Complex Numbers And Quadratic Equations

Question
CBSEENMA11015658

If z = x – i y and z1/3 = p+ iq , then fraction numerator begin display style open parentheses straight x over straight p plus straight y over straight q close parentheses end style over denominator left parenthesis straight p squared plus straight q squared right parenthesis end fraction is equal to 

  • 1

  • -2

  • 2

  • -1

Solution

B.

-2

D.

-1

straight z to the power of 1 divided by 3 end exponent space equals space straight p space plus space iq
left parenthesis straight x minus iy right parenthesis to the power of 1 divided by 3 end exponent space equals space left parenthesis straight p plus iq right parenthesis      therefore,(Qz = x − iy)
(x - iy) = (p + iq)3
⇒ (x - iy) = p3 +(iq)3 + 3p2qi + 3pq2i2
⇒ (x - iy) = p3 - iq3 + 3p2qi - 3pq2
⇒ (x - iy) = (p3 - 3pq2 ) + i (3p2 q - q3 ) On comparing both sides, we get
⇒ x = (p3 - 3pq2) and - y = 3p2 q - q3
⇒ x = p(p2 - 3q2 ) and y = q(q2 - 3p2 )

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