For example, 3+2i, -2+i√3 are complex numbers. = (cos315° + i.sin315°). 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 = 0, Z0 = 11/4 [cos 0 + i.sin0] = 1. Now consider a point in the second quadrant that is. Question 1. ..... (2). So, we can say now, i4n = 1 where n is any positive interger. Dear You can see the same point in the figure below. = cos90° + i.sin90°. Solving (6) and (7), we have b = ½ + i and a = i/2. Helpful for self-study and doubt clearance. = {cos 120° + i.sin120°} = $ - \frac{1}{2} + {\rm{i\: }}.\frac{{\sqrt 3 }}{2}$, When k = 2, Z2 = cos $\left( {\frac{{120 + 720}}{4}} \right)$ + i.sin $\left( {\frac{{120 + 720}}{4}} \right)$, = cos 210° + i.sin210° = $ - \frac{{\sqrt 3 }}{2} - {\rm{i}}.\frac{1}{2}$, When k = 3, Z3 = cos $\left( {\frac{{120 + 1080}}{4}} \right)$ + i.sin $\left( {\frac{{120 + 1080}}{4}} \right)$, = cos 300° + i.sin300° = $\frac{1}{2} - {\rm{i}}.\frac{{\sqrt 3 }}{2}$. Now let’s consider a point in the third quadrant as z = -2 – j3. Hence the required equation is x2 + x + 1 = 0. 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. 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. So, required roots are ± ($\sqrt 6 $ + i.$\sqrt 2 $). Illustration 2: Dividing f(z) by z - i, we obtain the remainder i and dividing it by z + i, we get remainder 1 + i. When, k = 3, Z3 = cos $\left( {\frac{{180 + 1080}}{4}} \right)$ + i.sin $\left( {\frac{{180 + 1080}}{4}} \right)$. When k = 2, Z2 = cos $\left( {\frac{{0 + 720}}{4}} \right)$ + i.sin $\left( {\frac{{0 + 720}}{4}} \right)$, When, k = 3, Z3 = cos $\left( {\frac{{0 + 1080}}{4}} \right)$ + i.sin $\left( {\frac{{0 + 1080}}{4}} \right)$, So, ${\rm{z}}_{\rm{k}}^6$ = r$\{ \cos \left( {\theta + {\rm{k}}.360} \right) + {\rm{i}}. Privacy Policy | 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). = $\frac{{\left( {{\rm{cos}}3\theta + {\rm{i}}. ‘a’ is called the real part, and ‘b’ is called the imaginary part of the complex number. Or, 3 $\left( {\frac{1}{2} + \frac{{{\rm{i}}\sqrt 3 }}{2}} \right)$ = $\frac{3}{2}$ + $\frac{{{\rm{i}}3\sqrt 3 }}{2}$. = 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. 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}$. Horizontal axis represents real part while the vertical axis represents imaginary part. = cos 300° + i.sin300° = $\frac{1}{2}$ - i.$\frac{{\sqrt 3 }}{2}$ = $\frac{{1 - {\rm{i}}\sqrt 3 }}{2}$. = $\sqrt 2 $$\left( { - \frac{1}{{\sqrt 2 }} - \frac{{\rm{i}}}{{\sqrt 2 }}} \right)$ = - 1 – i = - (1 + i). 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. Hence, the required remainder = az + b = ½ iz + ½ + i. 1. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $ - \frac{1}{{\sqrt 3 }}$ then θ= 150°. By a… Or, 3 $\left( { - \frac{1}{2} + \frac{{{\rm{i}}\sqrt 3 }}{2}} \right)$ = $ - \frac{3}{2}$ + $\frac{{{\rm{i}}3\sqrt 3 }}{2}$. It is the exclusive and best Telegu education portal established by Sakshi Media Group. Since both a and b are positive, which means number will be lying in the first quadrant. Click here for the Detailed Syllabus of IIT JEE Mathematics. It is defined as the combination of real part and imaginary part. r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {3 + 1} $ = 2. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{1}{{\sqrt 3 }}$ = 1, then θ= 30°. 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? Here, x = 4, y = 4$\sqrt 3 $, r = $\sqrt {{4^2} + {{\left( {4\sqrt 3 } \right)}^2}} $ = $\sqrt {16 + 48} $ = 8. 6. (7). Back to Solutions. Here, x = 0, y = 2, r = $\sqrt {0 + 4} $ = 2. But first equality of complex numbers must be defined. A similar problem was … ‘z’ will be 6 units in the right and 4 units upwards from the origin. = $\frac{1}{{\sqrt 2 }}$ + i.$\frac{1}{{\sqrt 2 }}$. Complex Numbers Class 11 solutions NCERT PDF are beneficial in several ways. Here, x = 0, y = 8, r = $\sqrt {0 + 64} $ = 8. {\rm{sin}}\theta } \right)}}{{{{\left( {{\rm{cos}}\theta + {\rm{i}}. basically the combination of a real number and an imaginary number = cos60° + i.sin60° = $\frac{1}{2}$ + i.$\frac{{\sqrt 3 }}{2}$. This form of representation is also called as the Cartesian or algebraic form of representation. Sitemap | 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. Extraction of square root of complex number. 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. = $\sqrt 2 $[cos60° + i.sin60°] = $\sqrt 2 $$\left[ {\frac{1}{2} + {\rm{i}}.\frac{{\sqrt 3 }}{2}} \right]$ = $\frac{1}{{\sqrt 2 }} + \frac{{{\rm{i}}\sqrt 3 }}{{\sqrt 2 }}$. Refer the figure to understand it pictorially. √b = √ab is valid only when atleast one of a and b is non negative. Example: Having introduced a complex number, the ways in which they can be combined, i.e. using askIItians. For a complex number z = x+iy, x is called the real part, denoted by Re z and y is called the imaginary part denoted by Im z. (-1)} = - 2i. = 2{cos 120° + i.sin120°} = 2.$\left( { - \frac{1}{2} + {\rm{i}}.\frac{{\sqrt 3 }}{2}} \right)$ = $ - {\rm{\: }}1{\rm{\: }}$+ i$\sqrt 3 $. = 2$\sqrt 2 $[cos30 + i.sin30] = 2$\sqrt 2 $$\left[ {\frac{{\sqrt 3 }}{2} + {\rm{i}}.\frac{1}{2}} \right]$ = $\sqrt 6 $ + i.$\sqrt 2 $. Or, zk = r1/4$\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 $\frac{{180 + 0}}{4}$]. (c) 0 if r is not a multiple of 3 and 3 if r is a multiple of 3. ir = ir 1. Class 8; Class 9; Class 10; Grade 11; Grade 12; Grade 11 Mathematics Solution. 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°. So, required roots are ± $\left( {\frac{1}{{\sqrt 2 }} + \frac{{{\rm{i}}\sqrt 3 }}{{\sqrt 2 }}} \right)$ = ± $\frac{1}{{\sqrt 2 }}$ (1 + i$\sqrt 3 $). $\left[ {\cos \frac{{180 + 720}}{3} + {\rm{i}}.\sin \frac{{180 + 720}}{3}} \right]$. {\rm{sin}}80\infty }}{{{\rm{cos}}20\infty + {\rm{i}}. For example: x = (2+3i) (3+4i), In this example, x is a multiple of two complex numbers. In addition to this, if a student faces any doubts concerning CH 2 Maths Class 12, he or she can go to the website and drop in their queries and download NCERT Book Solution for Class 12 Maths Chapter 2 PDF version. How do we locate any Complex Number on the plane? If z is purely real negative complex number then. ‘i’ (or ‘j’ in some books) in math is used to denote the imaginary part of any complex number. Look into the Previous Year Papers with Solutions to get a hint of the kinds of questions asked in the exam. Pay Now | When k = 2, Z2 = cos $\left( {\frac{{90 + 720}}{3}} \right)$ + i.sin $\left( {\frac{{90 + 720}}{3}} \right)$. 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 Solved and explained by expert mathematicians. = cos 60° + i.sin60° = $\frac{1}{2}$ + i.$\frac{{\sqrt 3 }}{2}$ = $\frac{1}{2}$(1 + i$\sqrt 3 $). Get Important questions for class 11 Maths Chapter 5 – Complex Numbers and Quadratic Equations here.Students can get different types of questions covered in this chapter. Practice JEE Main Important Topics Questions solved by our expert teachers helps to score good marks in IIT JEE Exams. Here x =$\frac{1}{{\sqrt 2 }}$, y = $ - \frac{1}{{\sqrt 2 }}$. Here x =$\frac{1}{2}$, y = $\frac{1}{2}$. One of our academic counsellors will contact you within 1 working day. = $\frac{1}{{\sqrt 2 }}$ (cos45° + i.sin45°). When k = 2, Z2 = cos $\left( {\frac{{0 + 720}}{6}} \right)$ + i.sin $\left( {\frac{{0 + 720}}{6}} \right)$. Here, z = - 2, y = - 2, r = $\sqrt {4 + 12} $ = 4. What is the application of Complex Numbers? Complex Numbers are the numbers which along with the real part also has the imaginary part included with it. = $\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 you can read Chapter 5 of Class 11 Maths NCERT Book. {\rm{sin}}2\theta }}$ = cos (2θ – 2θ) + i.sin(2θ – 2θ). Main application of complex numbers is in the field of electronics. So, ${\rm{z}}_{\rm{k}}^3$ = r$\{ \cos \left( {\theta + {\rm{k}}.360} \right) + {\rm{i}}. Digital NCERT Books Class 11 Maths pdf are always handy to use when you do not have access to physical copy. An imaginary number I (iota) is defined as √-1 since I = x√-1 we have i2 = –1 , 13 = –1, i4 = 1 1. It helps us to clearly distinguish the real and imaginary part of any complex number. Tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{{\frac{{\sqrt 3 }}{2}}}{{\frac{1}{2}}}$ = $\sqrt 3 $ then θ = 60°. Students can also make the best out of its features such as Job Alerts and Latest Updates. When k = 1, Z1 = cos $\left( {\frac{{180 + 360}}{4}} \right)$ + i.sin $\left( {\frac{{180 + 360}}{4}} \right)$. {\rm{sin}}3\theta } \right)\left( {{\rm{cos}}\theta - {\rm{i}}. Let z4 = $ - \frac{1}{2}$ + $\frac{{{\rm{i}}\sqrt 3 }}{2}$. CBSE Worksheets for Class 11 Maths: One of the best teaching strategies employed in most classrooms today is Worksheets. 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}$]. It will help you to save your precious time just before the examination. Ltd. Trigonometric Equations and General Values. Register yourself for the free demo class from √a . $\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. They will get back to you in case of doubts and clear that off in a very efficient manner. Here, x = 1, y = 1, r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {1 + 1} $ = $\sqrt 2 $. Complex numbers often are denoted by the letter z or by Greek letters like a (alpha). The complex number in the polar form = r(cosθ + i.sinθ). The notion of complex numbers increased the solutions to a lot of problems. Let us take few examples to understand that, how can we locate any point on complex or argand plane? name, Please Enter the valid The well-structured Intermediate portal of sakshieducation.com provides study materials for Intermediate, EAMCET.Engineering and Medicine, JEE (Main), JEE (Advanced) and BITSAT. = + ∈ℂ, for some , ∈ℝ The set of all the complex numbers are generally represented by ‘C’. 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]$. subject, comprising study notes, revision notes, video lectures, previous year solved questions etc. Q5. the imaginary numbers. That means complex numbers contains two different information included in it. Here, x = 0, y = 1, r = $\sqrt {0 + 1} $ = 1. You can get the knowledge of Recommended Books of Mathematics here. {\rm{sin}}\theta } \right)}^2}}}$, = $\frac{{{\rm{cos}}2\theta + {\rm{i}}. addition, multiplication, division etc., need to be defined. Again, ${\rm{\bar z}}$ = r(cosθ – i.sinθ) = r[cos (2π – θ) + i.sin(2π – θ)], So, Arg $\left( {{\rm{\bar z}}} \right)$ = 2π – θ = 2π – Arg (z). Since in third quadrant both a and b are negative and thus a = -2 and b = -3 in our example. These NCERT Solutions of Maths help the students in solving the problems quickly, accurately and efficiently. It provides EAMCET Mock tests, Online Practice Tests, EAMCET Bit banks, EAMCET Previous Solved Model Papers, and also it gives you the experts guidance. 6. (cos60° + i.sin60°), Z7 = [cos60° + i.sin60°]7 = cos (60 * 7) + i.sin(60 * 7). In electronics, already the letter ‘i’ is reserved for current and thus they started using ‘j’ in place of i for the imaginary part. MCQ Questions for Class 11 Maths with Answers were prepared based on the latest exam pattern. askiitians. A complex number is usually denoted by the letter ‘z’. = cos(80 – 20) + i.sin(80 – 20) = cos 60° + i.sin 60° = $\frac{1}{2}$ + i.$\frac{{\sqrt 3 }}{2}$. (2), (-2y + 6)2 + y2 – 6 (-2y + 6) + 2y + 1 = 0, Q1. 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. grade, Please choose the valid Tanθ= $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{{\sqrt 3 }}{{ - 1}}$ = $ - \sqrt 3 $ then θ = 120°. It provides the information on AP EAMCET and TS EAMCET Notifications, and EAMCET Counselling. Let now take the fourth (of fourth quadrant) and the last case where z = 5 – j6. 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. Find important formulae and previous year questions related to Complex Numbers for JEE Main and JEE Advanced 2019. {\rm{sin}}20\infty }}$. If ω1 = ω2 are the complex slopes of two lines, then. Also after the chapter, you can get links to Class 11 Maths Notes, NCERT Solutions, Important Question, Practice Papers, etc. When k = 1, $\sqrt {{{\rm{z}}_1}} $ = 2$\sqrt 2 $$\left[ {\cos \left( {\frac{{60 + 360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{60 + 360}}{2}} \right)} \right]$. Are all Real Numbers are Complex Numbers? a positive and b negative. So, required roots are ± $\frac{1}{{\sqrt 2 }}$(1 + i), ± $\frac{1}{{\sqrt 2 }}$(1 – i). Complex numbers often are denoted by the letter z or by Greek letters like a (alpha). Then f(z) = g(z) (z2 + 1) + az + b ..... (3), So, f(i) = g(i) (i2 + 1) + ai + b = ai + b .… (4), and f(-i) = g(-i) (i2 + 1) – ai + b = -ai + b .… (5), From (1) and (4), we have b + ai = i .… (6), from (2) and (5) we have b – ai = 1 + i …. Two mutually perpendicular axes are used to locate any complex point on the plane. If ‘a’ is the real part and ‘b’ represents imaginary part, then complex number is represented as z = a + ib where i, stands for iota which itself is a square root of negative unity. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{1}{{ - 1}}$ = - 1 then θ= 135°, Z14 = [$\sqrt 2 $(cos 135° + i.sin135°)]14, = ${\left( {\sqrt 2 } \right)^{14}}$ [cos(135 * 14) + i.sin (135 * 14)], = 27 [cos(90 * 21 + 0) + i.sin(90 * 21 + 0)], = 27 [sin0 + i.cos0] = 27 [0 + i.1] = 27.i, Let z = $\left( {\frac{1}{2} + {\rm{i}}.\frac{{\sqrt 3 }}{2}} \right)$, Here, x = $\frac{1}{2}$, y = $\frac{{\sqrt 3 }}{2}$, r = $\sqrt {{{\left( {\frac{1}{2}} \right)}^2} + {{\left( {\frac{{\sqrt 3 }}{2}} \right)}^2}} $ = 1. All the examples listed here are in Cartesian form. $\left[ {\cos \frac{{180 + 0}}{3} + {\rm{i}}.\sin \frac{{180 + 0}}{3}} \right]$. which means i can be assumed as the solution of this equation. Complex numbers are built on the concept of being able to define the square root of negative one. = cos 225° + i.sin225° = $ - \frac{1}{2}$ + i.$\frac{1}{2}$. Similarly, the remainder when f(z) is divided by (z + i) = f(- i) ….. (1), and f( -i) = 1 + i. When k = 1, $\sqrt {{{\rm{z}}_1}} $ = $\sqrt 2 $$\left[ {\cos \left( {\frac{{120 + 360}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{120 + 360}}{2}} \right)} \right]$. This is termed the algebra of complex numbers. = $\sqrt 2 ${cos 45° + i.sin45°} = $\sqrt 2 $.$\left( {\frac{1}{{\sqrt 2 }} + {\rm{i}}.\frac{1}{{\sqrt 2 }}} \right)$ = 1 + i. 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. The complex number in the polar form = r(cosθ + i.sinθ) Grade 12; PRACTICE. = cos 60° + i.sin60° = $\frac{1}{2}$ + i. CBSE Class 11 Maths Worksheet for students has been used by teachers & students to develop logical, lingual, analytical, and problem-solving capabilities. Here, z = -1, y = 0, r = $\sqrt {1 + 0} $ = 1. tanθ = $\frac{0}{{ - 1}}$ = 0 then θ= 180°. Or, $\sqrt {{{\rm{z}}_{\rm{k}}}} $ = $\sqrt 2 $$\left[ {\cos \frac{{120 + {\rm{k}}.360}}{2} + {\rm{i}}.\sin \frac{{120 + {\rm{k}}.360}}{2}} \right]$, When k = 0, $\sqrt {{{\rm{z}}_0}} $ = $\sqrt 2 $$\left[ {\cos \left( {\frac{{120 + 0}}{2}} \right) + {\rm{i}}.\sin \left( {\frac{{120 + 0}}{2}} \right)} \right]$. Contact Us | = cos300° + i.sin300° = $\frac{1}{2} - {\rm{i}}.\frac{{\sqrt 3 }}{2}$. z = -7 + j6, Here since a= -7 and b = 6 and thus will be lying in the second quadrant. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{1}{1}$ = 1 then θ= 45°. r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {1 + 1} $ = $\sqrt 2 $. 1. a. Soln: Or, (2 + 5i) + (1 + i) = 2 + 5i + 1 – i = 3 + 4i. 2cos45° - i.2sin45° = 2.$\frac{1}{{\sqrt 2 }}$ – i.2.$\frac{1}{{\sqrt 2 }}$ = $\sqrt 2 $ – i$\sqrt 2 $. You will see that, in general, you proceed as in real numbers, but using i 2 =−1 where appropriate. Sequence and Series and Mathematical Induction. Franchisee | We then write z = x +yi or a = a +bi. tanθ = $\frac{{\rm{y}}}{{\rm{x}}}$ = $\frac{{\frac{1}{2}}}{{\frac{1}{2}}}$ = 1 then θ= 45°. The real part of the complex number is represented by x, and the imaginary part of the complex number is represented by y. 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. r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {{2^2} + {2^2}} $ = $\sqrt {4 + 4} $ = 2$\sqrt 2 $. Home ; Grade 11 ; Mathematics ; Sukunda Pustak Bhawan ; Complex Number. Tanθ = $\frac{0}{{ - 1}}$ then θ = 180°. = (cos 30° + i.sin30°) = $\frac{{\sqrt 3 }}{2} + {\rm{i}}.\frac{1}{2}$. Or, $\sqrt {\frac{{1 - {\rm{i}}}}{{1 + {\rm{i}}}}} $ = $\sqrt {\frac{{1 - {\rm{i}}}}{{1 + {\rm{i}}}}{\rm{*}}\frac{{1 - {\rm{i}}}}{{1 - {\rm{i}}}}} $ = $\frac{{1 - {\rm{i}}}}{{\sqrt {1 - {{\rm{i}}^2}} }}$ = $\frac{{1 - {\rm{i}}}}{{\sqrt {1 + 1} }}$ = $\frac{1}{{\sqrt 2 }} - \frac{{\rm{i}}}{{\sqrt 2 }}$. Signing up with Facebook allows you to connect with friends and classmates already Chapter List. Prepared by the best teachers with decades of experience, these are the latest Class 11 Maths solutions that you will find. = 2$\sqrt 2 $$\left[ { - \frac{{\sqrt 3 }}{2} - {\rm{i}}.\frac{1}{3}} \right]$ = $ - \left( {\sqrt 6 + {\rm{i}}.\sqrt 2 } \right)$. Also, note that i + i2 + i3 + i4 = 0 or in + i2n + i3n + i4n= 0. Here, x = -1, y = 0, r = $\sqrt {1 + 0} $ = 1. Complex Number can be considered as the super-set of all the other different types of number. Find the square roots of … Register Now. Moreover, i is just not to distinguish but also has got some value. It’s an easier way as well. Consider a complex number z = 6 +j4 (‘i’ and ‘j’, both can be used for representing imaginary part), if we compare this number with z = a + jb form. {\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 $\frac{{90 + 0}}{3}$]. If z = -2 + j4, then Re(z) = -2 and Im(z) = 4. ©Copyright 2014 - 2021 Khulla Kitab Edutech Pvt. School Tie-up | Register online for Maths tuition on Vedantu.com to … Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.5 Additional Problems. Hello friends Complex Numbers Class 11th | Exercise 1.1 Q.24 | Part - 8This videos based on complex numbers class 11th Maharashtra Board new syllabus. {\rm{sin}}2\theta }}{{{\rm{cos}}2\theta + {\rm{i}}. (c) If ω1 = ω2 then the lines are not parallel. Terms & Conditions | Complex number has two parts, real part and the imaginary part. (a) If ω1 = ω2 then the lines are parallel. Solution: (i) Question 2. RD Sharma Solutions | Can we take the square-root of a negative number? and if a = 0, z = ib which is called as the Purely Imaginary Number. The imaginary part, therefore, is a real number! 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°. When k = 1, Z1 = cos $\left( {\frac{{0 + 360}}{4}} \right)$ + i.sin $\left( {\frac{{0 + 360}}{4}} \right)$. All the examples listed here are in Cartesian form. = cos(18 * 5) + i.sin(18 * 5) = cos90° + i.sin90° = 0 + i.1 = i, = cos(9 * 40) + i.sin(9 * 40) = cos 360° + i.sin360° = 1 + i.0 = 1. 74 EXEMPLAR PROBLEMS – MATHEMATICS 5.1.3 Complex numbers (a) A number which can be written in the form a + ib, where a, b are real numbers and i = −1 is called a complex number . 2 + i3, -5 + 6i, 23i, (2-3i), (12-i1), 3i are some of the examples of complex numbers. Sakshi EAMCET is provided by Sakshieducation.com. Complex numbers are defined as numbers of the form x+iy, where x and y are real numbers and i = √-1. This point will be lying 2 units in the left and 3 units downwards from the origin. A complex number is of the form i 2 =-1. 1/i = – i 2. With the help of the NCERT books, students can score well in the JEE Main entrance exam. 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) r = $\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} $ = $\sqrt {\frac{1}{4} + \frac{1}{4}} $ = $\frac{1}{{\sqrt 2 }}$. Remarks. 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. When k = 1, Z1 ={cos$\left( {\frac{{120 + 360}}{4}} \right)$ + i.sin $\left( {\frac{{120 + 360}}{4}} \right)$}. These values represent the position of the complex number in the two-dimensional Cartesian coordinate system. Then we can easily equate the two and get a = 6 and b = 4. {\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}}}$, = $\frac{{\cos \left( {3\theta - \theta } \right) + {\rm{i}}.\sin \left( {3\theta - \theta } \right)}}{{{{\left( {{\rm{cos}}\theta + {\rm{i}}. {\rm{sin}}\theta } \right)}^2}}}$, = $\frac{{\left( {{\rm{cos}}3\theta + {\rm{i}}. the imaginary numbers. = 2 {cos 270° + i.sin270°} = 2{0 + i. Point z is 7 units in the left and 6 units upwards from the origin. Updated to latest CBSE syllabus. This point will be lying 5 units in the right and 6 units downwards. 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. $\left( {{\rm{\bar z}}} \right)$ = 2π – Arg(z). Tanθ = $ - \frac{{2\sqrt 3 }}{{ - 2}}$ = $\sqrt 3 $ then θ = 240°. 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