| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Moderate -0.5 This is a two-part question where part (i) requires routine vector manipulation to find direction vectors and write a vector equation, and part (ii) involves finding a normal vector via cross product and substituting to get the Cartesian form. Both are standard textbook exercises for Further Pure students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.04b Plane equations: cartesian and vector forms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Vectors in plane: two of \(\begin{pmatrix}-4\\4\\1\end{pmatrix}, \begin{pmatrix}0\\6\\4\end{pmatrix} = 2\begin{pmatrix}0\\3\\2\end{pmatrix}, \begin{pmatrix}4\\2\\3\end{pmatrix}\) | M1 | Differences between two pairs. Any multiple |
| \(\mathbf{r} = \begin{pmatrix}1\\6\\2\end{pmatrix} + \lambda\begin{pmatrix}0\\3\\2\end{pmatrix} + \mu\begin{pmatrix}4\\2\\3\end{pmatrix}\) | A1 | Aef of parametric equation. Must have "\(\mathbf{r} = \ldots\)" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{pmatrix}0\\3\\2\end{pmatrix} \times \begin{pmatrix}4\\2\\3\end{pmatrix} = \begin{pmatrix}5\\8\\-12\end{pmatrix}\) | M1, A1 | Calculate vector product or multiple. M1 awarded where vector product has method shown or only one term wrong |
| \(\left(\mathbf{r} - \begin{pmatrix}1\\6\\2\end{pmatrix}\right) \cdot \begin{pmatrix}5\\8\\-12\end{pmatrix} = 0\) | M1 | Or Cartesian form \(= d\) with attempt to compute \(d\) |
| \(5x + 8y - 12z = 29\) | A1 | Aef of cartesian equation, isw |
| Alternate method: \(x, y, z\) in parametric form, both parameters in terms of e.g. \(x, y\), substitute into parametric form of \(z\) | M1, A1, M1A1 | EITHER method |
| Or: \(x, y, z\) in parametric form, 2 equations in \(x, y, z\) and one parameter, eliminate parameter | M1, A1, M1A1 | OR method |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Vectors in plane: two of $\begin{pmatrix}-4\\4\\1\end{pmatrix}, \begin{pmatrix}0\\6\\4\end{pmatrix} = 2\begin{pmatrix}0\\3\\2\end{pmatrix}, \begin{pmatrix}4\\2\\3\end{pmatrix}$ | M1 | Differences between two pairs. Any multiple |
| $\mathbf{r} = \begin{pmatrix}1\\6\\2\end{pmatrix} + \lambda\begin{pmatrix}0\\3\\2\end{pmatrix} + \mu\begin{pmatrix}4\\2\\3\end{pmatrix}$ | A1 | Aef of parametric equation. Must have "$\mathbf{r} = \ldots$" |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{pmatrix}0\\3\\2\end{pmatrix} \times \begin{pmatrix}4\\2\\3\end{pmatrix} = \begin{pmatrix}5\\8\\-12\end{pmatrix}$ | M1, A1 | Calculate vector product or multiple. M1 awarded where vector product has method shown or only one term wrong |
| $\left(\mathbf{r} - \begin{pmatrix}1\\6\\2\end{pmatrix}\right) \cdot \begin{pmatrix}5\\8\\-12\end{pmatrix} = 0$ | M1 | Or Cartesian form $= d$ with attempt to compute $d$ |
| $5x + 8y - 12z = 29$ | A1 | Aef of cartesian equation, isw |
| **Alternate method:** $x, y, z$ in parametric form, both parameters in terms of e.g. $x, y$, substitute into parametric form of $z$ | M1, A1, M1A1 | EITHER method |
| Or: $x, y, z$ in parametric form, 2 equations in $x, y, z$ and one parameter, eliminate parameter | M1, A1, M1A1 | OR method |
---1 The plane $\Pi$ passes through the points with coordinates $( 1,6,2 ) , ( 5,2,1 )$ and $( 1,0 , - 2 )$.\\
(i) Find a vector equation of $\Pi$ in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }$.\\
(ii) Find a cartesian equation of $\Pi$.\\
$2 G$ consists of the set $\{ 1,3,5,7 \}$ with the operation of multiplication modulo 8 .\\
\hfill \mbox{\textit{OCR FP3 2013 Q1 [6]}}