IIT-JEE 2011 · Paper 1 · Mathematics Q59

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Paragraph for Question Nos. 58 to 60

Let $a$ , $b$ and $c$ be three real numbers satisfying
$\begin{bmatrix}a&b&c\end{bmatrix}\begin{bmatrix}1&9&7\\8&2&7\\7&3&7\end{bmatrix}=\begin{bmatrix}0&0&0\end{bmatrix}\qquad\text{(E)}$

Let $\omega$ be a solution of $x^3-1=0$ with $\operatorname{Im}(\omega)>0$ . If $a=2$ with $b$ and $c$ satisfying (E), then the value of
$\frac3{\omega^a}+\frac1{\omega^b}+\frac3{\omega^c}$
is equal to

-2

-3

Answer and worked solution

Official answer: A

Values. $a=2$ gives $k=-2$ : $b=12$ , $c=-14$ .

Sum. $\dfrac3{\omega^2}+\dfrac1{\omega^{12}}+\dfrac3{\omega^{-14}}=3\omega+1+3\omega^{14}=3\omega+1+3\omega^2=1+3(-1)=-2$ .

(A)

Topics:Roots of unity Systems of linear equations

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IIT-JEE 2011 · Paper 1, Mathematics question 59