| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with plane |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question requiring routine techniques: finding a plane equation from three points using cross product, substituting a line equation into the plane to find intersection, and using the formula for angle between line and plane. All methods are textbook procedures with no novel insight required, though it involves more steps than typical A-level questions. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane4.04f Line-plane intersection: find point |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\begin{pmatrix}2\\3\\6\end{pmatrix} - \begin{pmatrix}1\\2\\1\end{pmatrix} = \begin{pmatrix}1\\1\\5\end{pmatrix}\) and \(\begin{pmatrix}4\\-1\\2\end{pmatrix} - \begin{pmatrix}1\\-1\\2\end{pmatrix} = \begin{pmatrix}3\\-3\\1\end{pmatrix}\) vectors in plane; \(\begin{pmatrix}1\\1\\5\end{pmatrix} \times \begin{pmatrix}3\\-3\\1\end{pmatrix} = \begin{pmatrix}16\\14\\-6\end{pmatrix} = 2\begin{pmatrix}8\\7\\-3\end{pmatrix}\) | M1* | or multiple(s) |
| \(\mathbf{r} \cdot \begin{pmatrix}8\\7\\-3\end{pmatrix} = \begin{pmatrix}1\\1\\5\end{pmatrix} \cdot \begin{pmatrix}8\\7\\-3\end{pmatrix}\) | M1dep* | for M1, method shown or 2 correct elements |
| \(8x + 7y - 3z = 19\) | A1 | AEF (Cartesian) |
| [5] | ||
| (ii) \(x = -1 + 4\lambda\), \(y = -2 + 3\lambda\), \(z = 6 - 2\lambda\) | M1 | |
| \(8(-1+4\lambda) + 7(-2+3\lambda) - 3(6-2\lambda) = 19 \Rightarrow \lambda = 1\) | M1 | solves and attempts substitution |
| intersect at \((3, 1, 4)\) | A1 | Accept vector form |
| [3] | ||
| (iii) \(\cos(\alpha) = \frac{\begin{vmatrix}8\\7\\-3\end{vmatrix} \cdot \begin{vmatrix}4\\3\\-2\end{vmatrix}}{\sqrt{8^2+7^2+3^2}\sqrt{4^2+3^2+2^2}} = \frac{59}{\sqrt{122}\sqrt{29}}\) | M1* | can be implied by 7.3° or 0.13 or \(\cos\alpha = 0.9919\) seen |
| \(\theta = \frac{1}{2}\pi - \alpha\) | M1dep* | can use \(\sin\theta\) |
| \(\theta \approx 1.44\) or \(\theta \approx 82.7°\) | A1 | consistent use of degrees or radians |
| [3] |
**(i)** $\begin{pmatrix}2\\3\\6\end{pmatrix} - \begin{pmatrix}1\\2\\1\end{pmatrix} = \begin{pmatrix}1\\1\\5\end{pmatrix}$ and $\begin{pmatrix}4\\-1\\2\end{pmatrix} - \begin{pmatrix}1\\-1\\2\end{pmatrix} = \begin{pmatrix}3\\-3\\1\end{pmatrix}$ vectors in plane; $\begin{pmatrix}1\\1\\5\end{pmatrix} \times \begin{pmatrix}3\\-3\\1\end{pmatrix} = \begin{pmatrix}16\\14\\-6\end{pmatrix} = 2\begin{pmatrix}8\\7\\-3\end{pmatrix}$ | M1* | or multiple(s)
$\mathbf{r} \cdot \begin{pmatrix}8\\7\\-3\end{pmatrix} = \begin{pmatrix}1\\1\\5\end{pmatrix} \cdot \begin{pmatrix}8\\7\\-3\end{pmatrix}$ | M1dep* | for M1, method shown or 2 correct elements
$8x + 7y - 3z = 19$ | A1 | AEF (Cartesian)
| [5] |
**(ii)** $x = -1 + 4\lambda$, $y = -2 + 3\lambda$, $z = 6 - 2\lambda$ | M1 |
$8(-1+4\lambda) + 7(-2+3\lambda) - 3(6-2\lambda) = 19 \Rightarrow \lambda = 1$ | M1 | solves and attempts substitution
intersect at $(3, 1, 4)$ | A1 | Accept vector form
| [3] |
**(iii)** $\cos(\alpha) = \frac{\begin{vmatrix}8\\7\\-3\end{vmatrix} \cdot \begin{vmatrix}4\\3\\-2\end{vmatrix}}{\sqrt{8^2+7^2+3^2}\sqrt{4^2+3^2+2^2}} = \frac{59}{\sqrt{122}\sqrt{29}}$ | M1* | can be implied by 7.3° or 0.13 or $\cos\alpha = 0.9919$ seen
$\theta = \frac{1}{2}\pi - \alpha$ | M1dep* | can use $\sin\theta$
$\theta \approx 1.44$ or $\theta \approx 82.7°$ | A1 | consistent use of degrees or radians
| [3] |3 The plane $\Pi$ passes through the points $( 1,2,1 ) , ( 2,3,6 )$ and $( 4 , - 1,2 )$.\\
(i) Find a cartesian equation of the plane $\Pi$.
The line $l$ has equation $\mathbf { r } = \left( \begin{array} { r } - 1 \\ - 2 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 4 \\ 3 \\ - 2 \end{array} \right)$.\\
(ii) Find the coordinates of the point of intersection of $\Pi$ and $l$.\\
(iii) Find the acute angle between $\Pi$ and $l$.
\hfill \mbox{\textit{OCR FP3 2015 Q3 [11]}}