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MHT CET202311 May 2023Morning ShiftMathematicsComplex NumberActual

If z =x+ i y and z ^ 1 / 3 = p + iq , where x, y, p , q R and i = -1 , then value of ( x p + y q ) is

Options

  1. Ap^2-q^2
  2. B4 (p^2-q^2 )
  3. Cp ^2+ q ^2
  4. D4 (p^2+q^2 )

Correct answer

B. 4 (p^2-q^2 )

Step-by-step solution

aligned & z ^ 1 3 = p + iq & z =( p + iq )^3 & x+ i y= p ^3+3 p ^2 qi +3 p ( iq )^2+( iq )^3 & x+ i y= ( p ^3-3 pq ^2 )+ (3 p ^2 q - q ^3 ) i & x= p ^3-3 pq ^2 and y=3 p ^2 q - q ^3 & & x p = p ^2-3 q ^2 and y q =3 p ^2- q ^2 & ( x p + y q )=4 p ^2-4 q ^2=4 ( p ^2- q ^2 ) aligned

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