| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Equation of plane through three points |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question requiring routine application of cross product to find a plane equation, then using the perpendicular distance formula and line-plane intersection. All techniques are textbook procedures with no novel insight required, making it slightly easier than average even for Further Maths. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04f Line-plane intersection: find point4.04j Shortest distance: between a point and a plane |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\overrightarrow{AB} = -8\mathbf{i} + 7\mathbf{j} - 5\mathbf{k}\), \(\overrightarrow{AC} = 3\mathbf{j} - 3\mathbf{k}\) | B1 | Finds direction vectors of two lines in the plane |
| \(\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -8 & 7 & -5 \\ 0 & 3 & -3 \end{vmatrix} = \begin{pmatrix} -6 \\ -24 \\ -24 \end{pmatrix} \sim \begin{pmatrix} 1 \\ 4 \\ 4 \end{pmatrix}\) | M1 A1 | Finds normal to the plane \(ABC\) |
| \((4) + 4(-4) + 4(1) = -8 \Rightarrow x + 4y + 4z = -8\) | M1 A1 | Substitutes point |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Setting up 3 equations using the points given | M1 | |
| \(x + 4y + 4z = -8\) | A1 A1 A1 A1 | |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{8}{\sqrt{1^2 + 4^2 + 4^2}} = 1.39\) | M1 A1 | Divides their constant by magnitude of their normal vector. \(\dfrac{8}{\sqrt{33}}\) CAO |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{r} = t\begin{pmatrix} 2 \\ 3 \\ -3 \end{pmatrix}\) | B1 | Equation of line \(OD\) |
| \(2t + 12t - 12t = -8\) | M1 | Substitutes into equation of plane |
| \((-8, -12, 12)\) | A1 | |
| Total: 3 |
## Question 2:
### Part 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\overrightarrow{AB} = -8\mathbf{i} + 7\mathbf{j} - 5\mathbf{k}$, $\overrightarrow{AC} = 3\mathbf{j} - 3\mathbf{k}$ | B1 | Finds direction vectors of two lines in the plane |
| $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -8 & 7 & -5 \\ 0 & 3 & -3 \end{vmatrix} = \begin{pmatrix} -6 \\ -24 \\ -24 \end{pmatrix} \sim \begin{pmatrix} 1 \\ 4 \\ 4 \end{pmatrix}$ | M1 A1 | Finds normal to the plane $ABC$ |
| $(4) + 4(-4) + 4(1) = -8 \Rightarrow x + 4y + 4z = -8$ | M1 A1 | Substitutes point |
**Alternative method 2(a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Setting up 3 equations using the points given | M1 | |
| $x + 4y + 4z = -8$ | A1 A1 A1 A1 | |
| **Total: 5** | | |
---
### Part 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{8}{\sqrt{1^2 + 4^2 + 4^2}} = 1.39$ | M1 A1 | Divides their constant by magnitude of their normal vector. $\dfrac{8}{\sqrt{33}}$ CAO |
| **Total: 2** | | |
---
### Part 2(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{r} = t\begin{pmatrix} 2 \\ 3 \\ -3 \end{pmatrix}$ | B1 | Equation of line $OD$ |
| $2t + 12t - 12t = -8$ | M1 | Substitutes into equation of plane |
| $(-8, -12, 12)$ | A1 | |
| **Total: 3** | | |
---2 The points $A , B , C$ have position vectors
$$4 \mathbf { i } - 4 \mathbf { j } + \mathbf { k } , \quad - 4 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } , \quad 4 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ,$$
respectively, relative to the origin $O$.
\begin{enumerate}[label=(\alph*)]
\item Find the equation of the plane $A B C$, giving your answer in the form $a x + b y + c z = d$.
\item Find the perpendicular distance from $O$ to the plane $A B C$.
\item The point $D$ has position vector $2 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k }$.
Find the coordinates of the point of intersection of the line $O D$ with the plane $A B C$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q2 [10]}}