| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Cartesian equation of a plane |
| Difficulty | Standard +0.2 Standard Core Pure question covering routine procedures: finding a Cartesian plane equation via dot product, finding angle between line and plane using sin formula, and substituting parametric line into plane equation. All parts are textbook exercises with no novel insight required. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04f Line-plane intersection: find point |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\mathbf{r} \cdot \begin{pmatrix}3\\-1\\2\end{pmatrix} = \begin{pmatrix}5\\-3\\-4\end{pmatrix} \cdot \begin{pmatrix}3\\-1\\2\end{pmatrix}\) | M1 | Attempts to apply formula \(\mathbf{r.n = a.n}\) |
| \(3x - y + 2z = 10\) | A1 | Correct Cartesian notation. e.g. \(3x-y+2z=10\) or \(-3x+y-2z=-10\). Do not allow final answer given as \(\mathbf{r}\cdot(3\mathbf{i}-\mathbf{j}+2\mathbf{k})=10\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{pmatrix}3\\-1\\2\end{pmatrix} \cdot \begin{pmatrix}-1\\-5\\3\end{pmatrix} = 8\) | B1 | \(\overrightarrow{OA}\cdot\mathbf{n}=8\) |
| \(\sqrt{(3)^2+(-1)^2+(2)^2}\cdot\sqrt{(-1)^2+(-5)^2+(3)^2}\cos\alpha = {-3+5+6}\) | M1 | Attempt to apply correct dot product formula between n and d |
| \(\theta = 90° - \arccos\left(\dfrac{8}{\sqrt{14}\cdot\sqrt{35}}\right)\) or \(\sin\theta = \dfrac{8}{\sqrt{14}\cdot\sqrt{35}}\) | M1 | Depends on previous M mark. Applies dot product formula to find angle between \(\Pi\) and \(l\) |
| \(\theta = 21.2°\) (1 dp) | A1* | cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3(7-\lambda)-(3-5\lambda)+2(-2+3\lambda)=10 \Rightarrow \lambda = \ldots\) | M1 | Substitutes \(l\) into \(\Pi\) and solves the resulting equation to give \(\lambda = \ldots\) |
| \(\lambda = -\dfrac{1}{2}\) | A1 | |
| \(\overrightarrow{OX} = \begin{pmatrix}7\\3\\-2\end{pmatrix} - \dfrac{1}{2}\begin{pmatrix}-1\\-5\\3\end{pmatrix} = \begin{pmatrix}\ldots\\\ldots\\\ldots\end{pmatrix}\) | M1 | Depends on previous M mark. Substitutes their \(\lambda\) into \(l\) and finds at least one of the coordinates |
| \(X(7.5, 5.5, -3.5)\) | A1ft | \((7.5, 5.5, -3.5)\) but follow through on their value of \(\lambda\) |
# Question 2:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{r} \cdot \begin{pmatrix}3\\-1\\2\end{pmatrix} = \begin{pmatrix}5\\-3\\-4\end{pmatrix} \cdot \begin{pmatrix}3\\-1\\2\end{pmatrix}$ | M1 | Attempts to apply formula $\mathbf{r.n = a.n}$ |
| $3x - y + 2z = 10$ | A1 | Correct Cartesian notation. e.g. $3x-y+2z=10$ or $-3x+y-2z=-10$. Do not allow final answer given as $\mathbf{r}\cdot(3\mathbf{i}-\mathbf{j}+2\mathbf{k})=10$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{pmatrix}3\\-1\\2\end{pmatrix} \cdot \begin{pmatrix}-1\\-5\\3\end{pmatrix} = 8$ | B1 | $\overrightarrow{OA}\cdot\mathbf{n}=8$ |
| $\sqrt{(3)^2+(-1)^2+(2)^2}\cdot\sqrt{(-1)^2+(-5)^2+(3)^2}\cos\alpha = {-3+5+6}$ | M1 | Attempt to apply correct dot product formula between **n** and **d** |
| $\theta = 90° - \arccos\left(\dfrac{8}{\sqrt{14}\cdot\sqrt{35}}\right)$ or $\sin\theta = \dfrac{8}{\sqrt{14}\cdot\sqrt{35}}$ | M1 | Depends on previous M mark. Applies dot product formula to find angle between $\Pi$ and $l$ |
| $\theta = 21.2°$ (1 dp) | A1* | cso |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3(7-\lambda)-(3-5\lambda)+2(-2+3\lambda)=10 \Rightarrow \lambda = \ldots$ | M1 | Substitutes $l$ into $\Pi$ and solves the resulting equation to give $\lambda = \ldots$ |
| $\lambda = -\dfrac{1}{2}$ | A1 | |
| $\overrightarrow{OX} = \begin{pmatrix}7\\3\\-2\end{pmatrix} - \dfrac{1}{2}\begin{pmatrix}-1\\-5\\3\end{pmatrix} = \begin{pmatrix}\ldots\\\ldots\\\ldots\end{pmatrix}$ | M1 | Depends on previous M mark. Substitutes their $\lambda$ into $l$ and finds at least one of the coordinates |
| $X(7.5, 5.5, -3.5)$ | A1ft | $(7.5, 5.5, -3.5)$ but follow through on their value of $\lambda$ |
---\begin{enumerate}
\item The plane $\Pi$ passes through the point $A$ and is perpendicular to the vector $\mathbf { n }$
\end{enumerate}
Given that
$$\overrightarrow { O A } = \left( \begin{array} { r }
5 \\
- 3 \\
- 4
\end{array} \right) \quad \text { and } \quad \mathbf { n } = \left( \begin{array} { r }
3 \\
- 1 \\
2
\end{array} \right)$$
where $O$ is the origin,\\
(a) find a Cartesian equation of $\Pi$.
With respect to the fixed origin $O$, the line $l$ is given by the equation
$$\mathbf { r } = \left( \begin{array} { r }
7 \\
3 \\
- 2
\end{array} \right) + \lambda \left( \begin{array} { r }
- 1 \\
- 5 \\
3
\end{array} \right)$$
The line $l$ intersects the plane $\Pi$ at the point $X$.\\
(b) Show that the acute angle between the plane $\Pi$ and the line $l$ is $21.2 ^ { \circ }$ correct to one decimal place.\\
(c) Find the coordinates of the point $X$.
\hfill \mbox{\textit{Edexcel CP AS Q2 [10]}}