| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2020 |
| Session | November |
| Marks | 11 |
| 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 normal vector, then using point-normal form for the plane equation. Parts (b) and (c) use standard distance and angle formulas. While it's a multi-part question requiring several techniques, each step follows a well-practiced procedure with no novel insight needed. Slightly above average difficulty due to being Further Maths content and requiring careful algebraic manipulation across three parts. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane4.04j Shortest distance: between a point and a plane |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\overrightarrow{AB} = -\mathbf{i} - 2\mathbf{j} - 2\mathbf{k}\), \(\overrightarrow{AC} = 3\mathbf{i} - \mathbf{j}\) | B1 | Finds direction vectors of two lines in the plane; \(\overrightarrow{BC} = 4\mathbf{i} + \mathbf{j} + 2\mathbf{k}\) |
| \(\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 2 \\ 3 & -1 & 0 \end{vmatrix} = \begin{pmatrix} -2 \\ -6 \\ 7 \end{pmatrix}\) | M1 A1 | Finds normal to the plane \(ABC\) |
| \(-2(-1) - 6(1) + 7(2) = 10 \Rightarrow -2x - 6y + 7z = 10\) | M1 A1 | Substitutes point |
| Alternative: Setting up 3 equations using points given | M1 | |
| \(-2x - 6y + 7z = 10\) | A1 A1 A1 A1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{10}{\sqrt{2^2+6^2+7^2}} = \frac{10}{\sqrt{89}}\) OE | M1 A1 | Divides by magnitude of normal vector; \(1.06\ldots\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & 1 & 2 \\ -2 & -1 & 0 \end{vmatrix} = \begin{pmatrix} 2 \\ -4 \\ 3 \end{pmatrix}\) | M1 A1 | Finds normal to the plane \(OAB\) |
| \(\begin{pmatrix} -2 \\ -6 \\ 7 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ -4 \\ 3 \end{pmatrix} = \sqrt{89}\sqrt{29}\cos\theta\) | M1 | Uses dot product correctly |
| \(36.2°\) | A1 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\overrightarrow{AB} = -\mathbf{i} - 2\mathbf{j} - 2\mathbf{k}$, $\overrightarrow{AC} = 3\mathbf{i} - \mathbf{j}$ | B1 | Finds direction vectors of two lines in the plane; $\overrightarrow{BC} = 4\mathbf{i} + \mathbf{j} + 2\mathbf{k}$ |
| $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 2 \\ 3 & -1 & 0 \end{vmatrix} = \begin{pmatrix} -2 \\ -6 \\ 7 \end{pmatrix}$ | M1 A1 | Finds normal to the plane $ABC$ |
| $-2(-1) - 6(1) + 7(2) = 10 \Rightarrow -2x - 6y + 7z = 10$ | M1 A1 | Substitutes point |
| **Alternative:** Setting up 3 equations using points given | M1 | |
| $-2x - 6y + 7z = 10$ | A1 A1 A1 A1 | OE |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{10}{\sqrt{2^2+6^2+7^2}} = \frac{10}{\sqrt{89}}$ OE | M1 A1 | Divides by magnitude of normal vector; $1.06\ldots$ |
## Question 4(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & 1 & 2 \\ -2 & -1 & 0 \end{vmatrix} = \begin{pmatrix} 2 \\ -4 \\ 3 \end{pmatrix}$ | M1 A1 | Finds normal to the plane $OAB$ |
| $\begin{pmatrix} -2 \\ -6 \\ 7 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ -4 \\ 3 \end{pmatrix} = \sqrt{89}\sqrt{29}\cos\theta$ | M1 | Uses dot product correctly |
| $36.2°$ | A1 | |
---4 The points $A , B , C$ have position vectors
$$- \mathbf { i } + \mathbf { j } + 2 \mathbf { k } , \quad - 2 \mathbf { i } - \mathbf { j } , \quad 2 \mathbf { i } + 2 \mathbf { k } ,$$
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 Find the acute angle between the planes $O A B$ and $A B C$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q4 [11]}}