| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Show lines intersect and find intersection point |
| Difficulty | Standard +0.3 This is a standard two-part vector line question requiring systematic equation solving to find intersection (equating components and checking consistency) and then applying the scalar product formula for angle between direction vectors. While it involves multiple steps and careful algebraic manipulation, it follows a well-practiced procedure with no novel insight required, making it slightly easier than average. |
| Spec | 4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Produce (at least 2) relevant equations | M1 | e.g. \(1+2\lambda = 6+\mu\), \(2+\lambda = 8+4\mu\), \(3\lambda = 1-5\mu\) |
| Eliminate either \(\lambda\) or \(\mu\) from 2 of them and solve for the other | M1 | soi by correct \((\lambda, \mu)\) |
| \(\lambda = 2\) and \(\mu = -1\) cao | A1 | or e.g. \(\lambda = 2\) from 2 different pairs |
| Check that \((\lambda, \mu) = (2, -1)\) satisfies all equations | B1 | This must be convincing; check unusual arguments; dep previous M1M1A1 earned |
| \(P\) is \((5, 4, 6)\) cao | A1 | Allow any reasonable vector notation |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Using \(\begin{pmatrix}2\\1\\3\end{pmatrix}\) and \(\begin{pmatrix}1\\4\\-5\end{pmatrix}\) | M1 | i.e. correct parts for direction vectors |
| Using \(\cos\theta = \frac{\mathbf{a.b}}{ | \mathbf{a} | |
| \(68.2°\ldots\) (not \(111.8\ldots\)) | A1 | or \(1.19\) radians |
| [3] |
# Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Produce (at least 2) relevant equations | M1 | e.g. $1+2\lambda = 6+\mu$, $2+\lambda = 8+4\mu$, $3\lambda = 1-5\mu$ |
| Eliminate either $\lambda$ or $\mu$ from 2 of them and solve for the other | M1 | soi by correct $(\lambda, \mu)$ |
| $\lambda = 2$ and $\mu = -1$ cao | A1 | or e.g. $\lambda = 2$ from 2 different pairs |
| Check that $(\lambda, \mu) = (2, -1)$ satisfies all equations | B1 | This must be convincing; check unusual arguments; dep previous M1M1A1 earned |
| $P$ is $(5, 4, 6)$ cao | A1 | Allow any reasonable vector notation |
| **[5]** | | |
---
# Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Using $\begin{pmatrix}2\\1\\3\end{pmatrix}$ and $\begin{pmatrix}1\\4\\-5\end{pmatrix}$ | M1 | i.e. correct parts for direction vectors |
| Using $\cos\theta = \frac{\mathbf{a.b}}{|\mathbf{a}||\mathbf{b}|}$ giving value $\frac{n}{\sqrt{a}\sqrt{b}}$ | M1 | For any 2 meaningful vectors; expect $\frac{-9}{\sqrt{14}\sqrt{42}}$ |
| $68.2°\ldots$ (not $111.8\ldots$) | A1 | or $1.19$ radians |
| **[3]** | | |
---4 The equations of two lines are
$$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \text { and } \mathbf { r } = 6 \mathbf { i } + 8 \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } ) .$$
(i) Show that these lines meet, and find the coordinates of the point of intersection.\\
(ii) Find the acute angle between these lines.
\hfill \mbox{\textit{OCR C4 2013 Q4 [8]}}