IIT-JEE 2011 · Paper 1 · Mathematics Q58

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

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

Answer and worked solution

Official answer: D

Solve (E). Column equations: $a+8b+7c=0$ , $9a+2b+3c=0$ , $7a+7b+7c=0$ . The last gives $a+b+c=0$ ; subtracting it from the first, $7b+6c=0$ . So $(a,b,c)=k(-1,-6,7)$ (check: $-9-12+21=0$ ).

On the plane. $2a+b+c=-k=1$ , so $k=-1$ : $(1,6,-7)$ and $7a+b+c=7+6-7=6$ .

(D)

Topics:Systems of linear equations

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IIT-JEE 2011 · Paper 1, Mathematics question 58