| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line of intersection of planes |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question requiring routine application of dot product formula for angle between planes and solving simultaneous equations to find the line of intersection. Both parts follow textbook methods with straightforward arithmetic, making it slightly easier than average even for Further Maths content. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\cos\theta = \frac{\begin{vmatrix}\begin{pmatrix}1\\2\\5\end{pmatrix}\cdot\begin{pmatrix}2\\-1\\3\end{pmatrix}\end{vmatrix}}{\sqrt{1^2+2^2+5^2}\sqrt{2^2+(-1)^2+3^2}} = \frac{15}{\sqrt{30}\sqrt{14}}\) | M1, A1 | Accept unsimplified |
| \(\theta = 0.750\) or \(43.0°\) | A1 | If zero, then sc1 for \(n_1 \cdot n_2 = 15\) seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{pmatrix}1\\2\\5\end{pmatrix}\times\begin{pmatrix}2\\-1\\3\end{pmatrix}=\begin{pmatrix}11\\7\\-5\end{pmatrix}\) | M1, A1 | M1 requires evidence of method for cross product or at least 2 correct values calculated |
| (eg) \(x=0 \Rightarrow 2y+5z=12, -y+3z=5 \Rightarrow y=1, z=2\) | M1 | Or any valid point e.g. \((-11/7, 0, 19/7)\), \((22/5, 19/5, 0)\) |
| \(\mathbf{r}=\begin{pmatrix}0\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}11\\7\\-5\end{pmatrix}\) | A1 | oe vector form; must have full equation including '\(\mathbf{r}=\)' |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\cos\theta = \frac{\begin{vmatrix}\begin{pmatrix}1\\2\\5\end{pmatrix}\cdot\begin{pmatrix}2\\-1\\3\end{pmatrix}\end{vmatrix}}{\sqrt{1^2+2^2+5^2}\sqrt{2^2+(-1)^2+3^2}} = \frac{15}{\sqrt{30}\sqrt{14}}$ | M1, A1 | Accept unsimplified |
| $\theta = 0.750$ or $43.0°$ | A1 | If zero, then sc1 for $n_1 \cdot n_2 = 15$ seen |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{pmatrix}1\\2\\5\end{pmatrix}\times\begin{pmatrix}2\\-1\\3\end{pmatrix}=\begin{pmatrix}11\\7\\-5\end{pmatrix}$ | M1, A1 | M1 requires evidence of method for cross product or at least 2 correct values calculated |
| (eg) $x=0 \Rightarrow 2y+5z=12, -y+3z=5 \Rightarrow y=1, z=2$ | M1 | Or any valid point e.g. $(-11/7, 0, 19/7)$, $(22/5, 19/5, 0)$ |
| $\mathbf{r}=\begin{pmatrix}0\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}11\\7\\-5\end{pmatrix}$ | A1 | oe vector form; must have full equation including '$\mathbf{r}=$' |
---1 Two planes have equations
$$x + 2 y + 5 z = 12 \text { and } 2 x - y + 3 z = 5$$
(i) Find the acute angle between the planes.\\
(ii) Find a vector equation of the line of intersection of the planes.
\hfill \mbox{\textit{OCR FP3 2013 Q1 [7]}}