| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2023 |
| Session | November |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line of intersection of planes |
| Difficulty | Standard +0.8 This is a comprehensive multi-part vectors question requiring: (a) finding a plane's Cartesian equation via cross product of direction vectors, (b) verifying parallelism by showing direction vector perpendicular to normal, (c) distance from line to plane calculation, and (d) finding line of intersection of two planes. While each individual step uses standard techniques, the question requires sustained accuracy across multiple methods and the final part involves solving simultaneous equations to find a point on both planes then using cross product of normals for direction. This is moderately challenging for Further Maths but follows established procedures. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane4.04f Line-plane intersection: find point4.04j Shortest distance: between a point and a plane |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\1 & -2 & -3\\3 & 0 & -1\end{vmatrix} = \begin{pmatrix}2\\-8\\6\end{pmatrix} \sim \begin{pmatrix}1\\-4\\3\end{pmatrix}\) | M1 A1 | Finds perpendicular to \(\Pi_1\) |
| \(1(1)-4(-1)+3(-2) = -1\) | M1 | Uses point on \(\Pi_1\) |
| \(x-4y+3z=-1\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{pmatrix}1\\-4\\3\end{pmatrix}\cdot\begin{pmatrix}1\\1\\1\end{pmatrix} = 1-4+3=0\) | M1 A1 | Shows dot product with direction of line is 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{\sqrt{1^2+4^2+3^2}}\begin{pmatrix}-4\\1\\3\end{pmatrix}\cdot\begin{pmatrix}1\\-4\\3\end{pmatrix}\) or \(\frac{1}{\sqrt{1^2+4^2+3^2}}\left(\begin{pmatrix}-3\\0\\1\end{pmatrix}\cdot\begin{pmatrix}1\\-4\\3\end{pmatrix}+1\right)\) | M1 A1 | Uses correct formula for distance from point on \(l\) to \(\Pi_1\): \(\frac{1}{\sqrt{1^2+4^2+3^2}}(-3\cdot1+0\cdot{-4}+1\cdot3+1)\) |
| \(\frac{1}{\sqrt{26}}\ (=0.196)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| States point common to both planes e.g. \(\begin{pmatrix}\frac{1}{15}\\\frac{4}{15}\\0\end{pmatrix}\) | B1 | \(\begin{pmatrix}\frac{5}{7}\\0\\\frac{-4}{7}\end{pmatrix}\) or \(\begin{pmatrix}0\\\frac{5}{17}\\\frac{1}{17}\end{pmatrix}\) or alternative |
| \(\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\1 & -4 & 3\\3 & 3 & 2\end{vmatrix} = \begin{pmatrix}-17\\7\\15\end{pmatrix}\) | M1 A1 | Finds direction of line |
| \(\mathbf{r} = \begin{pmatrix}\frac{5}{7}\\0\\\frac{-4}{7}\end{pmatrix} + \lambda\begin{pmatrix}-17\\7\\15\end{pmatrix}\) | A1 | OE |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\1 & -2 & -3\\3 & 0 & -1\end{vmatrix} = \begin{pmatrix}2\\-8\\6\end{pmatrix} \sim \begin{pmatrix}1\\-4\\3\end{pmatrix}$ | M1 A1 | Finds perpendicular to $\Pi_1$ |
| $1(1)-4(-1)+3(-2) = -1$ | M1 | Uses point on $\Pi_1$ |
| $x-4y+3z=-1$ | A1 | |
---
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{pmatrix}1\\-4\\3\end{pmatrix}\cdot\begin{pmatrix}1\\1\\1\end{pmatrix} = 1-4+3=0$ | M1 A1 | Shows dot product with direction of line is 0 |
---
## Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{\sqrt{1^2+4^2+3^2}}\begin{pmatrix}-4\\1\\3\end{pmatrix}\cdot\begin{pmatrix}1\\-4\\3\end{pmatrix}$ or $\frac{1}{\sqrt{1^2+4^2+3^2}}\left(\begin{pmatrix}-3\\0\\1\end{pmatrix}\cdot\begin{pmatrix}1\\-4\\3\end{pmatrix}+1\right)$ | M1 A1 | Uses correct formula for distance from point on $l$ to $\Pi_1$: $\frac{1}{\sqrt{1^2+4^2+3^2}}(-3\cdot1+0\cdot{-4}+1\cdot3+1)$ |
| $\frac{1}{\sqrt{26}}\ (=0.196)$ | A1 | |
---
## Question 5(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| States point common to both planes e.g. $\begin{pmatrix}\frac{1}{15}\\\frac{4}{15}\\0\end{pmatrix}$ | B1 | $\begin{pmatrix}\frac{5}{7}\\0\\\frac{-4}{7}\end{pmatrix}$ or $\begin{pmatrix}0\\\frac{5}{17}\\\frac{1}{17}\end{pmatrix}$ or alternative |
| $\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\1 & -4 & 3\\3 & 3 & 2\end{vmatrix} = \begin{pmatrix}-17\\7\\15\end{pmatrix}$ | M1 A1 | Finds direction of line |
| $\mathbf{r} = \begin{pmatrix}\frac{5}{7}\\0\\\frac{-4}{7}\end{pmatrix} + \lambda\begin{pmatrix}-17\\7\\15\end{pmatrix}$ | A1 | OE |5 The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { k } )$.
\begin{enumerate}[label=(\alph*)]
\item Find an equation for $\Pi _ { 1 }$ in the form $\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }$.\\
The line $l$, which does not lie in $\Pi _ { 1 }$, has equation $\mathbf { r } = - 3 \mathbf { i } + \mathbf { k } + t ( \mathbf { i } + \mathbf { j } + \mathbf { k } )$.
\item Show that $l$ is parallel to $\Pi _ { 1 }$.
\item Find the distance between $l$ and $\Pi _ { 1 }$.
\item The plane $\Pi _ { 2 }$ has equation $3 x + 3 y + 2 z = 1$.
Find a vector equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q5 [13]}}