| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| 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 three-part question on planes requiring routine techniques: finding a normal vector via cross product, writing Cartesian equations, and calculating the angle between planes using dot product of normals. While it involves multiple steps and two planes, each part follows textbook methods without requiring novel insight or particularly challenging algebraic manipulation. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -1 & 2 \\ 1 & 2 & 3 \end{vmatrix} = \begin{pmatrix} -7 \\ -1 \\ 3 \end{pmatrix}\) | M1A1 | |
| Cartesian equation of \(\Pi_1\) is \(7x + y - 3z = \text{const}\) | M1 | |
| \(7\times1 + 2 - 3\times1 = 6 \Rightarrow 7x + y - 3z = 6\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -2 & -1 \\ 2 & 3 & -1 \end{vmatrix} = \begin{pmatrix} 5 \\ -1 \\ 7 \end{pmatrix}\) | M1A1 | |
| \(5\times2 - (-3) + 7\times1 = 20 \Rightarrow 5x - y + 7z = 20\) | M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\cos\theta = \dfrac{\begin{pmatrix}7\\1\\-3\end{pmatrix}\cdot\begin{pmatrix}5\\-1\\7\end{pmatrix}}{\sqrt{49+1+9}\sqrt{25+1+49}}\) | M1M1 | |
| \(\cos\theta = \dfrac{13}{\sqrt{59}\sqrt{75}} \Rightarrow \theta = 78.7°\) or \(1.37\) rad | A1 |
## Question 9(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -1 & 2 \\ 1 & 2 & 3 \end{vmatrix} = \begin{pmatrix} -7 \\ -1 \\ 3 \end{pmatrix}$ | M1A1 | |
| Cartesian equation of $\Pi_1$ is $7x + y - 3z = \text{const}$ | M1 | |
| $7\times1 + 2 - 3\times1 = 6 \Rightarrow 7x + y - 3z = 6$ | A1 | |
## Question 9(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -2 & -1 \\ 2 & 3 & -1 \end{vmatrix} = \begin{pmatrix} 5 \\ -1 \\ 7 \end{pmatrix}$ | M1A1 | |
| $5\times2 - (-3) + 7\times1 = 20 \Rightarrow 5x - y + 7z = 20$ | M1A1 | |
## Question 9(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\cos\theta = \dfrac{\begin{pmatrix}7\\1\\-3\end{pmatrix}\cdot\begin{pmatrix}5\\-1\\7\end{pmatrix}}{\sqrt{49+1+9}\sqrt{25+1+49}}$ | M1M1 | |
| $\cos\theta = \dfrac{13}{\sqrt{59}\sqrt{75}} \Rightarrow \theta = 78.7°$ or $1.37$ rad | A1 | |9 The plane $\Pi _ { 1 }$ passes through the points $( 1,2,1 )$ and $( 5 , - 2,9 )$ and is parallel to the vector $\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$.\\
(i) Find the cartesian equation of $\Pi _ { 1 }$.\\
The plane $\Pi _ { 2 }$ contains the lines
$$\mathbf { r } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + \mu ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } ) .$$
(ii) Find the cartesian equation of $\Pi _ { 2 }$.\\
(iii) Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q9 [11]}}