Misc 12 - Let z1 = 2 - i, z2 = -2 + i. Find (i) Re (z1 z2) - Miscellaneous

Misc 12  - Chapter 5 Class 11 Complex Numbers - Part 2
Misc 12  - Chapter 5 Class 11 Complex Numbers - Part 3 Misc 12  - Chapter 5 Class 11 Complex Numbers - Part 4

Misc 12  - Chapter 5 Class 11 Complex Numbers - Part 5 Misc 12  - Chapter 5 Class 11 Complex Numbers - Part 6

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Misc 12 Let z1 = 2 i, z2 = -2 + i . Find Re (( _1 _2)/( _1 ) ) We need to find Re (( _1 _2)/( _1 ) ) i.e.Real part of (( _1 _2)/( _1 ) ) Lets first calculate (( _1 _2)/( _1 ) ) z1 = 2 z2 = 2 + ("z1" ) = 2 + (( _1 _2)/( _1 ) ) = ((2 ) ( 2 + ))/(2 + ) = (2( 2 + ) ( 2 + ))/(2 + ) = (2 ( 2) + 2 + ( ) ( 2) + ( ) )/(2 + ) = ( 4 + 2 + 2 2)/(2 + ) Putting i2 = 1 = ( 4 + 2 + 2 ( 1))/(2 + ) = ( 4 + 2 + 2 + 1)/(2 + ) = ( 4 + 1 + 2 + 2 )/(2 + ) = ( 3 + 4 )/(2 + ) Rationalizing = ( 3 + 4 )/(2 + ) (2 )/(2 ) = (( 3 + 4 ) ( 2 ))/(( 2 + ) ( 2 )) = ( 3 ( 2 ) + 4 ( 2 ))/(( 2 + ) ( 2 )) = ( 3 2 + ( 3) ( ) + 4 2 + 4 ( ))/(( 2 + ) ( 2 )) = ( 6 + 3 + 8 4 2)/(( 2 + ) ( 2 )) Using (a+b)(a-b) = a2 b2 = ( 6 + 3 + 8 4 2)/(22 2) Putting i2 = - 1 = ( 6 + 3 + 8 4 ( 1))/(4 ( 1) ) = ( 6 + 3 + 8 + 4)/(4 + 1) = ( 6 + 4 + 3 + 8 )/5 = ( 2 +11 )/5 = ( 2)/5 + 11/5 (( _1 _2)/( _1 ) ) = ( 2)/5 + 11/5 Re ((z1 z2)/("z1" ) ) = ( 2)/5 Misc 12 Let 1 = 2 , 2 = 2 + . Find (ii) Im (1/( _1 ( _1 ) )) We need to find Im (1/( _1 ( _1 ) )) i.e. imaginary part of (1/( _1 ( _1 ) )) Lets first calculate (1/( _1 ( _1 ) )) 1 = 2 ("z1" ) = 2 + 1/( _1 ( _1 ) ) = 1/(( 2 ) ( 2 + )) Using ( a + b ) ( a b ) = a2 b2 = 1/((2)2 ( )2) = 1/(4 ( 1) ) = 1/(4+1) = 1/5 = 1/5 + 0 = 1/5 + 0 So, imaginary part is 0 Hence, Im (1/( _1 ( _1 ) )) = 0

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.