QJEEM04Ma72d8: JEE Main 2004 Mathematics - Class 11 JEEM04Ma72d8 Solution: If and , then is equal to :

Question

Verified Solution • 2025–26 Session moderate

If

z = x - iy

and

{z^{{1 \over 3}}} = p + iq

, then

{{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}}

is equal to :

Accepted Answer

Given

{z^{{1 \over 3}}} = p + iq

\Rightarrow

z = (p + iq)³

= p³ + (iq)³ +3p(iq)(p + iq)

= p³ - iq³ +3ip²q - 3pq²

= p(p² - 3q²) - iq(q² - 3p²)

Given that

z = x - iy

\therefore

x - iy

= p(p² - 3q²) - iq(q² - 3p²)

By comparing both sides we get,

{x \over p} = {p^2} - 3{q^2}

and

{y \over q} = {q^2} - 3{p^2}

\therefore

{{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}}

=

{{{p^2} - 3{q^2} + {q^2} - 3{p^2}} \over {{p^2} + {q^2}}}

=

{{ - 2{q^2} - 2{p^2}} \over {{p^2} + {q^2}}}

=

{{ - 2\left( {{q^2} + {p^2}} \right)} \over {{p^2} + {q^2}}}

=

-2

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