Standard +0.3 This is a standard Further Maths question testing whether a line lies in, is parallel to, or intersects a plane. It requires checking if the direction vector is perpendicular to the normal (for parallel/lying in) and substituting the line equation into the plane equation. The arithmetic is straightforward with small integer values, making it slightly easier than average even for FM content.
2 A line \(l\) has equation \(\mathbf { r } = 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + t ( \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } )\) and a plane \(\Pi\) has equation \(8 x - 7 y + 10 z = 7\). Determine whether \(l\) lies in \(\Pi\), is parallel to \(\Pi\) without intersecting it, or intersects \(\Pi\) at one point.
**EITHER:** $(r-) = [3+t, 1+4t, -2+2t]$
$8(3+t) - 7(1+4t) + 10(-2+2t) = ?$
$(0) + (-3) = 3 \Rightarrow$ contradiction
$l$ is parallel to $\Pi$, no intersection | M1 | For parametric form of $l$ seen or implied
| M1 A1 | For substituting into plane equation
| A1 | For obtaining a contradiction
| B1 5 | For conclusion from correct working
**OR:** $[4, 2], [8, -7, 10] = 0$
$\Rightarrow l$ is parallel to $\Pi$ | M1 | For finding scalar product of direction vectors
| A1 | For correct conclusion
$(3, 1, -2)$ into $\Pi$
$\Rightarrow 24 - 7 - 20 \neq 7$ | M1 | For substituting point into plane equation
| A1 | For obtaining a contradiction
| B1 | For conclusion from correct working
**OR:** Solve $\frac{x-3}{1} = \frac{y-1}{4} = \frac{z+2}{2}$ and $8x - 7y + 10z = 7$
eg $y - 2z = 3$, $2y - 2 = 4z + 8$ | M1 A1 | For eliminating one variable
| M1 | For eliminating another variable
eg $4z + 4 = 4z + 8$
| A1 | For obtaining a contradiction
| B1 | For conclusion from correct working
2 A line $l$ has equation $\mathbf { r } = 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + t ( \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } )$ and a plane $\Pi$ has equation $8 x - 7 y + 10 z = 7$. Determine whether $l$ lies in $\Pi$, is parallel to $\Pi$ without intersecting it, or intersects $\Pi$ at one point.
\hfill \mbox{\textit{OCR FP3 2007 Q2 [5]}}