| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Line intersection verification |
| Difficulty | Standard +0.3 This is a structured multi-part question with clear signposting ('show that') that guides students through standard vector techniques: verifying a point lies on a line, checking perpendicularity via dot product, finding line intersection, and calculating distance ratios. While it requires multiple steps and careful algebraic manipulation, each part uses routine C4 methods without requiring novel insight or problem-solving strategies beyond textbook exercises. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Full justification that \(t = -1\). May be 'by inspection'. [No equations not satisfied by \(t=-1\) to be shown] | B1 | No other \(t\) to be mentioned |
| Consider scalar product \(\begin{pmatrix}-3\\0\\6\end{pmatrix}\cdot\begin{pmatrix}2\\2\\1\end{pmatrix}\) | M1 | |
| Show \(-6 + (0) + 6 = 0\) and state perpendicularity | A1 | |
| [If \(\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{ | \mathbf{a} |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(\mathbf{r} = v(-3\mathbf{i}+6\mathbf{k})\) and \(\ell_2\) | *M1 | or \((-3\mathbf{i}+6\mathbf{k}) + v(-3\mathbf{i}+6\mathbf{k})\) |
| Attempt to produce at least two relevant equations | M1dep* | |
| Solve two equations and produce \((v,s) = \left(\frac{1}{3}, -3\right)\) | A1 | \((v,s) = \left(-\frac{2}{3}, -3\right)\) |
| Demonstrate clearly that these satisfy the third equation | B1 | Numerical proof required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Method for finding \( | \overrightarrow{OB} | \) or \( |
| \( | \overrightarrow{OB} | = \sqrt{5}\) or \( |
| Obtain \(3:2\) | A1 | Answer \(3:2\) WW \(\rightarrow\) B3 |
## Question 10:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Full justification that $t = -1$. May be 'by inspection'. [No equations not satisfied by $t=-1$ to be shown] | B1 | No other $t$ to be mentioned |
| Consider scalar product $\begin{pmatrix}-3\\0\\6\end{pmatrix}\cdot\begin{pmatrix}2\\2\\1\end{pmatrix}$ | M1 | |
| Show $-6 + (0) + 6 = 0$ and state perpendicularity | A1 | |
| [If $\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$ quoted, ignore accuracy of work involving $|\mathbf{a}|$ and $|\mathbf{b}|$] | [3] | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $\mathbf{r} = v(-3\mathbf{i}+6\mathbf{k})$ and $\ell_2$ | *M1 | or $(-3\mathbf{i}+6\mathbf{k}) + v(-3\mathbf{i}+6\mathbf{k})$ |
| Attempt to produce at least two relevant equations | M1dep* | |
| Solve two equations and produce $(v,s) = \left(\frac{1}{3}, -3\right)$ | A1 | $(v,s) = \left(-\frac{2}{3}, -3\right)$ |
| Demonstrate clearly that these satisfy the third equation | B1 | Numerical proof required |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Method for finding $|\overrightarrow{OB}|$ or $|\overrightarrow{OA}|$ or $|\overrightarrow{AB}|$ | M1 | Method for finding $\overrightarrow{OB}$ or $\overrightarrow{BO}$ or $\overrightarrow{AB}$ or $\overrightarrow{BA}$ |
| $|\overrightarrow{OB}| = \sqrt{5}$ or $|\overrightarrow{OA}| = \sqrt{45}$ or $|\overrightarrow{BA}| = \sqrt{20}$ | A1 | $\overrightarrow{OB} = \begin{pmatrix}-1\\0\\2\end{pmatrix}$ or $\overrightarrow{BA} = \begin{pmatrix}-2\\0\\4\end{pmatrix}$ |
| Obtain $3:2$ | A1 | Answer $3:2$ WW $\rightarrow$ B3 |10 Lines $l _ { 1 }$ and $l _ { 2 }$ have vector equations
$$\mathbf { r } = - \mathbf { i } + 2 \mathbf { j } + 7 \mathbf { k } + t ( 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } + 9 \mathbf { j } - 4 \mathbf { k } + s ( \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )$$
respectively. The point $A$ has coordinates ( $- 3,0,6$ ) relative to the origin $O$.\\
(i) Show that $A$ lies on $l _ { 1 }$ and that $O A$ is perpendicular to $l _ { 1 }$.\\
(ii) Show that the line through $O$ and $A$ intersects $l _ { 2 }$.\\
(iii) Given that the point of intersection in part (ii) is $B$, find the ratio $| \overrightarrow { O A } | : | \overrightarrow { B A } |$.
\section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
\hfill \mbox{\textit{OCR C4 2012 Q10 [10]}}