Matrices & Determinants
Unique Solution of Overdetermined System and Distance from Plane
nta_pyq_2023_apr
Grade 12
Question:
Let the system $-x+2y-9z=7$, $-x+3y+7z=9$, $-2x+y+5z=8$, $-3x+y+13z=\lambda$ have a unique solution $x=\alpha,\ y=\beta,\ z=\gamma$. Then the distance of the point $(\alpha,\beta,\gamma)$ from the plane $2x-2y+z=\lambda$ is
Step-by-Step Solution
Key Concept: Solve the first three equations to get $(\alpha,\beta,\gamma)=(-3,2,0)$. Substitute into equation (4) to find $\lambda=11$.
$(\alpha,\beta,\gamma)=(-3,2,0)$, $\lambda=11$. Distance $=\frac{|-6-4-11|}{3}=7$.
Correct Answer: 2