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Question

$I f blank z to the power of 2 end exponent plus z plus 1 equals 0 comma$ where z is a complex number, then the value of $open parentheses z plus fraction numerator 1 over denominator z end fraction close parentheses to the power of 2 end exponent plus open parentheses z to the power of 2 end exponent plus fraction numerator 1 over denominator z to the power of 2 end exponent end fraction close parentheses to the power of 2 end exponent plus open parentheses z to the power of 1 end exponent plus fraction numerator 1 over denominator z to the power of 3 end exponent end fraction close parentheses to the power of 2 end exponent plus midline horizontal ellipsis plus open parentheses z to the power of 6 end exponent plus fraction numerator 1 over denominator z to the power of 6 end exponent end fraction close parentheses to the power of 2 end exponent text is end text$

18
54
6
12

The correct answer is: 12

$z to the power of 2 end exponent plus z plus 1 minus 0 rightwards arrow angle equals fraction numerator negative 1 plus-or-minus square root of 3 i over denominator 2 end fraction equals omega comma omega to the power of 2 end exponent$
Let z = w. Therefore,
$z plus fraction numerator 1 over denominator z end fraction equals omega plus fraction numerator 1 over denominator omega end fraction equals omega plus omega to the power of 2 end exponent equals negative 1$
$table row cell rightwards double arrow z to the power of 2 end exponent plus fraction numerator 1 over denominator L to the power of 2 end exponent end fraction equals omega to the power of 2 end exponent plus fraction numerator 1 over denominator omega to the power of 2 end exponent end fraction equals 0 to the power of 2 end exponent plus omega equals negative 1 end cell row cell rightwards double arrow z to the power of 3 end exponent plus fraction numerator 1 over denominator z to the power of 1 end exponent end fraction equals omega to the power of 3 end exponent plus fraction numerator 1 over denominator omega to the power of 2 end exponent end fraction equals 1 plus 1 equals 2 end cell end table$
$table row cell rightwards double arrow z to the power of 4 end exponent plus fraction numerator 1 over denominator z to the power of 4 end exponent end fraction equals omega to the power of 4 end exponent plus fraction numerator 1 over denominator omega to the power of 4 end exponent end fraction equals omega plus omega to the power of 2 end exponent equals negative 1 end cell row cell rightwards double arrow z to the power of 5 end exponent plus fraction numerator 1 over denominator z to the power of 5 end exponent end fraction equals omega to the power of 5 end exponent plus fraction numerator 1 over denominator omega to the power of 5 end exponent end fraction equals omega plus omega to the power of 2 end exponent equals negative 1 end cell row cell rightwards double arrow z to the power of 6 end exponent plus fraction numerator 1 over denominator z to the power of 6 end exponent end fraction equals omega to the power of 6 end exponent plus fraction numerator 1 over denominator omega to the power of 6 end exponent end fraction equals 1 plus 1 equals 2 end cell end table$
$text So, the required sum end text equals left parenthesis negative 1 right parenthesis to the power of 2 end exponent plus left parenthesis negative 1 right parenthesis to the power of 2 end exponent plus left parenthesis 2 right parenthesis to the power of 2 end exponent plus left parenthesis negative 1 right parenthesis to the power of 2 end exponent plus left parenthesis negative 1 right parenthesis to the power of 2 end exponent plus left parenthesis 2 right parenthesis to the power of 2 end exponent equals 1 plus$
1 + 4 + 1 + 1 + 4 = 12

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