QJEEAdv11Maac28: JEE Advanced 2011 Mathematics - Class 12 JEEAdv11Maac28 Solution: If the point P(a, b, c), with reference to (E), lies on the…

Given, $$[\begin{array}{ccc}a& b& c\end{array}{]}_{1\times 3}{\left[\begin{array}{ccc}1& 9& 7\\ 8& 2& 7\\ 7& 3& 7\end{array}\right]}_{3\times 3}=[\begin{array}{ccc}0& 0& 0\end{array}]$$

$$\Rightarrow \left[\begin{array}{c}a+8b+7c\\ 9a+2b+3c\\ 7a+7b+7c\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$

$\Rightarrow a + 8b + 7c = 0$ ..... (i)

$\Rightarrow 9a + 2b + 3c = 0$ .... (ii)

$\Rightarrow a + b + c = 0$ ....... (iii)

On multiplying Eq. (iii) by 2, then subtract from Eq. (ii), we get

$7a + c = 0$ ..... (iv)

Again multiplying Eq. (iii) by 3, then subtract from Eq. (ii), we get

$6a - b = 0$ ..... (v)

$\therefore$ $b = 6a$ and $c = - 7a$

As (a, b, c) lies on $2x + y + z = 1$

$\Rightarrow 2a + b + c = 1$

$\Rightarrow 2a + 6a - 7a = 1$

$\Rightarrow$ a = 1, b = 6 and c = $-$ 7

$\therefore$ $7a + b + c = 7 + 6 - 7 = 6$