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 b=6 , with a and c satisfying Eq.

JEE Advanced 2011Paper 1

JEE Advanced 2011 · Paper 1 · Mathematics · Matrices

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

b=6

, with

a and c

satisfying Eq. (E). If

\alpha

and

\beta

are the roots of the quadratic equation

a x^2+b x+c=0

, then

\sum_{n=0}^{\infty}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)^n is

B. 7

\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}

b=6, a=1

and

c=-7

\begin{array}{ll} \therefore& a x²+b x+c=0 \\ \Rightarrow & x²+6 x-7=0 \\ \Rightarrow & (x+7)(x-1)=0 \\ \therefore & x=1,-7 \\ \Rightarrow & \quad \sum_{n=0}^{\infty}\left(\frac{1}{1}-\frac{1}{7}\right)^n \Rightarrow \sum_{n=0}^{\infty}\left(\frac{6}{7}\right)^n \\ \Rightarrow & 1+\frac{6}{7}+\left(\frac{6}{7}\right)^n+\ldots \infty\\ & =\frac{1}{1-\frac{6}{7}}=\frac{1}{1 / 7}=7 \end{array}