Arihant 40 Days Crash Course for JEE Main Mathematics PDF

More ebooks at https://telegram.me/unacademyplusdiscounts  The Most Accepted CRASH COURSE PROGRAMME JEE Main in 40 DAYS MATHEMATICS ARIHANT PRAKASHAN (Series), MEERUT Arihant Prakashan (Series), Meerut All Rights Reserved © PUBLISHERS No part of this publication may be re-prod...

More ebooks at https://telegram.me/unacademyplusdiscounts  The Most Accepted CRASH COURSE PROGRAMME JEE Main in 40 DAYS MATHEMATICS ARIHANT PRAKASHAN (Series), MEERUT Arihant Prakashan (Series), Meerut All Rights Reserved © PUBLISHERS No part of this publication may be re-produced, stored in a retrieval system or distributed in any form or by any means, electronic, mechanical, photocopying, recording, scanning, web or otherwise without the written permission of the publisher. Arihant has obtained all the information in this book from the sources believed to be reliable and true. However, Arihant or its editors or authors or illustrators don’t take any responsibility for the absolute accuracy of any information published and the damages or loss suffered there upon. All disputes subject to Meerut (UP) jurisdiction only. ADMINISTRATIVE & PRODUCTION OFFICES Regd. Office ‘Ramchhaya’ 4577/15, Agarwal Road, Darya Ganj, New Delhi -110002 Tele: 011- 47630600, 43518550; Fax: 011- 23280316 Head Office Kalindi, TP Nagar, Meerut (UP) - 250002 Tele: 0121-2401479, 2512970, 4004199; Fax: 0121-2401648 SALES & SUPPORT OFFICES Agra, Ahmedabad, Bengaluru, Bareilly, Chennai, Delhi, Guwahati, Hyderabad, Jaipur, Jhansi, Kolkata, Lucknow, Meerut, Nagpur & Pune ISBN : 978-93-13199-32-8 Published by Arihant Publications (India) Ltd. For further information about the books published by Arihant log on to www.arihantbooks.com or email to [email protected] /arihantpub /@arihantpub Arihant Publications /arihantpub  PREFACE It is a fact that nearly 10 lacs students would be in the race with you in JEE Main, the gateway to some of the prestigious engineering and technology institutions in the country, requires that you take it seriously and head-on. A slight underestimation or wrong guidance will ruin all your prospects. You have to earmark the topics in the syllabus and have to master them in concept-driven-problem-solving ways, considering the thrust of the questions being asked in JEE Main. The book 40 Days JEE Main Mathematics serves the above cited purpose in perfect manner. At whatever level of preparation you are before the exam, this book gives you an accelerated way to master the whole JEE Main Physics Syllabus. It has been conceived keeping in mind the latest trend of questions, and the level of different types of students. The whole syllabus of Physics has been divided into day-wise-learning modules with clear groundings into concepts and sufficient practice with solved and unsolved questions on that day. After every few days you get a Unit Test based upon the topics covered before that day. On last three days you get three full-length Mock Tests, making you ready to face the test. It is not necessary that you start working with this book in 40 days just before the exam. You may start and finish your preparation of JEE Main much in advance before the exam date. This will only keep you in good frame of mind and relaxed, vital for success at this level. Salient Features Ÿ Concepts discussed clearly and directly without being superfluous. Only the required material for JEE Main being described comprehensively to keep the students focussed. Ÿ Exercises for each day give you the collection of only the Best Questions of the concept, giving you the perfect practice in less time. Ÿ Each day has two Exercises; Foundation Questions Exercise having Topically Arranged Questions & Progressive Question Exercise having higher Difficulty Level Questions. Ÿ All types of Objective Questions included in Daily Exercises (Single Option Correct, Assertion & Reason, etc). Ÿ Along with Daywise Exercises, there above also the Unit Tests & Full Length Mock Tests. Ÿ At the end, there are all Online Solved Papers of JEE Main 2019; January & April attempts. We are sure that 40 Days Physics for JEE Main will give you a fast way to prepare for Physics without any other support or guidance. Publisher  CONTENTS Preparing JEE Main 2020 Mathematics in 40 Days ! Day 1. Sets, Relations and Functions 1-9 Day 2. Complex Numbers 10-19 Day 3. Sequences and Series 20-30 Day 4. Quadratic Equation and Inequalities 31-44 Day 5. Matrices 45-54 Day 6. Determinants 55-67 Day 7. Binomial Theorem and Mathematical Induction 68-77 Day 8. Permutations and Combinations 78-86 Day 9. Unit Test 1 (Algebra) 87-94 Day 10. Real Function 95-103 Day 11. Limits, Continuity and Differentiability 104-116 Day 12. Differentiation 117-126 Day 13. Applications of Derivatives 127-137 Day 14. Maxima and Minima 138-149 Day 15. Indefinite Integrals 150-162 Day 16. Definite Integrals 163-175 Day 17. Area Bounded by the Curves 176-187 Day 18. Differential Equations 188-198 Day 19. Unit Test 2 (Calculus) 199-208 Day 20. Trigonometric Functions and Equations 209-221 Day 21. Properties of Triangle, Height and Distances 222-232 Day 22. Inverse Trigonometric Function 233-242 Day 23. Unit Test 3 (Trigonometry) 243-250 Day 24. Cartesian System of Rectangular Coordinates 251-262 Day 25. Straight Line 263-274 Day 26. The Circle 275-288 Day 27. Parabola 289-300 Day 28. Ellipse 301-313 Day 29. Hyperbola 314-325 Day 30. Unit Test 4 (Coordinate Geometry) 326-335 Day 31. Vector Algebra 336-350 Day 32. Three Dimensional Geometry 351-366 Day 33. Unit Test 5 (Vector & 3D Geometry) 367-374 Day 34. Statistics 375-383 Day 35. Probability 384-394 Day 36. Mathematical Reasoning 395-405 Day 37. Unit Test 6 (Statistics, Probability & Mathematical Reasoning) 406-411 Day 38. Mock Test 1 412-416 Day 39. Mock Test 2 417-423 Day 40. Mock Test 3 424-430 Online JEE Main Solved Papers 2019 1-32  SYLLABUS MATHEMATICS UNIT 1 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 2 Complex Numbers and Quadratic Equations 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, Quadratic equations in real and complex number system and their solutions. Relation between roots and co-efficients, nature of roots, formation of quadratic equations with given roots. UNIT 3 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 4 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 5 Mathematical Induction Principle of Mathematical Induction and its simple applications. UNIT 6 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 7 Sequences and Series Arithmetic and Geometric progressions, insertion of arithmetic, geometric means between two given numbers. Relation between AM and GM Sum upto n terms of special series: ∑ n, ∑ n2, ∑ n3. Arithmetico - Geometric progression. UNIT 8 Limit, Continuity and Differentiability Real valued functions, algebra of functions, polynomials, rational, trigonometric, logarithmic and exponential functions, inverse functions. Graphs of simple functions. Limits, continuity and differentiability. Differentiation of the sum, difference, product and quotient of two functions. Differentiation of trigonometric, inverse trigonometric, logarithmic exponential, composite and implicit functions derivatives of order upto two. Rolle's and Lagrange's Mean Value Theorems. Applications of derivatives: Rate of change of quantities, monotonic - increasing and decreasing functions, Maxima and minima of functions of one variable, tangents and normals. UNIT 9 Integral Calculus Integral as an anti - derivative. Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions. Integration by substitution, by parts and by partial fractions. Integration using trigonometric identities. Evaluation of simple integrals of the type dx , dx , dx , dx , x2 ± a2 2 x ± a 2 a2 – x2 a 2 – x 2 dx dx , (px + q) dx , , ax 2 + bx + c ax 2 + bx + c ax 2 + bx + c (px + q) dx , ax 2 + bx + c a 2 ± x 2 dx and x 2 – a 2 dx Integral as limit of a sum. Fundamental Theorem of Calculus. Properties of definite integrals. Evaluation of definite integrals, determining areas of the regions bounded by simple curves in standard form. UNIT 10 Differential Equations Ordinary differential equations, their order and degree. Formation of differential equations. Solution of differential equations by the method of separation of variables, solution of homogeneous and linear differential equations of the type dy +p (x) y = q(x) dx UNIT 11 Coordinate Geometry Cartesian system of rectangular coordinates in a plane, distance formula, section formula, locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes. Ÿ Straight Lines Various forms of equations of a line, intersection of lines, angles between two lines, conditions for concurrence of three lines, distance of a point from a line, equations of internal and external bisectors of angles between two lines, coordinates of centroid, orthocentre and circumcentre of a triangle, equation of family of lines passing through the point of intersection of two lines.  Ÿ Circles, Conic Sections Standard form of equation of a circle, general form of the equation of a circle, its radius and centre, equation of a circle when the end points of a diameter are given, points of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to a circle, equation of the tangent. Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard forms, condition for y = mx + c to be a tangent and point (s) of tangency. UNIT 12 Three Dimensional Geometry Coordinates of a point in space, distance between two points, section formula, direction ratios and direction cosines, angle between two intersecting lines. Skew lines, the shortest distance between them and its equation. Equations of a line and a plane in different forms, intersection of a line and a plane, coplanar lines. UNIT 13 Vector Algebra Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product. UNIT 14 Statistics and Probability Measures of Dispersion Calculation of mean, median, mode of grouped and ungrouped data. Calculation of standard deviation, variance and mean deviation for grouped and ungrouped data. Probability Probability of an event, addition and multiplication theorems of probability, Baye's theorem, probability distribution of a random variate, Bernoulli trials and Binomial distribution. UNIT 15 Trigonometry Trigonometrical identities and equations. Trigonometrical functions. Inverse trigonometrical functions and their properties. Heights and Distances. UNIT 16 Mathematical Reasoning Statements, logical operations and implies, implied by, if and only if. Understanding of tautology, contradiction, converse and contra positive.  HOW THIS BOOK IS USEFUL FOR YOU ? As the name suggest, this is the perfect book for your recapitulation of the whole syllabus, as it provides you a capsule course on the subject covering the syllabi of JEE Main, with the smartest possible tactics as outlined below: 1. REVISION PLAN The book provides you with a practical and sound revision plan. The chapters of the book have been designed day-wise to guide the students in a planned manner through day-by-day, during those precious 35-40 days. Every day you complete a chapter/a topic, also take an exercise on the chapter. So that you can check & correct your mistakes, answers with hints & solutions also have been provided. By 37th day from the date you start using this book, entire syllabus gets revisited. Again, as per your convenience/preparation strategy, you can also divide the available 30-35 days into two time frames, first time slot of 3 weeks and last slot of 1 & 1/2 week. Utilize first time slot for studies and last one for revising the formulas and important points. Now fill the time slots with subjects/topics and set key milestones. Keep all the formulas, key points on a couple of A4 size sheets as ready-reckner on your table and go over them time and again. If you are done with notes, prepare more detailed inside notes and go over them once again. Study all the 3 subjects every day. Concentrate on the topics that have more weightage in the exam that you are targeting. 2. MOCK TESTS Once you finish your revision on 37th day, the book provides you with full length mock tests for day 38th, 39th, & 40th, thereby ensures your total & full proof preparation for the final show. The importance of solving previous years' papers and 10-15 mock tests cannot be overemphasized. Identify your weaknesses and strengths. Work towards your strengths i.e., devote more time to your strengths to be 100% sure and confident. In the last time frame of 1 & 1/2 week, don't take-up anything new, just revise what you have studied before. Be exam- ready with quality mock tests in between to implement your winning strategy. 3. FOCUS TOPICS Based on past years question paper trends, there are few topics in each subject which have more questions in exam than other. So far Mathematics is concerned it may be summed up as below: Calculus, Trigonometry, Algebra, Coordinate Geometry & Vector 3D. More than 80% of questions are normally asked from these topics. However, be prepared to find a completely changed pattern for the exam than noted above as examiners keep trying to weed out 'learn by rot practice'. One should not panic by witnessing a new pattern , rather should be tension free as no one will have any upper hand in the exam. 4. IMPROVES STRIKE RATE AND ACCURACY The book even helps to improve your strike rate & accuracy. When solving practice tests or mock tests, try to analyze where you are making mistakes-where are you wasting your time; which section you are doing best. Whatever mistakes you make in the first mock test, try to improve that in second. In this way, you can make the optimum use of the book for giving perfection to your preparation. What most students do is that they revise whole of the syllabus but never attempt a mock and thus they always make mistake in main exam and lose the track. 5. LOG OF LESSONS During your preparations, make a log of Lesson's Learnt. It is specific to each individual as to where the person is being most efficient and least efficient. Three things are important - what is working, what's not working and how would you like to do in your next mock test. 6. TIME MANAGEMENT Most candidates who don't make it to good medical colleges are not good in one area- Time Management. And, probably here lies the most important value addition that's the book provides in an aspirant's preparation. Once the students go through the content of the book precisely as given/directed, he/she learns the tactics of time management in the exam. Realization and strengthening of what you are good at is very helpful, rather than what one doesn't know. Your greatest motto in the exam should be, how to maximize your scoring with the given level of preparation. You have to get about 200 plus marks out of a total of about 400 marks for admission to a good NIT (though for a good branch one needs to do much better than that). Remember that one would be doomed if s/he tries to score 400 in about 3 hours. 7. ART OF PROBLEM SOLVING The book also let you to master the art of problem solving. The key to problem solving does not lie in understanding the solution to the problem but to find out what clues in the problem leads you to the right solution. And, that's the reason Hints & Solutions are provided with the exercises after each chapter of the book. Try to find out the reason by analyzing the level of problem & practice similar kind of problems so that you can master the tricks involved. Remember that directly going though the solutions is not going to help you at all. 8. POSITIVE PERCEPTION The book put forth for its readers a 'Simple and Straightforward' concept of studies, which is the best possible, time-tested perception for 11th hour revision / preparation. The content of the book has been presented in such a lucid way so that you can enjoy what you are reading, keeping a note of your already stressed mind & time span. Cracking JEE Main is not a matter of life and death. Do not allow panic and pressure to create confusion. Do some yoga and prayers. Enjoy this time with studies as it will never come back. DAY ONE Sets, Relations and Functions Learning & Revision for the Day u Sets u Law of Algebra of Sets u Composition of Relations u Venn Diagram u Cartesian Product of Sets u Functions or Mapping u Operations on Sets u Relations u Composition of Functions Sets A set is a well-defined class or collection of the objects. Sets are usually denoted by the symbol A, B, C,... and its elements are denoted by a, b , c, … etc. If a is an element of a set A, then we write a ∈ A and if not then we write a ∉ A. Representations of Sets There are two methods of representing a set : In roster method, a set is described by listing elements, separated by commas, within curly braces{≠}. e.g. A set of vowels of English alphabet may be described as {a, e, i, o, u}. In set-builder method, a set is described by a property P ( x), which is possessed by all its PRED elements x. In such a case the set is written as { x : P ( x) holds} or { x| P ( x) holds}, which MIRROR Your Personal Preparation Indicator is read as the set of all x such that P( x) holds. e.g. The set P = {0, 1, 4 , 9, 16,...} can be written as P = { x2 | x ∈ Z }. u No. of Questions in Exercises (x)— u No. of Questions Attempted (y)— Types of Sets u No. of Correct Questions (z)— The set which contains no element at all is called the null set (empty set or void (Without referring Explanations) set) and it is denoted by the symbol ‘φ ’ or ‘{}’ and if it contains a single element, then it is u Accuracy Level (z / y × 100)— called singleton set. u Prep Level (z / x × 100)— A set in which the process of counting of elements definitely comes to an end, is called a finite set, otherwise it is an infinite set. In order to expect good rank in JEE, your Accuracy Level should be Two sets A and B are said to be equal set iff every element of A is an element of B and above 85 & Prep Level should be also every element of B is an element of A. i.e. A = B, if x ∈ A ⇔ x ∈ B. above 75. 2 40 ONE Equivalent sets have the same number of elements but not i.e. A ∩ B = { x : x ∈ A and x ∈ B}. exactly the same elements. A B U A set that contains all sets in a given context is called universal set (U). Let A and B be two sets. If every element of A is an element of B, then A is called a subset of B, i.e. A ⊆ B. If A is a subset of B and A ≠ B, then A is a proper subset of A∩B B. i.e. A ⊂ B. If A ∩ B = φ, then A and B are called disjoint sets. The null set φ is a subset of every set and every set is a subset of itself i.e. φ ⊂ A and A ⊆ A for every set A. They Let U be an universal set and A be a set such that A ⊂ U. are called improper subsets of A. Then, complement of A with respect to U is denoted by A′ or Ac or A or U − A. It is defined as the set of all those If S is any set, then the set of all the subsets of S is called elements of U which are not in A. the power set of S and it is denoted by P(S ). Power set of a given set is always non-empty. If A has n elements, then A′ U P( A) has 2 n elements. A NOTE The set { φ} is not a null set. It is a set containing one element φ. Whenever we have to show that two sets A and B are equal show that A ⊆ B and B ⊆ A. The difference A − B is the set of all those elements of A If a set A has m elements, then the number m is called which does not belong to B. cardinal number of set A and it is denoted by n( A). Thus, i.e. A − B = { x : x ∈ A and x ∉ B} n( A) = m. and B − A = { x : x ∈ B and x ∉ A}. U U Venn Diagram A B A B The combination of rectangles and circles is called Venn Euler diagram or Venn diagram. In Venn diagram, the universal set is represented by a rectangular region and a set is represented by circle on some closed geometrical figure. Where, A is the set and U is the universal set. A–B B–A The symmetric difference of sets A and B is the set U ( A − B) ∪ (B − A) and is denoted by A ∆ B. A i.e. A ∆ B = ( A − B) ∪ ( B − A) A B U Operations on Sets The union of sets A and B is the set of all elements which A∆B are in set A or in B or in both A and B. i.e. A ∪ B = { x : x ∈ A or x ∈ B} Law of Algebra of Sets A B U If A, B and C are any three sets, then 1. Idempotent Laws (i) A ∪ A = A (ii) A ∩ A = A 2. Identity Laws A∪B (i) A ∪ φ = A (ii) A ∩ U = A 3. Distributive Laws The intersection of A and B is the set of all those elements (i) A ∪ (B ∩ C) = ( A ∪ B) ∩ ( A ∪ C) that belong to both A and B. (ii) A ∩ (B ∪ C) = ( A ∩ B) ∪ ( A ∩ C) DAY 3 4. De-Morgan’s Laws Relations (i) ( A ∪ B)′ = A′ ∩ B ′ Let A and B be two non-empty sets, then relation R from A to B (ii) ( A ∩ B)′ = A′ ∪ B ′ is a subset of A × B, i.e. R ⊆ A × B. (iii) A − (B ∩ C) = ( A − B) ∪ ( A − C ) (iv) A − (B ∪ C) = ( A − B) ∩ ( A − C) If (a, b ) ∈ R, then we say a is related to b by the relation R and we write it as aRb. 5. Associative Laws Domain of R = {a :(a, b ) ∈ R} and range of R = {b : (a, b ) ∈ R}. (i) ( A ∪ B) ∪ C = A ∪ (B ∪ C) If n( A) = p and n(B) = q , then the total number of relations from A (ii) A ∩ (B ∩ C) = ( A ∩ B) ∩ C to B is 2 pq. 6. Commutative Laws (i) A ∪ B = B ∪ A (ii) A ∩ B = B ∩ A Types of Relations (iii) A ∆ B = B ∆ A Let A be any non-empty set and R be a relation on A. Then, Important Results on Operation of Sets (i) R is said to be reflexive iff ( a, a) ∈ R, ∀ a ∈ A. 1. A − B = A ∩ B ′ (ii) R is said to be symmetric iff 2. B − A = B ∩ A ′ (a, b ) ∈ R 3. A − B = A ⇔ A ∩ B = φ ⇒ (b , a) ∈ R, ∀ a, b ∈ A 4. ( A − B) ∪ B = A ∪ B (iii) R is said to be a transitive iff ( a, b) ∈ R and (b, c) ∈ R 5. ( A − B) ∩ B = φ ⇒ ( a, c) ∈ R, ∀ a, b, c ∈ A 6. A ⊆ B ⇔ B ′ ⊆ A ′ i.e. aRb and bRc ⇒ aRc, ∀ a, b, c ∈ A. 7. ( A − B) ∪ ( B − A) = ( A ∪ B) − ( A ∩ B) The relation I A = {(a, a) : a ∈ A} on A is called the identity 8. n ( A ∪ B) = n ( A) + n ( B) − n ( A ∩ B) relation on A. 9. n ( A ∪ B) = n ( A) + n ( B) R is said to be an equivalence relation iff ⇔ A and B are disjoint sets. (i) it is reflexive i.e. ( a, a) ∈ R, ∀ a ∈ A. 10. n ( A − B) = n ( A) − n ( A ∩ B) (ii) it is symmetric i.e. ( a, b) ∈ R ⇒ (b, a) ∈ R, ∀ a, b ∈ A 11. n ( A ∆ B) = n ( A) + n ( B) − 2n ( A ∩ B) (iii) it is transitive 12. n ( A ∪ B ∪ C ) = n ( A) + n ( B) + n (C ) − n ( A ∩ B) i.e. ( a, b) ∈ R and (b, c) ∈ R − n ( B ∩ C) − n ( A ∩ C) + n ( A ∩ B ∩ C) ⇒ ( a, c) ∈ R, ∀ a, b, c ∈ A 13. n ( A ′ ∪ B ′ ) = n ( A ∩ B) ′ = n (U ) − n ( A ∩ B) Inverse Relation 14. n ( A ′ ∩ B ′ ) = n ( A ∪ B) ′ = n (U ) − n ( A ∪ B) Let R be a relation from set A to set B, then the inverse of R, denoted by R −1 , is defined by Cartesian Product of Sets R −1 = {(b , a) : (a, b ) ∈ R}. Clearly, (a, b ) ∈ R ⇔ (b , a) ∈ R −1. Let A and B be any two non-empty sets. Then the NOTE The intersection of two equivalence relations on a set is an cartesian product A × B, is defined as set of all ordered equivalence relation on the set. pairs (a, b ) such that a ∈ A and b ∈ B. The union of two equivalence relations on a set is not necessarily i.e. an equivalence relation on the set. A × B = {(a, b ) : a ∈ A and b ∈ B} If R is an equivalence relation on a set A, then R −1 is also an B × A = {(b , a) : b ∈ B and a ∈ A} equivalence relation A. and A × A = {(a, b ) : a, b ∈ A}. A × B = φ, if either A or B is an empty set. If n ( A) = p and n (B) = q , then Composition of Relations n ( A × B) = n( A) ⋅ n(B) = pq. Let R and S be two relations from set A to B and B to C respectively, A × (B ∪ C) = ( A × B) ∪ ( A × C) then we can define a relation SoR from A to C such that A × (B ∩ C) = ( A × B) ∩ ( A × C) (a, c) ∈ SoR ⇔ ∃ b ∈ B such that (a, b ) ∈ R and (b , c) ∈ S. This relation A × (B − C) = ( A × B) − ( A × C) is called the composition of R and S. ( A × B) ∩ (C × D) = ( A ∩ C) × (B ∩ D). RoS ≠ SoR 4 40 ONE i.e. f : A → B is a many-one function, if it is not a one-one Functions or Mapping function. If A and B are two non-empty sets, then a rule f which f is said to be onto function or surjective function, if each associates each x ∈ A, to a unique member y ∈ B, is called a element of B has its pre-image in A. function from A to B and it is denoted by f : A → B. A B The set A is called the domain of f (D f ) and set B is called f a1 b1 the codomain of f (C f ). a2 b2 The set consisting of all the f -images of the elements of the a3 b3 domain A, called the range of f (R f ). NOTE A relation will be a function, if no two distinct ordered pairs Method to Check Onto Function have the same first element. Find the range of f ( x) and show that range of Every function is a relation but every relation is not necessarily a function. f ( x) = codomain of f ( x). The number of functions from a finite set A into finite set Any polynomial function of odd degree is always onto. B is { n(B )}n ( A). The number of onto functions that can be defined from a finite set A containing n elements onto a finite set B Different Types of Functions containing 2 elements = 2 n − 2. If n ( A) ≥ n (B), then number of onto function is 0. Let f be a function from A to B, i.e. f : A → B. Then, If A has m elements and B has n elements, where m < n, f is said to be one-one function or injective function, if then number of onto functions from A to B is different elements of A have different images in B. nm − nC1 (n − 1)m + nC2 (n − 2)m −..., m < n. A B f is said to be an into function, if there exists atleast one a1 f b1 element in B having no pre-image in A. i.e. f : A → B is an a2 b2 into function, if it is not an onto function. a3 b3 A B a1 f b1 a4 b4 a2 b2 a3 b3 Methods to Check One-One Function a4 b4 Method I If f ( x) = f ( y ) ⇒ x = y , then f is one-one. a5 b5 Method II A function is one-one iff no line parallel to X-axis meets the graph of function at more f is said to be a bijective function, if it is one-one as well as than one point. onto. The number of one-one function that can be defined from a NOTE If f : A → B is a bijective, then A and B have the same  n( B ) number of elements. Pn( A ) , if n (B) ≥ n ( A) finite set A into finite set B is . If n ( A) = n (B ) = m, then number of bijective map from A to 0, otherwise B is m!. f is said to be a many-one function, if two or more Composition of Functions elements of set A have the same image in B. Let f : A → B and g : B → C are two functions. Then, the A B composition of f and g, denoted by a1 f b1 gof : A → C, is defined as, a2 b2 gof ( x) = g[ f ( x)], ∀ x ∈ A. a3 b3 a4 b4 NOTE gof is defined only if f ( x ) is an element of domain of g. a5 b5 Generally, gof ≠ fog.  DAY PRACTICE SESSION 1 FOUNDATION QUESTIONS EXERCISE  1  (a) reflexive but neither symmetric nor transitive 1 If Q = x : x = , where y ∈ N , then  y  (b) symmetric and transitive 2 (c) reflexive and symmetric (a) 0 ∈Q (b) 1∈Q (c) 2∈Q (d) ∈Q (d) reflexive and transitive 3 2 If P ( A ) denotes the power set of A and A is the void set, 12 If g ( x ) = 1 + x and f {g ( x )} = 3 + 2 x + x , then f ( x ) is then what is number of elements in P {P {P {P ( A )}}}? equal to (a) 0 (b) 1 (c) 4 (d) 16 (a) 1 + 2 x 2 (b) 2 + x 2 (c) 1 + x (d) 2 + x 3 If X = {4n − 3n − 1: n ∈ N} and Y = {9 (n − 1): n ∈ N}; where N is the set of natural numbers,then X ∪ Y is equal to 13 Let f ( x ) = ax + b and g ( x ) = cx + d , a ≠ 0, c ≠ 0. Assume j JEE Mains 2014 a = 1, b = 2 , if ( fog ) ( x ) = ( gof ) ( x ) for all x. What can you (a) N (b) Y-X (c) X (d) Y say about c and d? (a) c and d both arbitrary (b) c = 1and d is arbitrary 4 If A, B and C are three sets such that A ∩ B = A ∩ C and (c) c is arbitrary and d = 1 (d) c = 1, d = 1 A ∪ B = A ∪ C, then 14 If R is relation from {11, 12 , 13 } to {8 , 10 , 12} defined by (a) A = C (b) B = C (c) A ∩ B = φ (d) A = B y = x − 3. Then, R −1 is 5 Suppose A1, A2,… , A 30 are thirty sets each having (a) {(8 , 11), (10, 13)} (b) {(11, 18), (13 , 10)} 5 elements and B1, B2,… , Bn are n sets each having (c) {(10, 13), (8 , 11)} (d) None of these 30 n 3 elements. Let ∪ A i = ∪ Bj = S and each element of S 15 Let R be a relation defined by R = {(4, 5), (1, 4), (4, 6), i =1 j =1 (7, 6), (3, 7)}, then R −1 OR is belongs to exactly 10 of Ai ’s and exactly 9 of Bj ’ s. The (a) {(1, 1), (4, 4), (4, 7), (7, 4), (7, 7), (3, 3)} value of n is equal to j NCERT Exemplar (b) {(1, 1), (4, 4), (7, 7), (3, 3)} (a) 15 (b) 3 (c) {(1, 5), (1, 6), (3, 6)} (c) 45 (d) None of these (d) None of the above 6 If A and B are two sets and A ∪ B ∪ C = U. Then, 16 Let A be a non-empty set of real numbers and f : A → A {( A − B ) ∪ (B − C ) ∪ (C − A )}′ is equal to be such that f (f ( x )) = x , ∀ x ∈ R. Then, f ( x ) is (a) A ∪ B ∪ C (b) A ∪ (B ∩ C) (a) a bijection (b) one-one but not onto (c) A ∩ B ∩ C (d) A ∩ (B ∪ C) (c) onto but not one-one (d) neither one-one nor onto 7 Let X be the universal set for sets A and B, if 17 The function f satisfies the functional equation n( A ) = 200, n(B ) = 300 and n( A ∩ B ) = 100, then  x + 59 n ( A′ ∩ B′) is equal to 300 provided n( X ) is equal to 3f ( x ) + 2f   = 10x + 30 for all real x ≠ 1. The value  x −1  (a) 600 (b) 700 (c) 800 (d) 900 of f (7) is 8 If n( A ) = 1000, n(B ) = 500, n( A ∩ B ) ≥ 1 and n( A ∪ B ) = P , (a) 8 (b) 4 (c) − 8 (d) 11 then 18 The number of onto mapping from the set A = {1, 2,...100} (a) 500 ≤ P ≤ 1000 (b) 1001 ≤ P ≤ 1498 to set B = {1, 2} is (c) 1000 ≤ P ≤ 1498 (d) 1000 ≤ P ≤ 1499 (a) 2100 − 2 (b) 2100 (c) 2 99 − 2 (d) 2 99 9 If n( A ) = 4, n(B ) = 3, n( A × B × C ) = 24, then n(C ) is equal to x −m 19 Let f : R − {n} → R be a function defined by f ( x ) = , (a) 2 (b) 288 (c) 12 (d) 1 x −n where m ≠ n. Then, 10 If R = {( 3 , 3) ,( 6, 6), ( 9, 9), (12 , 12), ( 6 , 12), ( 3 , 9), (3,12), (a) f is one-one onto (b) f is one-one into (3, 6)} is a relation on the set A = {3 , 6 , 9 , 12}. (c) f is many-one onto (d) f is many-one into The relation is 20 A function f from the set of natural numbers to integers (a) an equivalence relation n − 1 (b) reflexive and symmetric , when n is odd  (c) reflexive and transitive defined by f (n ) =  2 is n (d) only reflexive  − , when n is even  2 11 Let R = {( x , y ) : x , y ∈ N and x 2 − 4xy + 3y 2 = 0}, where (a) one-one but not onto (b) onto but not one-one N is the set of all natural numbers. Then, the relation R is (c) both one-one and onto (d) neither one-one nor onto j JEE Mains 2013 6 40 ONE 21 Let f : N → N defined by f ( x ) = x 2 + x + 1 , x ∈ N, then f is (c) S is an equivalence relation but R is not an equivalence (a) one-one onto (b) many-one onto relation (c) one-one but not onto (d) None of these (d) R and S both are equivalence relations 22 Let R be the real line. Consider the following subsets of 2 + x , x ≥ 0 24 If f ( x ) =  , then f (f ( x )) is given by the plane R × R. 4 − x , x < 0 S = {( x , y ): y = x + 1and 0 < x < 2}  4 + x, x≥0  4 + x, x ≥ 0 (a) f (f (x)) =  (b) f (f (x)) =  and T = {( x , y ): x − y is an integer}  6 − x, x 0, k ≠ 1 is a real Properties of Cube Roots of Unity | z − z1| number, then = k represents a circle. (i) 1 + ω + ω2 = 0 | z − z2| (ii) ω3 = 1 For k = 1, it represents perpendicular bisector of the 0 if n ≠ 3 m, m ∈ N (iii) 1 + ω n + ω2 n =  segment joining A(z1 ) and B (z2 ). 3 if n = 3 m, m ∈ N 8. If end points of diameter of a circle are A(z1 ) and B(z 2) and P(z) be any point on the circle, then equation of circle in nth Roots of Unity diameter form is By nth root of unity we mean any complex number z which (z − z1 ) (z − z2 ) + (z − z 2) (z − z1 ) = 0 satisfies the equation zn = 1. DAY 13 DAY PRACTICE SESSION 1 FOUNDATION QUESTIONS EXERCISE 1 4 1 Real part of is 11 If z − = 2, then the maximum value of | z | is 1 − cos θ + i sin θ z j AIEEE 2009 1 1 1 (a) − (b) (c) tanθ / 2 (d) 2 (a) 3 +1 (b) 5 +1 (c) 2 (d) 2 + 2 2 2 2 12 If z is a complex number such that z ≥ 2, then the 2 + 3i sin θ 1 2 A value of θ, for which is purely imaginary, is minimum value of z + 1 − 2i sin θ 2 j JEE Mains 2014 π π (d) sin  −1 3 −1 1  (a) (b) (c) sin  (a) is equal to 5/2 3 6 4  3 (b) lies in the interval (1, 2) 13 (c) is strictly greater than 5/2 3 ∑ (i n =1 n + i n + 1 ) is equal to (d) is strictly greater than 3/2 but less than 5/2 13 If | z1| = 2, | z 2 | = 3 then z1 + z 2 + 5 + 12i is less than or (a) i (b) i − 1 (c) −i (d) 0 equal to z −1 4 If is a purely imaginary number ( where, z ≠ −1), then (a) 8 (b) 18 (c) 10 (d) 5 z +1 14 If z < 3 − 1, then z + 2z cos α is 2 the value of | z | is (a) less than 2 (b) 3 + 1 (a) −1 (b) 1 (c) 2 (d) −2 (c) 3 − 1 (d) None of these z2 5 If z1 ≠ 0 and z 2 are two complex numbers such that is 15 The number of complex numbers z such that z1 z − 1 = z + 1 = z − i , is 2z1 + 3z 2 a purely imaginary number, then is equal to (a) 0 (b) 1 (c) 2 (d) ∞ 2z1 − 3z 2 2 j JEE Mains 2013 16 Number of solutions of the equation z + 7z = 0 is/are (a) 2 (b) 5 (c) 3 (d) 1 (a) 1 (b) 2 (c) 4 (d) 6 7−z 17 If z z + ( 3 − 4i )z + ( 3 + 4i )z = 0 represent a circle, the area 6 If f ( z ) = , where z = 1 + 2i , then | f ( z )| is equal to 1− z 2 of the circle in square units is |z | (a) 5 π (b) 10π (c) 25 π 2 (d) 25 π (a) (b) | z | 2  π  π (c) 2| z | (d) None of these 18 If z = 1 + cos   + i sin  , then {sin (arg( z ))} is equal to  5  5 7 If 8 iz 3 + 12z 2 − 18z + 27i = 0, then the value of | z | is 10 − 2 5 5 −1 (a) 3 / 2 (b) 2 / 3 (c) 1 (d) 3 / 4 (a) (b) 4 4 8 If a complex number z satisfies the equation 5+1 (c) (d) None of these z + 2 z + 1 + i = 0, then z is equal to j JEE Mains 2013 4 (a) 2 (b) 3 (c) 5 (d) 1 19 If z is a complex number of unit modulus and argument 9 If α and β are two different complex numbers such that 1 + z  θ, then arg   equals to β −α 1 + z  JEE Mains 2013 | α | = 1, | β | = 1, then the expression j is equal to 1 − αβ π (a) −θ (b) −θ (c) θ (d) π − θ 1 2 (a) (b) 1 2 20 Let z and ω are two non-zero complex numbers such that (c) 2 (d) None of these z = ω and arg z + arg ω = π, then z equals z −1 (a) ω (b) ω 10 If | z | = 1 and ω = (where z ≠ −1), then Re(ω ) is z +1 (c) − ω (d) − ω 1 21 If z − 1 = 1, then arg ( z ) is equal to (a) 0 (b) − 2 z+1 1 1 (a) arg (z) (b) arg (z + 1) 2 2 3 (c) 2 (d) None of these 1 z+1 (c) arg (z − 1) (d) None of these 2 14 40 15  1 22 Let z = cos θ + i sin θ. Then the value of ∑ Im ( z 2 m −1 ) at, 33 If Re   = 3 , then z lies on m =1 z θ = 2°, is (a) circle with centre onY-axis 1 1 1 1 (b) circle with centre on X-axis not passing through origin (a) (b) (c) (d) sin 2 ° 3 sin 2 ° 2 sin 2 ° 4 sin 2 ° (c) circle with centre on X-axis passing through origin (i ) 23 If z = (i )(i ) , where i = −1, then | z | is equal to (d) None of the above (a) 1 (b) e − π / 2 (c) 0 (d) e π / 2 34 If the imaginary part of ( 2z + 1) / (iz + 1) is −2, then the 8 locus of the point representing z in the complex plane is  π π 1 + i sin + cos  (a) a circle (b) a straight line 24  8 8  equals to (c) a parabola (d) None of these  1 − i sin π + cos π   8 8 z 35 If | z | = 1 and z ≠ ± 1, then all the values of lie on (a) 2 8 (b) 0 (c) −1 (d) 1 1− z 2 (a) a line not passing through the origin 25 If 1, α 1, α 2 , K , α n − 1 are the nth roots of unity, then (b) |z | = 2 (c) the X-axis ( 2 − α 1 )( 2 − α 2 ) K ( 2 − α n −1 ) is equal to (d) theY-axis (a) n (b) 2 n (c) 2 n + 1 (d) 2 n − 1 z 26 If ω( ≠ 1) is a cube root of unity and (1 + ω )7 = A + Bω. 36 If ω = and | ω | = 1, then z lies on i z− Then, ( A, B ) is equal to 3 (a) (11 ,) (b) (1, 0) (c) (−1, 1) (d) (0, 1) (a) a circle (b) an ellipse 27 If α , β ∈C are the distinct roots of the equation (c) a parabola (d) a straight line x 2 − x + 1 = 0, then α 101 + β107 is equal to j JEE Mains 2018 37 If z 1 and z 2 are two complex numbers such that (a) −1 (b) 0 (c) 1 (d) 2 z1 z 2 2 + = 1, then  25 1 z 2 z1 28 If x 2 + x + 1 = 0, then ∑  x r +  is equal to r=1  xr  (a) z 1, z 2 are collinear (b) z 1, z 2 and the origin form a right angled triangle (a) 25 (b) 25 ω (c) z 1, z 2 and the origin form an equilateral triangle (c) 25 ω2 (d) None of these (d) None of the above 29 Let ω be a complex number such that 2ω + 1 = z, 38 A complex number z is said to be unimodular, if z = 1. 1 1 1 Suppose z1 and z 2 are complex numbers such that where z = −3. If 1 −ω − 1 ω 2 = 3k, then k is equal to 2 z1 − 2z 2 is unimodular and z 2 is not unimodular. 1 ω2 ω7 j JEE Mains 2017 2 − z 1z 2 (a) −z (b) z (c) −1 (d) 1 Then, the point z1 lies on a j JEE Mains 2015 1+ ω ω2 1 + ω2 (a) straight line parallel to X −axis 30 The value − ω − (1 + ω 2 ) (1 + ω) , where ω is cube (b) straight line parallel toY −axis − 1 − (1 + ω 2 ) 1 + ω (c) circle of radius 2 (d) circle of radius 2 root of unity, is equal to (a) 2 ω (b) 3 ω2 (c) − 3 ω2 (d) 3ω 39 If | z 2 − 1 | = | z |2 +1, then z lies on 31 If a , b and c are integers not all equal and ω is a cube (a) a real axis (b) an ellipse root of unity (where, ω ≠ 1), then minimum value of (c) a circle (d) imaginary axis | a + bω + cω 2 | is equal to 3 1 40 Let z satisfy z = 1 and z = 1 − z j JEE Mains 2013 (a) 0 (b) 1 (c) (d) 2 2 Statement I z is a real number. iπ /3 32 Let ω = e , and a, b, c, x , y , z be non-zero complex Statement II Principal argument of z is π /3. numbers such that: (a) Statement I is true, Statement II is true; Statement II is a a + b + c = x ; a + b ω + cω = y ; a + b ω + c ω = z 2 2 correct explanation for statement I 2 2 2 (b) Statement I is true, Statement II is true; Statement II is x + y + z not a correct explanation for Statement I Then the value of 2 2 2 is: a + b + c (c) Statement I is true, Statement II is false (d) Statement I is false, Statement II is true (a) 1 (b) 2 (c) 3 (d) 4  DAY PRACTICE SESSION 2 PROGRESSIVE QUESTIONS EXERCISE 1 For positive integers n1 and n 2 the value of the expression 8 If a complex number z lies in the interior or on the (1 + i )n1 + (1 + i 3 )n1 + (1 + i 5 )n 2 + (1 + i 7 )n 2 boundary of a circle of radius 3 and centre at ( − 4 , 0), where i = −1, is a real number iff then the greatest and least value of | z + 1| are (a) n1 = n2 (b) n2 = n2 − 1 (c) n1 = n2 + 1 (d) ∀n1 and n2 (a) 5, 0 (b) 6, 1 (c) 6, 0 (d) None of these 9 If z is any complex number satisfying z − 3 − 2i ≤ 2, 2 z 2 If z ≠ 1 and is real, then the point represented by the z −1 then the minimum value of 2z − 6 + 5i is complex number z lies (a) 2 (b) 3 (c) 5 (d) 6 (a) on the imaginary axis 10 A man walks a distance of 3 units from the origin towards (b) either on the real axis or on a circle passing through the the North-East (N 45° E) direction. From there, he walks a origin distance of 4 units towards the North-West (N 45° W) (c) on a circle with centre at the origin direction to reach a point P. Then the position of P in the (d) either on the real axis or on a circle not passing through Argand plane is the origin (a) 3eiπ / 4 + 4i (b) (3 − 4i )eiπ / 4 2π 2π 3 Let ω be the complex number cos + i sin. Then the (c) (4 + 3i )eiπ / 4 (d) (3 + 4i )eiπ / 4 3 3 number of distinct complex numbers z satisfying 11 If 1, a1, a 2... a n −1 are n th roots of unity, then 1 1 1 z +1 ω ω2 + +... + equals to 1 − a1 1 − a 2 1 − a n −1 ω z + ω2 1 = 0 is equal to ω2 1 z +ω 2n − 1 n −1 n (a) (b) (c) (d) None of these n 2 n −1 (a) 0 (b) 1 (c) 2 (d) 4 2 2 12 For z ,ω ∈ C, if z ω − ω z = z − ω, then z is equal to 4 The locus of z = x + iy which satisfying the inequality log1/ 2 z − 1 > log1/ 2 z − i is given by (a) ω or ω (b) ω or ω /|ω | 2 (c) ω or ω /|ω | 2 (d) None of these (a) x + y < 0 (b) x − y > 0 (c) x − y < 0 (d) x + y > 0 10  2kπ 2kπ  5 Let z1 = 10 + 6i , z 2 = 4 + 6i. If z is any complex number 13 The value of ∑  sin + i cos  is  11 11  such that arg ( z − z1 ) / ( z − z 2 ) = π / 4, then z − 7 − 9i is k =1 equal to (a) 1 (b) −1 (c) − i (d) i (a) 18 (b) 3 2 (c) 3 / 2 (d) None of these 14 Let z1 and z 2 be roots of the equation z + pz + q = 0, 2 6 Let z = x + iy be a complex number where x and y are p, q ∈ C. Let A and B represents z1 and z 2 in the complex integers. Then the area of the rectangle whose vertices plane. If ∠AOB = α ≠ 0 and OA = OB; O is the origin, then are the roots of the equation zz 3 + zz 3 = 350 is p 2 / 4q is equal to (a) 48 (b) 32 (c) 40 (d) 80 (a) sin2 (α / 2) (b) tan2 (α / 2) (c) cos2 (α / 2) (d) None of these 7 If α + i β = cot −1( z ), where z = x + iy and α is a constant, 15 If 1, ω and ω 2 are the three cube roots of unity α , β, γ are then the locus of z is the cube roots of p, q < 0, then for any x , y , z the (a) x2 + y2 − x cot 2 α − 1 = 0  x α + y β + z γ expression   is equal to (b) x2 + y2 − 2 x cot α − 1 = 0  x β + y γ + z α (c) x2 + y2 − 2 x cot 2 α + 1 = 0 (d) x2 + y2 − 2 x cot 2 α − 1 = 0 (a) 1 (b) ω (c) ω 2 (d) None of these ANSWERS SESSION 1 1. (b) 2. (d) 3. (b) 4. (b) 5. (d) 6. (a) 7. (a) 8. (c) 9. (b) 10. (a) 11. (b) 12. (b) 13. (b) 14. (a) 15. (b) 16. (b) 17. (d) 18. (b) 19. (c) 20. (c) 21. (c) 22. (d) 23. (a) 24. (c) 25. (d) 26. (a) 27. (c) 28. (d) 29. (a) 30. (c) 31. (b) 32. (c) 33. (c) 34. (b) 35. (d) 36. (d) 37. (c) 38. (c) 39. (d) 40. (d) SESSION 2 1. (d) 2. (b) 3. (b) 4. (b) 5. (b) 6. (a) 7. (d) 8. (c) 9. (c) 10. (d) 11. (b) 12. (b) 13. (c) 14. (c) 15. (c) 16 40 Hints and Explanations SESSION 1 ⇒ x2 + y 2 = 1 10 Given,|z | = 1 1 ⇒ |z |2 = 1 ⇒ |z |= 1 1 Let z = ⇒ zz = 1 1 − cos θ + i sin θ 5 Given, z2 is a purely imaginary z−1 z−1 1 Now, 2Re(ω ) = ω + ω = + = z1 z+1 z+1 2sin2 (θ / 2) + 2i sin(θ / 2)cos(θ / 2) Let z = ni. Then, (z − 1)(z + 1) + (z − 1)(z + 1) 1 1 = = z2 z+1 2 2+ 3⋅ 2i sin(θ / 2) [cos(θ / 2) − i sin(θ / 2)] 2z1 + 3z2 z1 2 + 3ni = = 2zz − 2 cos(θ / 2) + i sin(θ / 2) 1 1 2z1 − 3z2 z2 2 − 3ni = =0 [Q zz = 1] = = + cot(θ / 2) 2 − 3⋅ z+1 2 2i sin(θ / 2) 2 2i z1 1 1 = − i ⋅ cot θ / 2 ∴ Re(ω ) = 0. 4 + 9n2 2 2 = =1 4 + 9n2   z − 4  + 4 11 |z | =     2 Let z = 2 + 3i sin θ is purely imaginary 7− z    z z 1 − 2i sin θ 6 Given, f (z ) = and z = 1 + 2i |z | ≤  z −  + 1 − z2 4 4 then we have ⇒ Re (z ) = 0 7 − (1 + 2i )  z  |z | ∴ f (z ) = 2 + 3i sin θ 1 − (1 + 2i )2 4 Consider, z = ⇒ |z | ≤ 2 + 1 − 2i sin θ 6 − 2i 6 − 2i |z| = = (2 + 3i sin θ)(1 + 2i sin θ) 1 − (1 − 4 + 4i ) 4 − 4i |z |2 − 2 |z | − 4 = ⇒ ≤0 (1 − 2i sin θ)(1 + 2i sin θ) 6 − 2i 1+ i 6 + 4i + 2 |z | = × = (2 − 6sin2 θ) + (4sin θ + 3sin θ )i 4(1 − i ) (1 + i ) 4(12 − i 2 ) Since,|z | > 0 = 1 + 4sin2 θ 8 + 4i 1 ⇒ |z |2 − 2 |z | − 4 ≤ 0 = = (2 + i ) Q Re(z ) = 0 4(2) 2 ⇒ [|z | − ( 5 + 1)] [|z |− (1 − 5)] ≤ 0 2 − 6sin2 θ ∴ =0 4+ 1 5 |z | ⇒ 1− 5 ≤ |z | ≤ 5+ 1 1 + 4sin2 θ Now,| f (z )| = = = 2 2 2 ⇒ 1 sin2 θ = ⇒ sinθ = ± 1 12 z ≥ 2 is the region on or outside circle [Qz = 1 + 2i , given ⇒|z |= 5] 3 3 whose centre is (0,0) and radius is 2. 1 −1  1  7 Given, 8 iz 3 + 12z 2 − 18z + 27i = 0 Minimum z + is distance of z, which ⇒ θ = ± sin   2  3 ⇒ 4z 2 (2iz + 3) + 9 i (2iz + 3) = 0 lie on circle z = 2 from  − , 0. 1 13 13 ⇒ (2iz + 3) (4z2 + 9 i ) = 0  2  ∑ (i + i n +1 ) = (1 + i ) ∑i n n 3 ⇒ 2iz + 3 = 0 or 4z2 + 9 i = 0 n =1 n =1 Y 3 i (1 − i 13 ) ∴ |z | = = (1 + i ) 2 1− i 8 We have, ( x + iy ) + 2 x + iy + 1 + i = 0 = i − 1 [Q i 13 = i , i 2 = −1] [put z = x + iy ] 4 Let z = x + iy A z−1 x + iy − 1 ( x − 1) + iy ⇒ ( x + iy ) + 2 ( x + 1)2 + y 2 + i = 0 X′ (–2, 0) (– 1 ,0 ( (0, 0) (2, 0) X = = ⇒ x + 2 ( x + 1) + y = 02 2 2 z + 1 x + iy + 1 ( x + 1) + iy ( x + 1) − iy and y+1= 0 × ( x + 1) − iy ⇒ x + 2 ( x + 1) + (−1)2 = 0 2 ( x − 1)( x + 1) − iy ( x − 1) + iy and y = −1 Y′ ( x + 1) − i 2 y 2 ⇒ x2 = 2[( x + 1)2 + 1] = ∴ Minimum z + 1 ( x + 1)2 − i 2 y 2 ⇒ x2 = 2 x2 + 4 x + 4 2 x2 − 1 + iy ( x + 1 − x + 1) + y 2 ⇒ x + 4 x + 4 = 0 ⇒ ( x + 2)2 = 0 2 =  1  = Distance of  − , 0 from (−2, 0) ⇒ x = −2 ( x + 1)2 + y 2  2  ∴ z = −2 − i ⇒ z = 4 + 1 = 5 z − 1 ( x2 + y 2 − 1) i (2 y ) 2 ⇒ = + β −α β −α =  −2 + 1  + 0 = 3 z + 1 ( x + 1)2 + y 2 ( x + 1)2 + y 2 9 =    2 2 z−1 1 − αβ β ⋅ β − αβ Since, is purely imaginary. Alternate Method z+1  Q |β | = 1  We know,|z1 + z2 |≥||z1 |−|z2|| z − 1 and |β|2 = ββ = 1 ∴ Re   =0    z + 1 ∴ z+ 1 ≥ |z| − 1 = |z|− 1 x + y −1 2 2 β −α 1 |β − α| |β − α| 2 2 2 ⇒ =0 = = = =1 ( x + 1)2 + y 2 β (β − α ) |β| | β − α| |β − α| 1 3 ≥ z− = ⇒ x + y −1 = 0 2 2 [Q|z|=|z|] 2 2 DAY 17 ∴ z+ 1 ≥ 3 20 Let z = ω = r and let arg ω = θ ⇒ − ω2 = A + Bω ⇒ 1 + ω = Α + Βω 2 2 [Qω14 = ω12 ⋅ ω2 = ω2 ] Then, ω = r (cos θ + i sin θ) = re i θ 1 3 On comparing both sides, we get ∴ Minimum value of z + is ⋅ and arg z = π − θ 2 2 A = 1, B = 1 Hence, z = r (cos( π − θ) + i sin( π − θ)) 13 Fact: z1 + z2 +... + z n = r (− cos θ + i sin θ) 27 α,β are the roots of x2 − x + 1 = 0 = − r (cos θ − i sin θ) Q Roots of x2 − x + 1 = 0 are − ω, − ω 2 ≤ z1 + z2 +... + z n z = −ω ∴ z1 + z2 + (5 + 12i ) ∴ Let α = −ω and β = − ω 2 21 Given,|z − 1| = 1 ⇒ z − 1 = e i θ , ≤ z1 + z2 + 5 + 12i ⇒ α101 + ?