1. Chapter 5 Class 11 Complex Numbers
2. Serial order wise

Transcript

Misc 12 Let z1 = 2 โ i, z2 = -2 + i . Find Re ((๐ง_1 ๐ง_2)/(๐ง_1 )ย ฬ ) We need to find Re ((๐ง_1 ๐ง_2)/(๐ง_1 )ย ฬ ) i.e.Real part of ((๐ง_1 ๐ง_2)/(๐ง_1 )ย ฬ ) Lets first calculate ((๐ง_1 ๐ง_2)/(๐ง_1 )ย ฬ ) z1 = 2 โ ๐ z2 = โ2 + ๐ ("z1" )ย ฬ = 2 + ๐ ((๐ง_1 ๐ง_2)/(๐ง_1 )ย ฬ ) = ((2 โ ๐) (โ2 + ๐))/(2 + ๐) = (2(โ2 + ๐) โ๐ (โ2 + ๐))/(2 + ๐) = (2 ร (โ 2) + 2 ร ๐ + (โ ๐) ร (โ 2) + (โ ๐) ร ๐)/(2 + ๐) = (โ4 + 2๐ + 2๐ โ ๐2)/(2 + ๐) Putting i2 = โ 1 = (โ4 + 2๐ + 2๐ โ (โ1))/(2 + ๐) = (โ4 + 2๐ + 2๐ + 1)/(2 + ๐) = (โ 4 + 1 + 2๐ + 2๐ )/(2 + ๐) = (โ3 + 4๐)/(2 + ๐) Rationalizing = (โ 3 + 4๐)/(2 + ๐) ร (2 โ ๐)/(2 โ ๐) = (( โ3 + 4๐) ( 2 โ ๐ ))/(( 2 + ๐ ) ( 2 โ ๐ )) = (โ 3 ( 2 โ ๐ ) + 4๐ ( 2 โ ๐ ))/(( 2 + ๐ ) ( 2 โ ๐ )) = (โ 3 ร 2 + (โ3) ร (โ๐) + 4๐ ร 2 + 4๐ ร (โ๐))/(( 2 + ๐ ) ( 2 โ ๐ )) = (โ 6 + 3๐ + 8๐ โ 4๐2)/(( 2 + ๐ ) ( 2 โ ๐ )) Using (a+b)(a-b) = a2 โ b2 = (โ 6 + 3๐ + 8๐ โ 4๐2)/(22 โ ๐2) Putting i2 = - 1 = (โ 6 + 3๐ + 8๐ โ 4 ( โ1))/(4 โ (โ1) ) = (โ 6 + 3๐ + 8๐ + 4)/(4 + 1) = (โ 6 + 4 + 3๐ + 8๐ )/5 = (โ 2 +11๐)/5 = (โ 2)/5 + ๐ 11/5 โ ((๐ง_1 ๐ง_2)/(๐ง_1 )ย ฬ ) = (โ 2)/5 + ๐ 11/5 โ Re ((z1 z2)/("z1" )ย ฬ ) = (โ2)/5 Misc 12 Let ๐ง1 = 2 โ ๐, ๐ง2 = โ 2 + ๐ . Find (ii) Im (1/(๐ง_1 (๐ง_1 )ย ฬ )) We need to find Im (1/(๐ง_1 (๐ง_1 )ย ฬ )) i.e. imaginary part of (1/(๐ง_1 (๐ง_1 )ย ฬ )) Lets first calculate (1/(๐ง_1 (๐ง_1 )ย ฬ )) ๐ง1 = 2 โ ๐ ("z1" )ย ฬ = 2 + ๐ 1/(๐ง_1 (๐ง_1 )ย ฬ ) = 1/(( 2 โ ๐ ) ( 2 + ๐ )) Using ( a + b ) ( a โ b ) = a2 โ b2 = 1/((2)2 โ (๐)2) = 1/(4 โ ( โ1) ) = 1/(4+1) = 1/5 = 1/5 + 0 = 1/5 + 0๐ So, imaginary part is 0 Hence, Im (1/(๐ง_1 (๐ง_1 )ย ฬ )) = 0