Paragraph: Let a , b and c be three real numbers satisfying [ array lll a & b & c array ][ array lll 1 & 9 & 7 8 & 2 & 7 7 & 3 & 7 array ]=[ array lll 0 & 0 & 0 array ] Question: Let ω be a solution of x^3-1=0 with

JEE Advanced 2011Paper 1

JEE Advanced 2011 · Paper 1 · Mathematics · Complex Number

Paragraph: Let a , b and c be three real numbers satisfying [ array...

Paragraph: Let

a

, b and

c

be three real numbers satisfying \left[

\begin{array}{lll}a & b & c\end{array}

\right]\left[

\begin{array}{lll}1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7\end{array}

\right]=\left[

\begin{array}{lll}0 & 0 & 0\end{array}$$

\right]

Question: Let

\omega

be a solution of

x^3-1=0

with

\operatorname{Im}(\omega)>0

. If

a=2

, with

b and csatisfying Eq. (E), then the value of

\frac{3}{\omega^a}+\frac{1}{\omega^b}

+

\frac{3}{\omega^c}

A. $-2$

\begin{aligned}& \text{ Given, }[a b c]_{1 \times3}\left[\begin{array}{lll}1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7 \end{array}\right]_{3 \times3}=\left[\begin{array}{lll}0 & 0 & 0 \end{array}\right] \\ & \Rightarrow \quad\left[\begin{array}{c}a+8 b+7 c \\ 9 a+2 b+3 c \\ 7 a+7 b+7 c \end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0 \end{array}\right] \\ & \Rightarrow \quad a+8 b+7 c=0 \\ & \Rightarrow 9 a+2 b+3 c=0 \\ & \Rightarrow \quad a+b+c=0 \\ & \end{aligned}

On multiplying Eq. (iii) by 2 , then subtract from Eq. (ii), we get 7 a+c=0 Again multiplying Eq. (iii) by 3 , then subtract from Eq. (ii), we get

\begin{array}{ll}& 6 a-b=0 \\ \therefore \quad & b=6 a \text{ and }c=-7 a \end{array}

a=2, b=12

and

c=-14

\begin{aligned} \therefore& \frac{3}{\omega^a}+\frac{1}{\omega^b}+\frac{3}{\omega^c}\\ \Rightarrow & \frac{3}{\omega^2}+\frac{1}{\omega^{12}}+\frac{3}{\omega^{-14}}=\frac{3}{\omega^2}+1+3 \omega^2 \\ & =3 \omega+1+3 \omega^2 \\ & =1+3\left(\omega+\omega^2\right) \\ & =1-3=-2 \end{aligned}