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Misc 13 Find the modulus and argument of the complex number ( 1 + 2i)/(1 β 3i) . First we solve ( 1 + 2π)/(1 β 3π) Let π§ = ( 1 + 2π)/(1 β 3π) Rationalizing the same = ( 1 + 2π)/(1 β 3π) x ( 1 + 3π)/(1 + 3π) = ((1 + 2π) ( 1 + 3π))/((1 β 3π) ( 1 + 3π)) = (1 (1 + 3π) + 2π (1 + 3π))/((1 β 3π) (1 + 3π) ) = ( 1 + 3π+ 2π+ 6π2)/((1 β 3π) (1 + 3π) ) Using ( a β b ) ( a + b ) = a2 - b2 = ( 1+ 5π+ 6π2)/((1)2 β (3π)2) = (1 + 5π+ 6π2)/(1 β 9π2) Putting π2 = - 1 = (1 + 5π + 6 (β1))/(1 β 9 (β1)) = (1 + 5π β 6 )/(1 + 9 ) = ( \β 5 + 5π)/10 = ( 5 (β1 +π))/10 = (β 1+ π)/2 = (β 1)/2+π ( 1/2 ) Thus, π§ = (β 1)/2+π ( 1/2 ) Method 1 To find Modulus Now we have z = (β 1)/2 + π (1/2) Complex number z is of the form x + π y Here x = (β 1)/2 and y = 1/2 Modulus of z = |z| = β(π₯^2+π¦2) = β(( (β 1)/(2 ))^2+( 1/(2 ))^2 ) = β(1/4+1/4) = β(2/4) = β(1/2) = 1/β2 Modulus of z = 1/β2 Method 2 To calculate modulus of z Given z = (β1)/2 + π (1/2) Let π§=π (cosβ‘ΞΈ+π sinΞΈ) Here r is modulus, and ΞΈ is argument Form (1) and (2) (β1)/2 + π (1/2) = π (cosβ‘ΞΈ+π sinΞΈ) (β1)/2 + π (1/2) = r cos ΞΈ + πr sin ΞΈ Comparing real part (β 1)/2 = r cos ΞΈ Adding (3) and (4) 1/4 + 1/4 = r2 cos2 ΞΈ + r2 sin2 ΞΈ (1+1)/4 = r2 (cos2 ΞΈ+ sin2 ΞΈ) 2/4 = r2 Γ 1 1/2 = r2 1/β2 = r Modulus of z = 1/β2 Finding argument (β1)/2 + π (1/2) = r cos ΞΈ + πr sin ΞΈ Hence, sin ΞΈ = 1/β2 & cos ΞΈ = (β 1)/β2 Here, sin ΞΈ is positive and cos ΞΈ is negative, Hence, ΞΈ lies in IVth quadrant Argument = 180Β° β 45Β° = 135Β° = 135Β° Γ π/180o = ( 3 π)/4 So argument of z = ( 3 π)/4

Miscellaneous

Misc 1
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Misc 2

Misc 3

Misc 4 Important

Misc 5 (i) Deleted for CBSE Board 2022 Exams

Misc 5 (ii) Important

Misc 6

Misc 7

Misc 8 Important

Misc 9

Misc 10 Important

Misc 11

Misc 12

Misc 13 Important Deleted for CBSE Board 2022 Exams You are here

Misc 14

Misc 15 Important

Misc 16

Misc 17 Important

Misc 18 Important

Misc 19

Misc 20 Important

Chapter 5 Class 11 Complex Numbers (Term 1)

Serial order wise

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.