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NCERT Class XI Mathematics - Complex Numbers and Quadratic Equations - Solutions

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Question : 35 of 52
Marks: +1, -0
Reduce (11−4i−21+i)\left(\frac{1}{1-4i} - \frac{2}{1+i}\right) to the standard form.
Solution:  
We have , (11−4i−21+i)\left(\frac{1}{1-4i} - \frac{2}{1+i}\right) = 1+i−2(1−4i)(1−4i)(1+i)\frac{1+i-2(1-4i)}{(1-4i)(1+i)}
= 1+i−2+8i1+i−4i+4\frac{1+i-2+8i}{1+i-4i+4} = −1+9i5−3i\frac{-1+9i}{5-3i}
⇒ (11−4i−21+i)(3−4i5+i)\left(\frac{1}{1-4i} - \frac{2}{1+i}\right) \left(\frac{3-4i}{5+i}\right) = (−1+9i5−3i)(3−4i5+i)\left(\frac{-1+9i}{5-3i}\right) \left(\frac{3-4i}{5+i}\right)
= −3+4i+27i+3625+5i−15i+3\frac{-3+4i+27i+36}{25+5i-15i+3} = 33+31i28−10i×28+10i28+10i\frac{33+31i}{28-10i} \times \frac{28+10i}{28+10i} = 924+330i+868i−310784+100\frac{924+330i+868i-310}{784+100}
= 614+1198i884\frac{614+1198i}{884} = 614884\frac{614}{884} + 1198884\frac{1198}{884} i = 307442+599442\frac{307}{442} + \frac{599}{442} i
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