| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Parallel and perpendicular planes |
| Difficulty | Standard +0.3 This is a straightforward Further Maths vectors question testing standard techniques: (a) requires showing a vector is perpendicular to the normal (dot product = 0), and (b) uses the standard formula for angle between planes via their normals. Both parts are direct applications of learned methods with minimal problem-solving required, though the Further Maths context places it slightly above average A-level difficulty. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04d Angles: between planes and between line and plane |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | (i + 4j + 2k).( 2i + j − 3k) |
| Answer | Marks |
|---|---|
| hence parallel to the plane | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 3.2a | scalar product of (i + 4j + 2k) and (2i + j − 3k) |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (b) | ( 2 3 ) .( 3 ) i + j − k − i j − k |
| Answer | Marks |
|---|---|
| = 36.3 | M1 |
| Answer | Marks |
|---|---|
| [4] | 1.1a |
| Answer | Marks |
|---|---|
| 1.1 | finding angle between normals |
Question 2:
2 | (a) | (i + 4j + 2k).( 2i + j − 3k)
= 12 +41 + 2(−3) = 0
so 2i + j − 3k is perpendicular to normal to the plane,
hence parallel to the plane | M1
A1
A1
[3] | 3.1a
1.1
3.2a | scalar product of (i + 4j + 2k) and (2i + j − 3k)
= 0
justification given
2 | (b) | ( 2 3 ) .( 3 ) i + j − k − i j − k
c o s =
2 2 1 2 ( 3 2 ) 2 1 ( 1 ) 2 ( 3 ) 2 + + − + − + −
2 − 1 + 9 1 0
= [ = ]
1 4 1 1 1 4 1 1
= 36.3 | M1
M1
A1
A1
[4] | 1.1a
1.1
1.1
1.1 | finding angle between normals
formula correct
36 or 0.63 rads or better2
\begin{enumerate}[label=(\alph*)]
\item Show that the vector $\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$ is parallel to the plane $2 \mathrm { x } + \mathrm { y } - 3 \mathrm { z } = 10$.
\item Determine the acute angle between the planes $2 x + y - 3 z = 10$ and $x - y - 3 z = 3$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2022 Q2 [7]}}