI 3 ,
❙ Example23. Findrealvaluesof x and y forwhich thecomplexnumbers 3 2 ixy and xyi 2 4 , where i 1,areconjugatetoeachother.
Sol. Given,34 22 ixyxyi 34 22 ixyxyi
On comparing real and imaginary parts, we get xy 2 3…(i) and xy 2 4…(ii)
From Eq. (ii), we get x y 2 4
Then,43 y y puttinginEq.(i) x y 2 4 yy 2 340()() yy410 y 41, For y 4, x 2 1 x 1 For y 1, x 2 4[impossible] x 1, y 4
❙ Example24. If x 524,findthevalueof xxxx 432 935 4.
Sol. Since, x 524 xi 54 ()() xi 5422 xx 2 102516 xx 2 10410…(i)
Thelength OP iscalledmodulusofthecomplexnumber z denotedby z , i.e. OPrzxy() 22 andif(,)(,), xy 00 then iscalledtheargumentor amplitudeof z,
i.e. tan 1 y x [anglemadeby OP withpositive X-axis] or arg()tan(/) zyx 1
Also,argumentofacomplexnumberisnotunique,since if isavalueoftheargument,soalsois2n ,where nI Butusually,wetakeonlythatvalueforwhich 02 Anytwoargumentsofacomplexnumberdiffer by2n
PrincipalValueoftheArgument
Thevalue oftheargumentwhichsatisfiestheinequality iscalledthe principalvalue oftheargument. If zxiyxy(,), xyR , and i 1,then arg()tan z y x 1 alwaysgivestheprincipalvalue.It dependsonthequadrantinwhichthepoint(,) xy lies.
(i) (,) xy first quadrant xy00 ,.
The principal value of arg()tan z y x 1 It is an acute angle and positive.
(ii) (,) xy second quadrant xy00 ,. The principal value of arg() z tan 1 y x
It is an obtuse angle and positive. (iii) (,) xy third quadrant xy00 ,.
The principal value of arg() z tan 1 y x
It is an obtuse angle and negative. (iv) (,) xy fourth quadrant xy00 , The principal value of arg() z tan 1 y x
It is an acute angle and negative.
❙ Example27. Findtheprincipalvaluesofthe argumentsof zi 1 22 , zi 2 33 , zi 3 44 and zi 4 55 , where i 1
Sol. Since, zzz123 ,,and z 4 liesinI,II,IIIandIVquadrants respectively.Theprincipalvaluesoftheargumentsare givenby
(x) zzzzzz 121212 Chap01ComplexNumbers 11 y x (,) xy Y X Realaxis X ′ Y ′ Imaginary axis O θ y x (,) xy Y X Realaxis X ′ Y ′ Imaginary axis O θ y x (,) xy Y X Realaxis X ′ Y ′ Imaginary axis
zze i 11 1 || and zze i 22 2 || , where 12 , R and i 1
zzzeze ii 1212 12 zze i 12 12 ()
zzi 121212 (cos()sin())
Thus, zzzz 1212 and arg() zz1212 arg()arg() zz 12
(ii) DivisionofTwoComplexNumbers Lettwocomplexnumbersbe
zze i 11 1 and zze i 22 2 , where 12 , R and i 1
z z ze ze z z e i i i 1 2 1 2 1 2 1 2 12 () z z i 1 2 1212 (cos()sin())
Thus, z z z z z 1 2 1 2 2 0 ,() and arg z z 1 2 12 arg()arg() zz 12
(iii) LogarithmofaComplexNumber
log()log() ee i zze loglog() ee i ze
log e zi logarg() e ziz
So,thegeneralvalueof log() e z log() e zni 2 (arg) z .
❙ Example42. If m and x aretworealnumbersand i 1, provethat e xi xi mix m 2 1 1 1 1 cot .
Sol. Letcot, 1 x thencot x
LHS e xi xi mix m 2 1 1 1 cot e i i mi m 2 1 1 cot cot
e ii ii mi m 2 (cot) (cot) e i i mi m 2 cossin cossin
e e e mi i i m 2 ee miim22()
ee mimi22 e 0 1RHS
❙ Example43. If z and w aretwonon-zerocomplex numberssuchthat zw and arg()arg() zw , provethat z w
Sol. Letarg(), w thenarg() z zzziz (cos(arg)sin(arg)) zi(cos()sin()) zi(cossin) zi(cossin)
wwiw (cos(arg)sin(arg)) wwiw (cos(arg)sin(arg)) wwiw (cos(arg)sin(arg)) w
❙ Example44. Express () 1 i i ,(where, i 1)inthe form AiB
Sol. Let AiBi i () 1
On taking logarithm both sides, we get log()log() eeAiBii 1
i i e log21 22
ii e logcossin 2 44
ie e i log() 2 4 ie ee i (loglog) 2 4 i i e 1 2 2 4 log i e 2 2 4 log AiBe i e 2 2 4 log ee i e 4 log / 2 12 eiee 4 (cos(log)sin(log)) //221212 eieee 44 coslogsinlog 1 2 1 2
❙ Example45. If sin(log), e i iaib where i 1, find a and b, henceandfind cos(log). e i i
Sol. aibi e i sin(log)sin(log) ii e sin((logarg)) iiii e sin((log())) ii e 12 sin((( ii 0 / 2)))sin(21 ab10 , Now,cos(log)sin(log) e i e iii 1 2 11110 2 ()()
Aliter
Q iee iii ()22
sin(log)sin(log) e i e ie 2 sinlog 2 e e
sin(21 aib [given] ab10 , andcos(log)cos(log) e i e ie 2 coslog 2 e e cos 2 0
❙ Example46. Findthegeneralvalueof log() 2 5i , where i 1.
Sol. logloglog 2 5 5 2 i i e e 1 2 552 log{logarg()} e e iiini 1 2 5 2 2 log{log}, e e i ninI
