| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2009 |
| Session | January |
| Marks | 10 |
| 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 lines question requiring routine techniques: finding a second line equation from two points, solving simultaneous equations for intersection, and using the scalar product formula for angles. While multi-step, each component is textbook-standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10c Magnitude and direction: of vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt direction vector between the 2 given points | M1 | |
| State eqn of line using format \((\mathbf{r}) = (\text{either end}) + s(\text{dir vec})\) | M1 | \(s\) can be \(t\) |
| Produce 2/3 eqns containing \(t\) and \(s\) | M1 | 2 different parameters |
| Solve giving \(t=3\), \(s=-2\) or \(2\) or \(-1\) or \(1\) | A1 | |
| Show consistency | B1 | |
| Point of intersection \(= (5,9,-1)\) | A1 (6) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct method for scalar product of 'any' 2 vectors | M1 | Vectors from this question |
| Correct method for magnitude of 'any' vector | M1 | Vector from this question |
| Use \(\cos\theta = \frac{\mathbf{a.b}}{ | \mathbf{a} | |
| \(62.2\ (62.188157\ldots)\ \ \ 1.09\ (1.0853881)\) | A1 (4) | |
| Total: 10 |
# Question 7(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt direction vector between the 2 given points | M1 | |
| State eqn of line using format $(\mathbf{r}) = (\text{either end}) + s(\text{dir vec})$ | M1 | $s$ can be $t$ |
| Produce 2/3 eqns containing $t$ and $s$ | M1 | 2 different parameters |
| Solve giving $t=3$, $s=-2$ or $2$ or $-1$ or $1$ | A1 | |
| Show consistency | B1 | |
| Point of intersection $= (5,9,-1)$ | A1 **(6)** | |
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# Question 7(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct method for scalar product of 'any' 2 vectors | M1 | Vectors from this question |
| Correct method for magnitude of 'any' vector | M1 | Vector from this question |
| Use $\cos\theta = \frac{\mathbf{a.b}}{|\mathbf{a}||\mathbf{b}|}$ for the correct 2 vectors $\begin{pmatrix}1\\4\\-2\end{pmatrix}$ & $\begin{pmatrix}2\\-1\\3\end{pmatrix}$ | M1 | Vects may be mults of dvs |
| $62.2\ (62.188157\ldots)\ \ \ 1.09\ (1.0853881)$ | A1 **(4)** | |
| **Total: 10** | | |
---7 (i) Show that the straight line with equation $\mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 4 \\ - 2 \end{array} \right)$ meets the line passing through ( $9,7,5$ ) and ( $7,8,2$ ), and find the point of intersection of these lines.\\
(ii) Find the acute angle between these lines.
\hfill \mbox{\textit{OCR C4 2009 Q7 [10]}}