| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Cartesian equation of a plane |
| Difficulty | Standard +0.3 This is a standard Further Maths question on vector planes requiring routine application of cross product, dot product for plane equation, and scalar triple product for volume. All steps are algorithmic with no novel insight needed, making it slightly easier than average A-level difficulty overall, though harder than typical single-maths questions. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04g Vector product: a x b perpendicular vector8.04a Vector product: definition, magnitude/direction, component form8.04e Scalar triple product: volumes of tetrahedra and parallelepipeds |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\overrightarrow{AB} = -2\mathbf{i}+6\mathbf{j}+4\mathbf{k}\) and \(\overrightarrow{AC} = -5\mathbf{i}+5\mathbf{j}+2\mathbf{k}\) | B1 | Both \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) correct |
| \(\overrightarrow{AB}\times\overrightarrow{AC} = \begin{vmatrix}-2 & 6 & 4\\-5 & 5 & 2\end{vmatrix}\) | M1 | Applies cross product to their \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\); at least two correct components if no method seen |
| \(= -8\mathbf{i}-16\mathbf{j}+20\mathbf{k}\) | A1 | Correct vector |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Uses \(\mathbf{n} = -8\mathbf{i}-16\mathbf{j}+20\mathbf{k}\) with point on plane to find \(p\) | M1 | Uses their \(\mathbf{n}\) (any multiple of \(\overrightarrow{AB}\times\overrightarrow{AC}\)) and any point on plane; use of \(\overrightarrow{AB}\) or \(\overrightarrow{AC}\) is M0 |
| \(\mathbf{r}.(-8\mathbf{i}-16\mathbf{j}+20\mathbf{k}) = 28\) or \(\mathbf{r}.(2\mathbf{i}+4\mathbf{j}-5\mathbf{k}) = -7\) | A1 | Correct equation; accept any multiples e.g. \(\mathbf{r}.(-8\mathbf{i}-16\mathbf{j}+20\mathbf{k})=28\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\overrightarrow{AD}.\left(\overrightarrow{AB}\times\overrightarrow{AC}\right) = {``}\overrightarrow{AD}{''}.(-8\mathbf{i}-16\mathbf{j}+20\mathbf{k})\) | M1 | Attempts suitable scalar triple product; must include complete method using all necessary vectors |
| \(\overrightarrow{AD} = 5\mathbf{i}+9\mathbf{j}+4\mathbf{k}\) | B1 | Correct \(\overrightarrow{AD}\) if using \(\overrightarrow{AD}.(\overrightarrow{AB}\times\overrightarrow{AC})\), or all vectors correct if using different product |
| \(\text{Volume} = \frac{1}{6}\left | \overrightarrow{AD}.(\overrightarrow{AB}\times\overrightarrow{AC})\right | = \frac{1}{6} |
| \(= \frac{52}{3}\) o.e. \(17\frac{1}{3}\) | A1 | Correct exact answer |
# Question 3:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\overrightarrow{AB} = -2\mathbf{i}+6\mathbf{j}+4\mathbf{k}$ and $\overrightarrow{AC} = -5\mathbf{i}+5\mathbf{j}+2\mathbf{k}$ | **B1** | Both $\overrightarrow{AB}$ and $\overrightarrow{AC}$ correct |
| $\overrightarrow{AB}\times\overrightarrow{AC} = \begin{vmatrix}-2 & 6 & 4\\-5 & 5 & 2\end{vmatrix}$ | **M1** | Applies cross product to their $\overrightarrow{AB}$ and $\overrightarrow{AC}$; at least two correct components if no method seen |
| $= -8\mathbf{i}-16\mathbf{j}+20\mathbf{k}$ | **A1** | Correct vector |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses $\mathbf{n} = -8\mathbf{i}-16\mathbf{j}+20\mathbf{k}$ with point on plane to find $p$ | **M1** | Uses their $\mathbf{n}$ (any multiple of $\overrightarrow{AB}\times\overrightarrow{AC}$) and any point on plane; use of $\overrightarrow{AB}$ or $\overrightarrow{AC}$ is M0 |
| $\mathbf{r}.(-8\mathbf{i}-16\mathbf{j}+20\mathbf{k}) = 28$ or $\mathbf{r}.(2\mathbf{i}+4\mathbf{j}-5\mathbf{k}) = -7$ | **A1** | Correct equation; accept any multiples e.g. $\mathbf{r}.(-8\mathbf{i}-16\mathbf{j}+20\mathbf{k})=28$ |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\overrightarrow{AD}.\left(\overrightarrow{AB}\times\overrightarrow{AC}\right) = {``}\overrightarrow{AD}{''}.(-8\mathbf{i}-16\mathbf{j}+20\mathbf{k})$ | **M1** | Attempts suitable scalar triple product; must include complete method using all necessary vectors |
| $\overrightarrow{AD} = 5\mathbf{i}+9\mathbf{j}+4\mathbf{k}$ | **B1** | Correct $\overrightarrow{AD}$ if using $\overrightarrow{AD}.(\overrightarrow{AB}\times\overrightarrow{AC})$, or all vectors correct if using different product |
| $\text{Volume} = \frac{1}{6}\left|\overrightarrow{AD}.(\overrightarrow{AB}\times\overrightarrow{AC})\right| = \frac{1}{6}|(5\mathbf{i}+9\mathbf{j}+4\mathbf{k}).(-8\mathbf{i}-16\mathbf{j}+20\mathbf{k})|$ | **M1** | Use of volume $= \frac{1}{6}\left|\text{their } \overrightarrow{AD}.(\overrightarrow{AB}\times\overrightarrow{AC})\right|$ |
| $= \frac{52}{3}$ o.e. $17\frac{1}{3}$ | **A1** | Correct exact answer |
---\begin{enumerate}
\item The points $A , B$ and $C$, with position vectors $\mathbf { a } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \mathbf { b } = \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }$ and $\mathbf { c } = - 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }$ respectively, lie on the plane $\Pi$\\
(a) Find $\overrightarrow { A B } \times \overrightarrow { A C }$\\
(b) Find an equation for $\Pi$ in the form r.n $= p$
\end{enumerate}
The point $D$ has position vector $8 \mathbf { i } + 7 \mathbf { j } + 5 \mathbf { k }$\\
(c) Determine the volume of the tetrahedron $A B C D$
\hfill \mbox{\textit{Edexcel FP1 2020 Q3 [9]}}