Z = a + ib is the algebraic form in which ‘a’ represents real part and ‘b’ represents imaginary part. Ltd. Trigonometric Equations and General Values. All the examples listed here are in Cartesian form. The first value represents the real part of the complex number, and the second value represents its imaginary part. Let 2=−බ ∴=√−බ Just like how ℝ denotes the real number system, (the set of all real numbers) we use ℂ to denote the set of complex numbers. = $\frac{1}{{\sqrt 2 }}$ (cos45° + i.sin45°). (b) If ω1 + ω2 = 0 then the lines are parallel. Horizontal axis represents real part while the vertical axis represents imaginary part. ‘a’ is called the real part, and ‘b’ is called the imaginary part of the complex number. (c) If ω1 = ω2 then the lines are not parallel. {\rm{sin}}3\theta } \right)\left\{ {\cos \left( { - \theta } \right) + {\rm{i}}.\sin \left( { - \theta } \right)} \right\}}}{{{{\left( {{\rm{cos}}\theta  + {\rm{isin}}\theta } \right)}^2}}}$, = $\frac{{\cos \left( {3\theta  - \theta } \right) + {\rm{i}}.\sin \left( {3\theta  - \theta } \right)}}{{{{\left( {{\rm{cos}}\theta  + {\rm{i}}. If z = -2 + j4, then Re(z) = -2 and Im(z) = 4. = $\frac{1}{{\sqrt 2 }} - {\rm{i}}.\frac{1}{{\sqrt 2 }}$ = $ - \left( { - \frac{1}{{\sqrt 2 }} + {\rm{i}}.\frac{1}{{\sqrt 2 }}} \right)$. A complex number is usually denoted by the letter ‘z’. Question 1. When, k = 3, Z3 = cos $\left( {\frac{{0 + 1080}}{6}} \right)$ + i.sin $\left( {\frac{{0 + 1080}}{6}} \right)$, When, k = 4, Z4 = cos $\left( {\frac{{0 + 1440}}{6}} \right)$ + i.sin $\left( {\frac{{0 + 1440}}{6}} \right)$, = cos240° + i.sin240° = $ - \frac{1}{2} + {\rm{i}}.\frac{{\sqrt 3 }}{2}$, When, k = 5, Z5 = cos $\left( {\frac{{0 + 1800}}{6}} \right)$ + i.sin $\left( {\frac{{0 + 1800}}{6}} \right)$. 2 + i3, -5 + 6i, 23i, (2-3i), (12-i1), 3i are some of the examples of complex numbers. {\rm{sin}}(\theta  + {\rm{k}}.360\} $, Or, zk = r1/6$\left\{ {\cos \frac{{0 + {\rm{k}}.360}}{4} + {\rm{i}}.\sin \frac{{0 + {\rm{k}}.360}}{4}} \right\}$. If ω1 = ω2 are the complex slopes of two lines, then. Moreover, i is just not to distinguish but also has got some value. = (cos 30° + i.sin30°) = $\frac{{\sqrt 3 }}{2} + {\rm{i}}.\frac{1}{2}$. Or, $\frac{{1 + {\rm{i}}}}{{1 - {\rm{i}}}}$ = $\frac{{1 + {\rm{i}}}}{{1 - {\rm{i}}}}$ * $\frac{{1 + {\rm{i}}}}{{1 + {\rm{i}}}}$ = $\frac{{1 + 2{\rm{i}} + {{\rm{i}}^2}}}{{{1^2} - {{\rm{i}}^2}}}$ = $\frac{{1 + 2{\rm{i}} - 1}}{{1 + 1}}$ = I = 0 + i. r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {0 + 1} $ = 1. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{1}{0}$ = -1  then θ= 90°. (7). This chapter provides detailed information about the complex numbers and how to represent complex numbers in the Arg and plane. NCERT Books Free PDF Download for Class 11th to 12th "NCERT Books Free PDF Download for Class 11th to 12th" plays an important role in the JEE Main Preparation because mostly the questions in the JEE Main exam will be asked directly from the NCERT books. Register online for Maths tuition on Vedantu.com to … Remainder when f(z) is divided by (z – i) = f(i). On multiplying these two complex number we can get the value of x. z2 + 2z + 3 = 0 is also an example of complex equation whose solution can be any complex number. = $\frac{{\left( {{\rm{cos}}3\theta  + {\rm{i}}. news feed!”. $\left[ {\cos \frac{{\theta  + {\rm{k}}.360}}{3} + {\rm{i}}.\sin \frac{{\theta  + {\rm{k}}.360}}{3}} \right]$, Or, ${\rm{z}}_0^{\frac{1}{3}}$ = 1. Similarly, for z = 3+j5, Re(z) = 3 and Im(z) = (5). If ‘a’ is the real part and ‘b’ represents imaginary part, then complex number is represented as z = a + ib where i, stands for iota which itself is a square root of negative unity. $\left[ {\cos \frac{{180 + 0}}{3} + {\rm{i}}.\sin \frac{{180 + 0}}{3}} \right]$. {\rm{sin}}\theta } \right)}}{{{{\left( {{\rm{cos}}\theta  + {\rm{i}}. CBSE Class 11 Mathematics Worksheet - Complex Numbers and Quadratic Equation (1) CBSE,CCE and NCERT students can refer to the attached file. Solved and explained by expert mathematicians. How do we locate any Complex Number on the plane? In electronics, already the letter ‘i’ is reserved for current and thus they started using ‘j’ in place of i for the imaginary part. A complex number z is usually written in the form z = x + yi, where x and y are real numbers, and i is the imaginary unit that has the property i 2 = -1. name, Please Enter the valid r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {0 + 4} $ = 2. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{2}{0}$ = ∞, then θ= 90°. One of our academic counsellors will contact you within 1 working day. Express your answer in Cartesian form (a+bi): (a) z3 = i z3 = ei(π 2 +n2π) =⇒ z = ei(π 2 +n2π)/3 = ei(π 6 +n2π 3) n = 0 : z = eiπ6 = cos π 6 +isin π 6 = 3 2 + 1 i n = 1 : z = ei56π = cos 5π 6 +isin 5π Also, BYJU’S provides step by step solutions for all NCERT problems, thereby ensuring students understand them and clear their exams with flying colours. Complex Number itself has many ways in which it can be expressed. Similarly, the remainder when f(z) is divided by (z + i) = f(- i)   ….. (1), and f( -i) = 1 + i. √b = √ab is valid only when atleast one of a and b is non negative. Hence, the equation becomes x2 – (ω + ω2)x + ω ω2 = 0. Since, z2 + 1 is a quadratic expression, therefore remainder when f(z) is divided by z2 + 1 will be in general a linear expression. To read more, Buy study materials of Complex Numbers comprising study notes, revision notes, video lectures, previous year solved questions etc. 2 + i3, -5 + 6i, 23i, (2-3i), (12-i1), 3i are some of the examples of complex numbers. Look into the Previous Year Papers with Solutions to get a hint of the kinds of questions asked in the exam. Having introduced a complex number, the ways in which they can be combined, i.e. Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.5 Additional Problems. Students can also make the best out of its features such as Job Alerts and Latest Updates. Here, z = - 2, y = - 2, r = $\sqrt {4 + 12} $ = 4. Or, $\frac{{\rm{i}}}{{1 + {\rm{i}}}}$ = $\frac{{\rm{i}}}{{1 + {\rm{i}}}}$ * $\frac{{1 - {\rm{i}}}}{{1 - {\rm{i}}}}$ = $\frac{{{\rm{i}} - {{\rm{i}}^2}}}{{{1^2} - {{\rm{i}}^2}}}$ = $\frac{{{\rm{i}} - \left( { - 1} \right)}}{{1\left( { - 1} \right)}}$ = $\frac{{{\rm{i}} + 1}}{2}$ = $\frac{1}{2} + \frac{{\rm{i}}}{2}$. r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {{2^2} + {2^2}} $ = $\sqrt {4 + 4} $ = 2$\sqrt 2 $. The complex number in the polar form = r(cosθ + i.sinθ). Find the modulus and argument of the following complex numbers and convert them in polar form. Remarks. ‘z’ will be 6 units in the right and 4 units upwards from the origin. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{1}{1}$ = 1 then θ= 45°. r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {\frac{1}{4} + \frac{1}{4}} $ = $\frac{1}{{\sqrt 2 }}$. When k = 1, $\sqrt {{{\rm{z}}_1}} $ = $\sqrt 2 $$\left[ {\cos \left( {\frac{{90 + 360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{90 + 360}}{2}} \right)} \right]$. $\left[ {\cos \frac{{180 + 720}}{3} + {\rm{i}}.\sin \frac{{180 + 720}}{3}} \right]$. This point will be lying 2 units in the left and 3 units downwards from the origin. So, Zk = r  [cos (θ + k.360) + i.sin(θ + k.360)], Or, ${\rm{z}}_{\rm{k}}^{\frac{1}{2}}$ = [8{cos (60 + k.360) + i.sin (60 + k.360)}]1/2, = 81/2$\left[ {\cos \left( {\frac{{60 + {\rm{k}}.360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{60 + {\rm{k}}.360}}{2}} \right)} \right]$, When k = 0, $\sqrt {{{\rm{z}}_0}} $ = 2$\sqrt 2 $$\left[ {\cos \left( {\frac{{60 + 0}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{60 + 0}}{2}} \right)} \right]$. The well-structured Intermediate portal of sakshieducation.com provides study materials for Intermediate, EAMCET.Engineering and Medicine, JEE (Main), JEE (Advanced) and BITSAT. Since in third quadrant both a and b are negative and thus a = -2 and b = -3 in our example. Here, x = -1, y = 1, r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {{{\left( { - 1} \right)}^2} + {1^2}} $ = $\sqrt 2 $. 2. Let g(z) be the quotient and az + b the remainder when g(z) is divided by z2 + 1. We have provided Complex Numbers and Quadratic Equations Class 11 Maths MCQs Questions with Answers to help students understand the … {\rm{sin}}80\infty }}{{{\rm{cos}}20\infty  + {\rm{i}}. Here, x = 1, y = 1, r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {1 + 1} $ = $\sqrt 2 $. So, required roots are ± (- 1 + i$\sqrt 3 $). $\frac{{\sqrt 3 }}{2}$. Complete JEE Main/Advanced Course and Test Series. Tanθ= $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{{\sqrt 3 }}{{ - 1}}$ = $ - \sqrt 3 $ then θ = 120°. So, required roots are $\sqrt 3 $ + i, $ - \sqrt 3 $ + i , - 2i. Here x =$\frac{1}{2}$, y = $\frac{1}{2}$. CBSE Worksheets for Class 11 Maths: One of the best teaching strategies employed in most classrooms today is Worksheets. = 12−8−15+102 9−6+6−42 = 12−23+10(−1) 9−4(−1) =2−23 13 = − Graphical Representation A complex number can be represented on an Argand diagram by plotting the real part on the -axis and the imaginary part on the y-axis. Free PDF Download of JEE Main Complex Numbers and Quadratic Equations Important Questions of key topics. Chapter 3 Complex Numbers 56 Activity 1 Show that the two equations above reduce to 6x 2 −43x +84 =0 when perimeter =12 and area =7.Does this have real solutions? Grade 12; PRACTICE. SPI 3103.2.1 Describe any number in the complex number system. Tanθ = $\frac{{ - 1}}{0}$  then θ = 270°. Then find the equation whose roots are a19 and b7. So, ${\rm{z}}_{\rm{k}}^3$ = r$\{ \cos \left( {\theta  + {\rm{k}}.360} \right) + {\rm{i}}. subject, comprising study notes, revision notes, video lectures, previous year solved questions etc. NCERT Solutions For Class 11 Maths: The NCERT Class 11 Maths book contains 16 chapters each with their exercises that help students practice the concepts. ©Copyright 2014 - 2021 Khulla Kitab Edutech Pvt. = cos 45° + i.sin45° = $\frac{1}{{\sqrt 2 }}$ + i.$\frac{1}{{\sqrt 2 }}$. Here you can read Chapter 5 of Class 11 Maths NCERT Book. Students looking for NCERT Solutions for Class 11 Maths can download the same from this article. grade, Please choose the valid (b) If z = a + ib is the complex number, then a and b are called real and imaginary parts, respectively, of the complex number and written as R e (z) = a, Im (z) = b. (c) 0 if r is not a multiple of 3 and 3 if r is a multiple of 3. Careers | When k = 1, $\sqrt {{{\rm{z}}_1}} $ = $\sqrt 2 $$\left[ {\cos \left( {\frac{{120 + 360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{120 + 360}}{2}} \right)} \right]$. Tanθ = $ - \frac{{2\sqrt 3 }}{{ - 2}}$ = $\sqrt 3 $ then θ = 240°. Thus we can say that all real numbers are also complex number with imaginary part zero. If z is purely real negative complex number then. Free PDF download of Class 11 Maths revision notes & short key-notes for Chapter-5 Complex Numbers and Quadratic Equations to score high marks in exams, prepared by expert mathematics teachers from latest edition of CBSE books. = cos13° + i.sin135° = $ - \frac{1}{2}$ + i.$\frac{1}{2}$. There are also different ways of representation for the complex number, which we shall learn in the next section. We know from the above discussion that, Complex Numbers can be represented in four different ways. All educational material on the website has been prepared by the best teachers having more than 20 years of teaching experience in various schools. Complex Numbers are the numbers which along with the real part also has the imaginary part included with it. It’s an easier way as well. Natural Numbers: Whole Numbers: Integers: Rational Numbers: Irrational Numbers: Types of Rational Numbers: Terminating Decimal Fractions; Recurring and Non-terminating Decimal Fractions: Concept of Radicals and Radicands: Base and Exponent: Definition of a Complex Number: Conjugate of a Complex Number: Section 2: (Exercise No : 2.1) If a = a + bi is a complex number, then a is called its real part, notation a = Re(a), and b is called its imaginary part, notation b = Im(a). Complex Numbers (a + bi) Natural (Counting) Numbers Whole Numbers Integers Rational Numbers Real Numbers Irrational #’s Imaginary #’s Complex Numbers are written in the form a + bi, where a is the real part and b is the imaginary part. Q5. Prepared by the best teachers with decades of experience, these are the latest Class 11 Maths solutions that you will find. = 2 {cos 270° + i.sin270°} = 2{0 + i. For a complex number z = x+iy, x is called the real part, denoted by Re z and y is called the imaginary part denoted by Im z. These values represent the position of the complex number in the two-dimensional Cartesian coordinate system. 6. When k = 2, Z2 = cos $\left( {\frac{{90 + 720}}{3}} \right)$ + i.sin $\left( {\frac{{90 + 720}}{3}} \right)$. When k = 1, Z1 = 2 {cos$\left( {\frac{{90 + 360}}{3}} \right)$ + i.sin $\left( {\frac{{90 + 360}}{3}} \right)$}. FAQ's | When k = 1, Z1 = cos $\left( {\frac{{0 + 360}}{6}} \right)$ + i.sin $\left( {\frac{{0 + 360}}{6}} \right)$. Extraction of square root of complex number. The real part of the complex number is represented by x, and the imaginary part of the complex number is represented by y. We then write z = x +yi or a = a +bi. A complex number is usually denoted by the letter ‘z’. Also after the chapter, you can get links to Class 11 Maths Notes, NCERT Solutions, Important Question, Practice Papers, etc. You can get the knowledge of Recommended Books of Mathematics here. To find the value of in (n > 4) first, divide n by 4.Let q is the quotient and r is the remainder.n = 4q + r where o < r < 3in = i4q + r = (i4)q , ir = (1)q . Complex number has two parts, real part and the imaginary part. Privacy Policy | Now consider a point in the second quadrant that is. Here, x = 0, y = 2, r = $\sqrt {0 + 4} $ = 2. Hence, Arg. 3. = $\sqrt 2 ${cos 45° + i.sin45°} = $\sqrt 2 $.$\left( {\frac{1}{{\sqrt 2 }} + {\rm{i}}.\frac{1}{{\sqrt 2 }}} \right)$ = 1 + i. √a . = 1 (cos315° + i.sin315°). Or, zk = r1/4$\left\{ {\cos \frac{{\theta  + {\rm{k}}.360}}{4} + {\rm{i}}.\sin \frac{{\theta  + {\rm{k}}.360}}{4}} \right\}$, = 1$\left\{ {\cos \frac{{120 + {\rm{k}}.360}}{4} + {\rm{i}}.\sin \frac{{120 + {\rm{k}}.360}}{4}} \right\}$, When k = 0, Z0 = [cos $\frac{{120 + 0}}{4}$ + i.sin $\frac{{120 + 0}}{4}$]. So, required roots are ± $\left( {\frac{{\sqrt 3 }}{2} + \frac{1}{2}{\rm{i}}} \right)$, ± $\left( {\frac{1}{2} - \frac{{\sqrt 3 }}{2}{\rm{i}}} \right)$. = 2{cos 120° + i.sin120°} = 2.$\left( { - \frac{1}{2} + {\rm{i}}.\frac{{\sqrt 3 }}{2}} \right)$ = $ - {\rm{\: }}1{\rm{\: }}$+ i$\sqrt 3 $. = cos90° + i.sin90°. Any integral power of ‘i’ (iota) can be expressed as, Q2. NCERT Solutions For Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations are prepared by the expert teachers at BYJU’S. Or, zk = r1/4$\left\{ {\cos \frac{{180 + {\rm{k}}.360}}{4} + {\rm{i}}.\sin \frac{{180 + {\rm{k}}.360}}{4}} \right\}$, When k = 0, Z0 = 1 [cos $\frac{{180 + 0}}{4}$ + i.sin $\frac{{180 + 0}}{4}$]. = 210 [-cos0 + i.sin0] = 210 [-1 + i.0] = - 210. 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. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{{\frac{1}{2}}}{{\frac{1}{2}}}$ = 1  then θ= 45°. It will help you to save your precious time just before the examination. If a = a + bi is a complex number, then a is called its real part, notation a = Re(a), and b is called its imaginary part, notation b = Im(a). Preparing for entrance exams? Tutor log in | The set of all the complex numbers are generally represented by ‘C’. Concepts of complex numbers, addition, subtraction, multiplication, division of complex numbers. Here, x = -1, y = 0, r = $\sqrt {1 + 0} $ = 1. which means i can be assumed as the solution of this equation. Natural Numbers: Whole Numbers: Integers: Rational Numbers: Irrational Numbers: Types of Rational Numbers: Terminating Decimal Fractions; Recurring and Non-terminating Decimal Fractions: Concept of Radicals and Radicands: Base and Exponent: Definition of a Complex Number: Conjugate of a Complex Number: Section 2: (Exercise No : 2.1) So, z = r (cosθ + i.sinθ) = $\sqrt 2 $(cos 45° + i.sin45°), Or, z20 = [$\sqrt 2 $(cos 45° + i.sin45°)]20, = ${\left( {\sqrt 2 } \right)^{20}}$[cos(45 * 20) + i.sin (45 * 20)], = 210 [cos(90 * 10 + 0) + i.sin (90 * 10 + 0)]. Two mutually perpendicular axes are used to locate any complex point on the plane. {\rm{sin}}\theta } \right)}^2}}}$, = $\frac{{{\rm{cos}}2\theta  + {\rm{i}}. When k = 2, Z2 = cos $\left( {\frac{{0 + 720}}{6}} \right)$ + i.sin $\left( {\frac{{0 + 720}}{6}} \right)$. CBSE Class 11 Maths Worksheet for students has been used by teachers & students to develop logical, lingual, analytical, and problem-solving capabilities. basically the combination of a real number and an imaginary number ‘i’ (or ‘j’ in some books) in math is used to denote the imaginary part of any complex number. Or, 2 $\left( { - \frac{{\sqrt 3 }}{2} + \frac{{{\rm{i}}.1}}{2}} \right)$ = $ - \sqrt 3 $ + i. Then we can easily equate the two and get a = 6 and b = 4. Home ; Grade 11 ; Mathematics ; Sukunda Pustak Bhawan ; Complex Number. Here, z = -1, y = 0, r = $\sqrt {1 + 0} $ = 1. tanθ = $\frac{0}{{ - 1}}$ = 0 then θ= 180°. These NCERT Solutions of Maths help the students in solving the problems quickly, accurately and efficiently. 1. It provides the information on AP EAMCET and TS EAMCET Notifications, and EAMCET Counselling. MCQ Questions for Class 11 Maths with Answers were prepared based on the latest exam pattern. (cos60° + i.sin60°), Z7 = [cos60° + i.sin60°]7  = cos (60 * 7) + i.sin(60 * 7). Argument of a Complex Number Argument of a... Complex Number System Indian mathematician... n th Roots of Unity In general, the term root of... About Us | Illustration 3: Find all complex numbers z for which arg [(3z-6-3i)/(2z-8-6i)] = π/4 and |z-3+4i| = 3. Complex Number can be considered as the super-set of all the other different types of number. = 2$\sqrt 2 $$\left[ { - \frac{{\sqrt 3 }}{2} - {\rm{i}}.\frac{1}{3}} \right]$ = $ - \left( {\sqrt 6  + {\rm{i}}.\sqrt 2 } \right)$. Let 2=−බ ∴=√−බ Just like how ℝ denotes the real number system, (the set of all real numbers) we use ℂ to denote the set of complex numbers. = + ∈ℂ, for some , ∈ℝ and if a = 0, z = ib which is called as the Purely Imaginary Number. A complex number is a number that comprises a real number part and an imaginary number part. Or, ${\rm{z}}_1^{\frac{1}{3}}$ = $\left[ {\cos \frac{{180 + 360}}{3} + {\rm{i}}.\sin \frac{{180 + 360}}{3}} \right]$, Or, ${\rm{z}}_2^{\frac{1}{3}}$ = 1. Or, $\frac{1}{{{{\left( {\rm{z}} \right)}^{\rm{n}}}}}$ = z-n = (cosθ + i.sinθ)-n = cos(-n)θ + i.sin(-n)θ, Now, zn – $\frac{1}{{{{\rm{z}}^{\rm{n}}}}}$ = cosnθ + i.sinnθ – cosnθ + i.sinnθ. = - (- 1 + i$\sqrt 3 $). The sum of four consecutive powers of I is zero.In + in+1 + in+2 + in+3 = 0, n ∈ z 1. With the help of the NCERT books, students can score well in the JEE Main entrance exam. So, we can say now, i4n = 1 where n is any positive interger. A similar problem was … This is termed the algebra of complex numbers. {\rm{sin}}20\infty }}$. An imaginary number I (iota) is defined as √-1 since I = x√-1 we have i2 = –1 , 13 = –1, i4 = 1 1. Contact Us | Signing up with Facebook allows you to connect with friends and classmates already (cos90° + i.sin90°). Complex Numbers have wide verity of applications in a variety of scientific and related areas such as electromagnetism, fluid dynamics, quantum mechanics, vibration analysis, cartography and control theory. Let us have a look at the types of questions asked in the exam from this topic: Illustration 1: Let a and b be roots of the equation x2 + x + 1 = 0. Out of which, algebraic or rectangular form is one of the form. Practice JEE Main Important Topics Questions solved by our expert teachers helps to score good marks in IIT JEE Exams. When k = 1, $\sqrt {{{\rm{z}}_1}} $ = 2 $\left[ {\cos \left( {\frac{{240 + 360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{240 + 360}}{2}} \right)} \right]$, = $\sqrt 2 $$\left( {\frac{1}{2} - \frac{{{\rm{i}}\sqrt 3 }}{2}} \right)$ = $1 - {\rm{i}}\sqrt 3 $. When k = 2, Z2 = cos $\left( {\frac{{180 + 720}}{4}} \right)$ + i.sin $\left( {\frac{{180 + 720}}{4}} \right)$. Complex numbers often are denoted by the letter z or by Greek letters like a (alpha). = 2(cos 30° + i.sin30°) = $2\left( {\frac{{\sqrt 3 }}{2} + {\rm{i}}.\frac{1}{2}} \right)$ = $\sqrt 3 $ + i. ‘a’ is called the real part, and ‘b’ is called the imaginary part of the complex number. With the help of the NCERT books, students can score well in the JEE Main entrance exam. {\rm{sin}}2\theta }}$ = cos (2θ – 2θ) + i.sin(2θ – 2θ). = cos 120° + i.sin120° = $ - \frac{1}{2}$ + i.$\frac{{\sqrt 3 }}{2}$. When k = 1, $\sqrt {{{\rm{z}}_1}} $ = 2$\sqrt 2 $$\left[ {\cos \left( {\frac{{60 + 360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{60 + 360}}{2}} \right)} \right]$. Sequence and Series and Mathematical Induction. {\rm{sin}}2\theta }}{{{\rm{cos}}2\theta  + {\rm{i}}. What is the application of Complex Numbers? = cos60° + i.sin60° = $\frac{1}{2}$ + i.$\frac{{\sqrt 3 }}{2}$. Find the square roots of … Or, $\sqrt {{{\rm{z}}_{\rm{k}}}} $ = $\sqrt 4 $$\left[ {\cos \frac{{120 + {\rm{k}}.360}}{2} + {\rm{i}}.\sin \frac{{120 + {\rm{k}}.360}}{2}} \right]$, When k = 0, $\sqrt {{{\rm{z}}_0}} $ = 2 $\left[ {\cos \left( {\frac{{240 + 0}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{240 + 0}}{2}} \right)} \right]$. Yes of course, but to understand this question, let’s go into more deep of complex numbers, Consider the equation x2+1 = 0, If we try to get its solution, we would stuck at x = √(-1) so in Complex Number we assume that ​√(-1) =i or i2 =-1. Thus, we can also write z = Re(z) + i Im(z). = $\frac{1}{{\sqrt 2 }}$ + i.$\frac{1}{{\sqrt 2 }}$. 2cos45° - i.2sin45° = 2.$\frac{1}{{\sqrt 2 }}$ – i.2.$\frac{1}{{\sqrt 2 }}$ = $\sqrt 2 $ – i$\sqrt 2 $. When k = 1, Z1 ={cos$\left( {\frac{{120 + 360}}{4}} \right)$ + i.sin $\left( {\frac{{120 + 360}}{4}} \right)$}. = 64 [cos 90° + i.sin90°] = 64 [0 + i.1] = 64i. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $ - \frac{1}{1}$ = -1  then θ= 315°. Terms & Conditions | = (cos315° + i.sin315°). We then write z = x +yi or a = a +bi. That means complex numbers contains two different information included in it. Tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{{\frac{{\sqrt 3 }}{2}}}{{\frac{1}{2}}}$ = $\sqrt 3 $  then θ = 60°. Complex Numbers Class 11 solutions NCERT PDF are beneficial in several ways. The notion of complex numbers increased the solutions to a lot of problems. Check the below NCERT MCQ Questions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations with Answers Pdf free download. Also, note that i + i2 + i3 + i4 = 0 or in + i2n + i3n + i4n= 0. It is the exclusive and best Telegu education portal established by Sakshi Media Group. Pay Now | EAMCET- National Eligibility cum Entrance Test: Study material, Mock Tests, Online Tests, Practice Bits, Model tests, Experts Advise etc., number, Please choose the valid Complex numbers are defined as numbers of the form x+iy, where x and y are real numbers and i = √-1. Find every complex root of the following. Point z is 7 units in the left and 6 units upwards from the origin. Complex numbers are built on the concept of being able to define the square root of negative one. You can assign a value to a complex number in one of the following ways: 1. SPI 3103.2.1 Describe any number in the complex number system. Refer the figure to understand it pictorially. Sakshi EAMCET is provided by Sakshieducation.com. You can see the same point in the figure below. Here, x = 4, y = 4$\sqrt 3 $, r = $\sqrt {{4^2} + {{\left( {4\sqrt 3 } \right)}^2}} $ = $\sqrt {16 + 48} $ = 8. Also browse for more study materials on Mathematics here. the imaginary numbers. Complex numbers are often denoted by z. Use Coupon: CART20 and get 20% off on all online Study Material, Complete Your Registration (Step 2 of 2 ), Free webinar on Robotics (Block Chain) Learn to create a Robotic Device Using Arduino. = cos(80 – 20) + i.sin(80 – 20) = cos 60° + i.sin 60° = $\frac{1}{2}$ + i.$\frac{{\sqrt 3 }}{2}$. They will get back to you in case of doubts and clear that off in a very efficient manner. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $ - \frac{1}{{\sqrt 3 }}$ then θ= 150°. When k = 0, Z0 = 11/6 [cos 0 + i.sin0] = 1. using askIItians. This form of representation is also called as the Cartesian or algebraic form of representation. Here, z = 0, y = 1, r = $\sqrt {{0^2} + {1^2}} $ = 1, So, z = 1(cosθ + i.sinθ) = 1. r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {\frac{1}{2} + \frac{1}{2}} $ = 1. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{{ - \frac{1}{{\sqrt 2 }}}}{{\frac{1}{{\sqrt 2 }}}}$ = -1  then θ= 315°. By calling the static (Shared in Visual Basic) Complex.FromPolarCoordinatesmethod to create a complex number from its polar coordinates. = $\sqrt 2 $[cos60° + i.sin60°] = $\sqrt 2 $$\left[ {\frac{1}{2} + {\rm{i}}.\frac{{\sqrt 3 }}{2}} \right]$ = $\frac{1}{{\sqrt 2 }} + \frac{{{\rm{i}}\sqrt 3 }}{{\sqrt 2 }}$. It provides EAMCET Mock tests, Online Practice Tests, EAMCET Bit banks, EAMCET Previous Solved Model Papers, and also it gives you the experts guidance. Here, x = $ - \frac{1}{2}$, y = $\frac{{\sqrt 3 }}{2}$, r = $\sqrt {\frac{1}{4} + \frac{3}{4}} $ = 1. tanθ = $\frac{{\frac{{\sqrt 3 }}{2}}}{{ - \frac{1}{2}}}$ = $ - \sqrt 3 $ then θ = 120°. When k = 0, Z0 = 11/4 [cos 0 + i.sin0] = 1. The notion of complex numbers increased the solutions to a lot of problems. Blog | When, k = 3, Z3 = cos $\left( {\frac{{180 + 1080}}{4}} \right)$ + i.sin $\left( {\frac{{180 + 1080}}{4}} \right)$. {\rm{sin}}(\theta  + {\rm{k}}.360\} $, Or, zk = r1/3$\left\{ {\cos \frac{{\theta  + {\rm{k}}.360}}{3} + {\rm{i}}.\sin \frac{{\theta  + {\rm{k}}.360}}{3}} \right\}$, = 81/3$\left\{ {\cos \frac{{90 + {\rm{k}}.360}}{3} + {\rm{i}}.\sin \frac{{90 + {\rm{k}}.360}}{3}} \right\}$, When k = 0, Z0 = 2 [cos $\frac{{90 + 0}}{3}$ + i.sin $\frac{{90 + 0}}{3}$]. = cos315° + i.sin315° = $\frac{1}{{\sqrt 2 }} - {\rm{i}}.\frac{1}{{\sqrt 2 }}$. 1/i = – i 2. Click hereto get an answer to your question ️ For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 - 3 - 4 i| = 5 , the minimum value of |z1 - z2| is Register yourself for the free demo class from The complex number in the polar form = r(cosθ + i.sinθ) = $\sqrt 2 $$\left[ { - \frac{1}{2} - {\rm{i}}.\frac{{\sqrt 3 }}{2}} \right]$ = $ - \left( {\frac{1}{{\sqrt 2 }} + \frac{{{\rm{i}}\sqrt 3 }}{{\sqrt 2 }}} \right)$. Here, x = 1, y = 0, r = $\sqrt {1 + 0} $ = 1, So, ${\rm{z}}_{\rm{k}}^4$ = r$\{ \cos \left( {\theta  + {\rm{k}}.360} \right) + {\rm{i}}. School Tie-up | The well-structured Intermediate portal of sakshieducation.com provides study materials for Intermediate, EAMCET.Engineering and Medicine, JEE (Main), JEE (Advanced) and BITSAT. Here, x = 0, y = 8, r = $\sqrt {0 + 64} $ = 8. Find the remainder upon the division of f(z) by z2 + 1. Tanθ=${\rm{\: }}\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{{4\sqrt 3 }}{4}$ = $\sqrt 3 $ then θ = 60°. z = -7 + j6, Here since a= -7 and b = 6 and thus will be lying in the second quadrant. (a) If ω1 = ω2 then the lines are parallel. 4. By passing two Doublevalues to its constructor. i is called as Iota in Complex Numbers. = cos 60° + i.sin60° = $\frac{1}{2}$ + i. Register Now. (-1)} = - 2i. Helpful for self-study and doubt clearance. = + ∈ℂ, for some , ∈ℝ Step by step solutions. Chapters. 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