Discussion Forum

1)

Let a, b and c be three real numbers satisfying [a b c]  $ \begin{bmatrix}1 & 9 &7 \\8 & 2&7 \\7&3&7 \end{bmatrix}=\begin{bmatrix}0 & 0 &0 \end{bmatrix}$

If the point P(a, b, c), with reference to Eq. (E), lies on the plane 2x + y +z = 1, then the value of 7 a + b + c is

Answer:

Option D

Explanation:

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

 $\Rightarrow$ $\begin{bmatrix}a+8b+7c  \\9a+2b+3c \\7a+7b+7c\end{bmatrix}=\begin{bmatrix}0 & 0 &0 \end{bmatrix}$

$\Rightarrow$  a+8b+7c=0....(i)

$\Rightarrow$  9a+2b+3c=0 .....(ii)

$\Rightarrow$  a+b+c=0 .....(iii)

 On multiplying Eq.(iii) bt=y 2 , then substract from Eq.(ii) we get

 7a+c=0...(iv)

 Again multiplying Eq,(iii) by 3 , then substract 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