(c) -1 The set of all the complex numbers are generally represented by ‘C’. CBSE Class 7 Mathematics Worksheets / Part 1/Yearly Examination, CBSE Important Extra Questions for Class 12 Sociology Chapter Wise, Lesson 117 – Parts of the Sentence – Transitive and Intransitive Verbs, दृष्टांत कौशल (Skill of Illustration with Examples), उद्दीपन परिवर्तन कौशल (Skill of Stimulus Variation), मानचित्र अध्ययन कौशल (Skill of Map Reading). ⇒ [(1 – 1 + 2i)/{1 + 1}]n = 1 (d) a parabola. Let z be a complex number such that |z| = 4 and arg(z) = 5π/6, then z = CBSE Class 11th Math 5 – Complex Numbers and Quadratic Equations MCQs. ⇒ x = 3/4. (d) a² = 4b, Answer: (c) a² = 3b = 12i MCQ from maths chapter Complex numbers. The number can also be written in the form of i 2 =-1. (d) 4, Answer: (d) 4 Get Important questions for class 11 Maths Chapter 5 – Complex Numbers and Quadratic Equations here.Students can get different types of questions covered in this chapter. Chapter 10 Class 11 Straight Lines. Now, (1 + ω – ω²)7 = (-ω² – ω²)7 So, √(-144) = 12i, Question 18. ... Chapter 5 Class 11 Complex Numbers. (a) 1 The value of i-999 is Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations Multiple Choice Questions and Answers for Board, JEE, NEET, AIIMS, JIPMER, IIT-JEE, AIEE and other competitive exams. ⇒ arg (-z) – arg (z) = π {since sin π + i cos π = -1, So arg(-1) = π}, Question 16. (c) 2√3 – 2i Note that since 1, ω, and ω² are the cube roots of unity (the three cube roots of 1), they are the three solutions to x³ = 1 (note: ω and ω² are the two complex solutions to this) The value of √(-144) is Download JEE Mathematics Complex Numbers MCQs Set A in pdf, Complex Numbers chapter wise Multiple Choice Questions free Check the below NCERT MCQ Questions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations with Answers Pdf free download. Find real θ such that (3 + 2i × sin θ)/(1 – 2i × sin θ) is imaginary = 1/i999 Chapter 9 Class 11 Sequences and Series. 3 – 4sin² θ = 0 Hint: Solution: Given, 4x + i (3x – y) = 3 + i (–6) …. ⇒ |Z| = √(25 + 16) 11TH CLASS MCQs on complex number. (b) θ = nπ ± π/3 where n is an integer So, x – 1 = 1 ± i√3 (a) x = 7/2, y = 2/3 Complex numbers Ex 13.1 Q1(i) Complex numbers Ex 13.1 Q1(ii) Complex numbers Ex 13.1 Q1(iii) Complex numbers Ex 13.1 Q1(iv) Complex numbers Ex 13.1 Q1(v) Parshant Kumar. This contains 30 Multiple Choice Questions for JEE Test: Complex Numbers (mcq) to study with solutions a complete question bank. (d) -√3 + i, Answer: (a) -2√3 + 2i (a) 41 Given, {(1 + i)/(1 – i)}n = 1 (d) -i, Answer: (c) i (1) By equating the real and the imaginary parts of equation (1), 4x = 3, 3x – y = –6, Now, 4x = 3. Introduction to Complex Numbers, Class 11 Mathematics NCERT Solutions. Chapter 8 Class 11 Binomial Theorem. If you have any queries regarding CBSE Class 11 Maths Complex Numbers and Quadratic Equations MCQs Multiple Choice Questions with Answers, drop a comment below and we will get back to you soon. If we let ω = (-1 – i√3)/2 and ω₂ = (-1 + i√3)/2 Given, (1 – w + w²)×(1 – w² + w4)×(1 – w4 + w8) × …………… to 2n factors This test is Rated positive by 87% students preparing for JEE.This MCQ test is related to JEE syllabus, prepared by JEE teachers. शिक्षण प्रक्रिया में शिक्षक को अनेक कार्य एक साथ करने पड़ते हैं जैसे लिखना, प्रश्न पूछना, स्पष्ट करना, प्रदर्शन करना... शिक्षण कौशल ( Teaching Skill) (a) (b) p. (c) … It goes to show that it is a solution to the equation x 2 +1=0. शिक्षण प्रक्रिया में शिक्षक को अनेक कार्य एक साथ करने पड़ते हैं जैसे लिखना, प्रश्न पूछना, स्पष्ट करना, प्रदर्शन करना आदि... शिक्षण कौशल ( Teaching Skill) ⇒ sin x = cos x and sin 2x = cos 2x {comparing real and imaginary part} (a) 2n ⇒ [{(1 + i)²}/{(1 – i²)}]n = 1 Hint: Now, in is 1 when n = 4 Now, sin x + i cos 2x = sin x – i cos 2x ⇒ x = {1 ± √(-3)}/2 CBSE guide notes are the comprehensive notes which covers the latest syllabus of CBSE and NCERT. (3 + 2i × sin θ)/(1 – 2i × sin θ) = {(3 – 4sin² θ) + 8i × sin θ}/(1 + 4sin² θ) …………. The test will consist of only objective type multiple choice questions requiring students to mouse-click their correct choice of the options against the related question number. English Mathematics. The test will consist of only objective type multiple choice questions requiring students to mouse-click their correct choice of the options against the related question number. Here you can get Class 11 Important Questions Maths based on NCERT Text book for Class XI. Chapter 11 Class 11 Conic Sections. A number that can be expressed in the form a + ib is known as the complex number; where a and b are the real numbers and i is the imaginary part of the complex number. Brief Discussion on Sets Part - … then 1 – 2ω and 1 – 2ω² yield the two complex solutions to (x – 1)³ + 8 = 0 = 1/(i996 × i³) ⇒ in = 1 ⇒ sin² θ = 3/4 Check out the Important Questions for Class 11 Maths Exam. 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Click here to join our FB Page and FB Group for Latest update and preparation tips and queries. = √(-1) × √(144) (a) a² = b Conjugate of this number = sin x – i cos 2x These CBSE NCERT Class 11 Complex Numbers and Quadratic Equation workbooks and question banks have been made by teachers of StudiesToday for benefit of Class 11 … Given ω is an imaginary cube root of unity. The modulus of 5 + 4i is (d) – 1, 1 + 2ω, 1 + 2ω², Answer: (c) – 1, 1 – 2ω, 1 – 2ω² Hint: Tips and Tricks on Logarithm. Let z = x + iy ⇒ (1 + ω – ω2)7 = -128 ω12 × ω2 {Since w4 = w, w8 = w2} Free PDF download of RD Sharma Solutions for Class 11 Maths Chapter 13 - Complex Numbers solved by Expert Mathematics Teachers on Vedantu.com. Question 1. Provide me latest MCQ Questions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations Free PDF Download so I can prepare for exams. We hope the given NCERT MCQ Questions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations with Answers Pdf free download will help you. Answer: (a) a circleHint:Let w = 2/zNow, |w| = |2/z|=> |w| = 2/|z|=> |w| = 2This shows that w lies on a circle with center at the origin and radius 2 units. Chapter 7 Class 11 Permutations and Combinations. ⇒ (z1 + z2)² – 2z1 × z2 = z1 × z2 Answer: (a) a circle ⇒ x = {1 ± √(1 – 4)}/2 . Complex Numbers and Quadratic Equations Class 11 MCQs Questions with Answers. Practicing these CBSE NCERT MCQ Questions of Class 11 Maths with Answers Pdf will guide students to do a quick revision for all the concepts present in each chapter and prepare for final exams. Now, put x = -w, we get For your final exams, Ncert Maths Exemplar for class 11 can help you a lot in order to prepare. Register for online coaching for IIT JEE (Mains & Advanced), NEET, Engineering and Medical entrance exams. Introduction to Complex Numbers, Class 11 Mathematics NCERT Solutions. (c) -2i Step by step Solutions of ML Aggarwal ISC Class-11 Mathematics with Exe1.1, Exe-1.2, Exe-1.3, Exe-1.4, Exe-1.5, Exe-1.6, Exe-1.7, and Chapter Test Questions. (d) -128 ω², Answer: (d) -128 ω² The value of √(-16) is Class 11 Maths Formulas: Complex Numbers And Quadratic Equations. Given, arg (z) < 0 |Z| = √(5² + 4²) Now, |w| = |2/z| No negative marking for incorrect choice. Chapter 6 Class 11 Linear Inequalities. = [{(1 + i) × (1 + i)}/{(1 – i) × (1 + i)}]n ⇒ [(1 + i² + 2i)/{1 – (-1)}]n = 1 There will be total 15 MCQ in this test. Hint: = i4/i3 {since i4 = 1} Class 11 Mathematics notes on Chapter 5 Complex Numbers And Quadratic Equations class 11 Notes Mathematics are also available for download in CBSE Guide website. Hindi Mathematics. Hint: Given x + 1/x = 1 Now, x³ = 1 when x³ – 1 = (x – 1)(x² + x + 1) = 0, giving x = 1 and ⇒ x = -w, -w² ⇒ z = 4(-√3/2 + i/2) If {(1 + i)/(1 – i)}n = 1 then the least value of n is (b) -13i ⇒ x = {1 ± √(-1)×√3}/2 ⇒ x = (-1 ± i√3)/2 (c) 128 ω² If the cube roots of unity are 1, ω, ω², then the roots of the equation (x – 1)³ + 8 = 0 are Complex Numbers ML Aggarwal ISC Class-11 Maths Understanding Solutions Chapter-5. Given, (a) a circle (b) – 1, -1, – 1 (c) 3 ⇒ (-a)² = 3b We have provided Complex Numbers and Quadratic Equations Class 11 Maths MCQs Questions with Answers to help students understand the concept very well. Hint: These questions unlike CBSE Ncert Textbook are a bit more difficult as well as advanced. MCQ Questions for Class 11 Maths with Answers were prepared based on the latest exam pattern. The above Practice worksheets for Class 11 Complex Numbers and Quadratic Equation have been designed as per latest NCERT CBSE and KVS guidelines and 2021 syllabus. Complex Numbers and Quadratic Equations Class 11 NCERT Book: If you are looking for the best books of Class 11 Maths then NCERT Books can be a great choice to begin your preparation. Hint: Hint: => |w| = 2/|z| Jan 12,2021 - Multiple Correct MCQ Of Complex Numbers, Past Year Questions JEE Advance, Class 11, Maths | 10 Questions MCQ Test has questions of JEE preparation. ⇒ z12 + z22 + z32 = z1 × z2 + z2 × z3 + z1 × z3 The test consists 25 questions. Hint: = in Maths Important Questions Class 11 are given below. 2/7 complex number. Chapter 5 Class 11 Complex Numbers. So, √(-144) = 12i, Question 3: Question 5. (d) None of these, Answer: (b) θ = nπ ± π/3 where n is an integer Then r = 4 and θ = 5π/6 (a) 128 ω Chapter 7 Class 11 … So, the value of x = 7/2 and y = 2/3, Question 13. = i × 12 {Since √(-1) = i} Let Z = 5 + 4i If ω is an imaginary cube root of unity, then (1 + ω – ω²)7 equals ⇒ x = {-(-1) ± √(1² – 4 × 1 × 1)}/(2 × 1) (b) 22n The above NCERT CBSE and KVS worksheets for Class 11 Complex Numbers and Quadratic Equation will help you to improve marks by clearing Complex Numbers and Quadratic Equation concepts and also improve problem solving skills. Complex numbers Ex 13.1 Q1(i) Complex numbers Ex 13.1 Q1(ii) Complex numbers Ex 13.1 Q1(iii) Complex numbers Ex 13.1 Q1(iv) Complex numbers Ex 13.1 Q1(v) = 5i + 6i + 6i Now, in is real when n = 2 {since i2 = -1 } We have picked up the questions one by one from each chapter, and solved them chapterwise . u³ + 8 = (u + 2)(u² – 2u + 4) = 0. ⇒ z = -2√3 + 2i, Question 8: Compare real and imaginary part, we get ⇒ a² = 3b, Question 10: No negative marking for incorrect choice. 0. Given, √(-16) = √(16) × √(-1) Multiple Choice Questions from chapter 5 complex numbers. Share. (d) a hyperbola. So, the solutions occur when u = -2 (giving -2 = x – 1 ⇒ x = -1), or when: Kindly give Chapter 5 Complex Numbers and Quadratic Equations Class 11 Maths MCQs Questions with Answers quickly as it is very essential. The questions includes 1 mark questions, 2 mark questions, 3 mark questions, 4 mark questions, 5 mark questions and other questions as per the latest CBSE curriculum for the current session. Hint: The solved questions answers in this Test: Complex Numbers quiz give you a good mix of easy questions and tough questions. ⇒ (1 + ω – ω2)7 = -128 (ω3)4 ω2 These short solved questions or quizzes are provided by Gkseries. (d) 2i, Answer: (b) 4i If you have any queries regarding CBSE Class 11 Maths Complex Numbers and Quadratic Equations MCQs Multiple Choice Questions with Answers, drop a comment below and we will get back to you soon. Given, z1 and z2 be two roots of the equation z² + az + b = 0 So, z = 4(cos 5π/6 + i × sin 5π/6) 11+12 – Physics; 11+12 – Chemistry; 11+12 – Mathematics; 11 – Physics; ... > Multiple Choice Questions from chapter 5 complex numbers. = (2²)n {since w³ = 1} The curve represented by Im(z²) = k, where k is a non-zero real number, is (3 + 2i × sin θ)/(1 – 2i × sin θ) = {(3 + 2i × sin θ)×(1 – 2i × sin θ)}/(1 – 4i² × sin² θ) 1. x² + x + 1 = 0 11+12 – Physics; ... > MCQ from maths chapter Complex numbers. (a) n1 = n2 … = 5i + 3×2i + 2×3i {since √(-1) = i} For a complex number z = p + iq, p is known as the real part, represented by Re z and q is known as the imaginary part, it is represented by Im z of complex number z. Let z = r(cos θ + i × sin θ) Here we have provided NCERT Exemplar Problems Solutions along with NCERT Exemplar Problems Class 11.. For positive integers n1, n2 the value of expression + here is a real number, if and only if. (a) 1 (a) θ = nπ ± π/2 where n is an integer The complex numbers sin x + i cos 2x are conjugate to each other for ⇒ z² = x² – y² + 2xy This chapter provides detailed information about the complex numbers and how to represent complex numbers in the Arg and plane. This shows that w lies on a circle with center at the origin and radius 2 units. and 7 – 2x = 0 Complex Numbers Class 11 Examples. (c) – 1, 1 – 2ω, 1 – 2ω² Hint: ⇒ 4sin² θ = 3 (c) a parabola if z lies on |z| = 1, then 2/z lies on (b) 1 = [{(1 + i)²}/{(1 – i²)}]n ⇒ xy = k/2 which is a hyperbola. All Chapter-13 Exercise Questions with Solutions to help you to revise complete Syllabus and Score More marks. (c) √41 (a) 12i (b) -12i Let z1 and z2 be two roots of the equation z² + az + b = 0, z being complex. = w2000 + 1/w2000 So, the modulus of 5 + 4i is √41. = (-2w) × (-2w²) × (-2w) × (-2w²)× …………… to 2n factors = 1/{(i4)249 × i3} Question 1. The value of √(-25) + 3√(-4) + 2√(-9) is All questions, including examples and miscellaneous have been solved and divided into different Concepts, with questions ordered from easy to difficult.The topics of the chapter includeSolvingQuadratic equationwhere root is in negativ We have provided Complex Numbers and Quadratic Equations Class 11 Maths MCQs Questions with Answers to help students understand the concept very well. Chapter 6 Class 11 Linear Inequalities. 1. = i × 12 {Since √(-1) = i} ⇒ x = 2 ± i√3 (a) 1 Complex Numbers extends the concept of one dimensional real numbers to the two dimensional complex numbers in which two dimensions comes from real part and the imaginary part. Given, Im(z²) = k ⇒ (z1 + z2)² = 2z1 × z2 + z1 × z2 Class 11. (c) i we will cover this topic from basic also we will try to solve MCQ based on chapter. = w² + 1/w² {since w³ = 1} Question 12.The value of x and y if (3y – 2) + i(7 – 2x) = 0(a) x = 7/2, y = 2/3(b) x = 2/7, y = 2/3(c) x = 7/2, y = 3/2(d) x = 2/7, y = 3/2, Answer: (a) x = 7/2, y = 2/3Hint:Given, (3y – 2) + i(7 – 2x) = 0Compare real and imaginary part, we get3y – 2 = 0⇒ y = 2/3and 7 – 2x = 0⇒ x = 7/2So, the value of x = 7/2 and y = 2/3, Question 13.Find real θ such that (3 + 2i × sin θ)/(1 – 2i × sin θ) is imaginary(a) θ = nπ ± π/2 where n is an integer(b) θ = nπ ± π/3 where n is an integer(c) θ = nπ ± π/4 where n is an integer(d) None of these, Answer: (b) θ = nπ ± π/3 where n is an integerHint:Given,(3 + 2i × sin θ)/(1 – 2i × sin θ) = {(3 + 2i × sin θ)×(1 – 2i × sin θ)}/(1 – 4i² × sin² θ)(3 + 2i × sin θ)/(1 – 2i × sin θ) = {(3 – 4sin² θ) + 8i × sin θ}/(1 + 4sin² θ) …………. So, i-999 = i, Question 9. Hint: (c) a² = 3b The solved questions answers in this Test: Complex Numbers quiz give you a good mix of easy questions and tough questions. What are Complex Numbers? ⇒ sin θ = ±√3/2 MCQ Questions for Class 11 Maths with Answers were prepared based on the latest exam pattern. Similar Classes. (a) -4i Given, (3y – 2) + i(7 – 2x) = 0 (c) 23n Question 1. KPK MCQs (c) -π/2 CBSE Class 11th Math 5 – Complex Numbers and Quadratic Equations MCQs If α and β are the roots of the equation 3 x 2 + 2x + 7 = 0, then 1/α+1/β=____. (a) a pair of striaght line शिक्षण प्रक्रिया में शिक्षक को अनेक कार्य एक साथ करने पड़ते हैं जैसे लिखना, प्रश्न पूछना, स्पष्ट करना, प्रदर्शन करना... MCQ Questions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic... MCQ Questions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations with Answers, Click here to join our FB Page and FB Group for Latest update and preparation tips and queries, https://www.facebook.com/groups/tetsuccesskey/, Child Development and Pedagogy (English Medium), Teaching of Social Studies (English Medium), Teaching of Social Studies (Hindi Medium), सी टी ई टी नोट्स: बाल विकास एवं शिक्षाशास्त्र, सामान्य ज्ञान – General Knowledge In Hindi, MCQ Questions for Class 11 Maths Chapter 4 Principle of Mathematical Induction with Answers, Maths GK Questions With Answers for Competitive Exams -Part 1. (b) -41 ⇒ [{(1 + i) × (1 + i)}/{(1 – i) × (1 + i)}]n = 1 Hint: ⇒ y = 2/3 Anuj Seth. Let w = 2/z (b) x = 0 ⇒ θ = nπ ± π/3 where n is an integer, Question 14. Given, √(-25) + 3√(-4) + 2√(-9) (a) -1, -1 + 2ω, – 1 – 2ω² = √{(-1) × (25)} + 3√{(-1) × 4} + 2√{(-1) × 9} Given complex number = sin x + i cos 2x This mock test of Test: Complex Numbers for JEE helps you for every JEE entrance exam. (d) no value of x, Answer: (d) no value of x MCQ Questions for Class 11 Maths with Answers were prepared based on the latest exam pattern. = w² + w³ /w² (b) 2 Instructions:-All the questions are compulsory. If arg (z) < 0, then arg (-z) – arg (z) = Hint: Check the below NCERT MCQ Questions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations with Answers Pdf free download. = (2² w³)×(2² w³)×(2² w³) …………… to 2n factors Hint: = -1 {since 1 + w + w² = 0} This contains 30 Multiple Choice Questions for JEE Test: Complex Numbers (mcq) to study with solutions a complete question bank. Answer: (d) a hyperbola Given Complex number: (1 – i) … 3y – 2 = 0 (b) -π (b) 2 (a) 0 302k watch mins. Register online for Maths tuition on Vedantu.com to … Example 1: If 4x + i (3x – y) = 3 + i (– 6), where x and y are real numbers, then find the values of x and y. So, the roots of (x – 1)³ + 8 are -1, 1 – 2ω, and 1 – 2ω², Question 19. Complex Numbers and Quadratic Equations Class 11 MCQs Questions with Answers. => |w| = 2 All Chapter 13 - Complex Numbers Ex 13.3 Questions with Solutions for RD Sharma Class 11 Maths to help you to revise complete Syllabus and Score More marks. The exemplar questions constitute of variety of questions such as Multiple-Choice Questions (MCQs), Short Answer Questions (SA) and Long Answer Questions (LA). We hope the given NCERT MCQ Questions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations with Answers Pdf free download will help you. MCQ Questions for Class 11 Maths with Answers were prepared based on the latest exam pattern. We have provided Complex Numbers and Quadratic Equations Class 11 Maths MCQs Questions with Answers to help students understand the concept very well. 1Now, equation 1 is imaginary if3 – 4sin² θ = 0⇒ 4sin² θ = 3⇒ sin² θ = 3/4⇒ sin θ = ±√3/2⇒ θ = nπ ± π/3 where n is an integer, Question 14.If {(1 + i)/(1 – i)}n = 1 then the least value of n is(a) 1(b) 2(c) 3(d) 4, Answer: (d) 4Hint:Given, {(1 + i)/(1 – i)}n = 1⇒ [{(1 + i) × (1 + i)}/{(1 – i) × (1 + i)}]n = 1⇒ [{(1 + i)²}/{(1 – i²)}]n = 1⇒ [(1 + i² + 2i)/{1 – (-1)}]n = 1⇒ [(1 – 1 + 2i)/{1 + 1}]n = 1⇒ [2i/2]n = 1⇒ in = 1Now, in is 1 when n = 4So, the least value of n is 4, Question 15.If arg (z) < 0, then arg (-z) – arg (z) =(a) π(b) -π(c) -π/2(d) π/2, Answer: (a) πHint:Given, arg (z) < 0Now, arg (-z) – arg (z) = arg(-z/z)⇒ arg (-z) – arg (z) = arg(-1)⇒ arg (-z) – arg (z) = π {since sin π + i cos π = -1, So arg(-1) = π}, Question 16.if x + 1/x = 1 find the value of x2000 + 1/x2000 is(a) 0(b) 1(c) -1(d) None of these, Answer: (c) -1Hint:Given x + 1/x = 1⇒ (x² + 1) = x⇒ x² – x + 1 = 0⇒ x = {-(-1) ± √(1² – 4 × 1 × 1)}/(2 × 1)⇒ x = {1 ± √(1 – 4)}/2⇒ x = {1 ± √(-3)}/2⇒ x = {1 ± √(-1)×√3}/2⇒ x = {1 ± i√3}/2 {since i = √(-1)}⇒ x = -w, -w²Now, put x = -w, we getx2000 + 1/x2000 = (-w)2000 + 1/(-w)2000= w2000 + 1/w2000= w2000 + 1/w2000= {(w³)666 × w²} + 1/{(w³)666 × w²}= w² + 1/w² {since w³ = 1}= w² + w³ /w²= w² + w= -1 {since 1 + w + w² = 0}So, x2000 + 1/x2000 = -1, Question 17.The value of √(-144) is(a) 12i(b) -12i(c) ±12i(d) None of these, Answer: (a) 12iHint:Given, √(-144) = √{(-1)×144}= √(-1) × √(144)= i × 12 {Since √(-1) = i}= 12iSo, √(-144) = 12i, Question 18.If the cube roots of unity are 1, ω, ω², then the roots of the equation (x – 1)³ + 8 = 0 are(a) -1, -1 + 2ω, – 1 – 2ω²(b) – 1, -1, – 1(c) – 1, 1 – 2ω, 1 – 2ω²(d) – 1, 1 + 2ω, 1 + 2ω², Answer: (c) – 1, 1 – 2ω, 1 – 2ω²Hint:Note that since 1, ω, and ω² are the cube roots of unity (the three cube roots of 1), they are the three solutions to x³ = 1 (note: ω and ω² are the two complex solutions to this)If we let u = x – 1, then the equation becomesu³ + 8 = (u + 2)(u² – 2u + 4) = 0.So, the solutions occur when u = -2 (giving -2 = x – 1 ⇒ x = -1), or when:u² – 2u + 4 = 0,which has roots, by the Quadratic Formula, to be u = 1 ± i√3So, x – 1 = 1 ± i√3⇒ x = 2 ± i√3Now, x³ = 1 when x³ – 1 = (x – 1)(x² + x + 1) = 0, giving x = 1 andx² + x + 1 = 0⇒ x = (-1 ± i√3)/2If we let ω = (-1 – i√3)/2 and ω₂ = (-1 + i√3)/2then 1 – 2ω and 1 – 2ω² yield the two complex solutions to (x – 1)³ + 8 = 0So, the roots of (x – 1)³ + 8 are -1, 1 – 2ω, and 1 – 2ω², Question 19. 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