(i) Find the real values of $x$ and $y$ for which $z_{1} = 9 y^{2} - 4 - 10 i x$ is complex conjugate of each other.
(ii) Find the value of $x^{4} - x^{3} + x^{2} + 3 x - 5$ if $x = 2 + 3 i$

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Solution

1. $_{1} = z$
$9 y^{2} - 4 + 10 i x = 9 y^{2} - 4 - 10 i x$
$20 i x = 0$
$x = 0$
$\forall x = 0$ & $y ϵ R$ , this holds true
2. $x^{4} - x^{3} + x^{2} + 3 x - 5 = P$ (Let)
$x = 2 + 3 i$
$x - 2 = 3 i$
${(x - 2)}^{2} = 9 i^{2}$
$x^{2} - 4 x + 4 = - 9$
$x^{2} - 4 x + 13 = 0$
(Image)
$P = (x^{2} - 4 x + 13) (x^{2} + 3 x) + (- 36 x - 5)$
Remainder $=$ Answer $= - 36 x - 5$
$= - 36 (2 + 3 i) - 5 = - 77 - 108 i$

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(i) Find the real values of x and y for which z1=9y2−4−10ix is complex conjugate of each other.(ii) Find the value of x4−x3+x2+3x−5 if x=2+3i

1. ¯z1=z9y2−4+10ix=9y2−4−10ix20ix=0x=0∀x=0 & yϵR, this holds true2. x4−x3+x2+3x−5=P (Let)x=2+3ix−2=3i(x−2)2=9i2x2−4x+4=−9x2−4x+13=0(Image)P=(x2−4x+13)(x2+3x)+(−36x−5)Remainder=Answer=−36x−5=−36(2+3i)−5=−77−108i

(i) Find the real values of $x$ and $y$ for which $z_{1} = 9 y^{2} - 4 - 10 i x$ is complex conjugate of each other.
(ii) Find the value of $x^{4} - x^{3} + x^{2} + 3 x - 5$ if $x = 2 + 3 i$