| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Parallel and perpendicular lines |
| Difficulty | Standard +0.3 This is a straightforward application of standard vector line techniques: checking if lines intersect (solving simultaneous equations) and testing for parallel/perpendicular relationships using direction vectors. While it requires multiple steps and careful algebraic manipulation, the methods are routine for C4 level with no novel insight required. Slightly easier than average due to the predictable structure and clear signposting of what to check. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1-s=2+t\); \(4+2s=8+3t\); \(1+2s=2+5t\) | B1 | For all three equations; NB third equation may appear later, or with values already substituted; or M1 for one value (of \(s\) or \(t\)) found from one pair of equations |
| Value of either \(s\) or \(t\) obtained from valid method | M1 | eqns (i) and (ii): \(s=0.2\), \(t=-1.2\); A1 for substitution of this value (of \(s\) or \(t\)) in third equation and obtaining the other parameter |
| Correct pair of values | A1 | eqns (i) and (iii): \(s=-\frac{4}{7}\), \(t=-\frac{3}{7}\); eqns (ii) and (iii): \(s=4.25\), \(t=1.5\); NB \((0.2,-0.12)\) or \((-\frac{4}{7},-\frac{12}{7})\) or \((4.25,-5.25)\) if \(s\) found first; \((-2.5,-1.2)\) or \((\frac{19}{14},-\frac{3}{7})\) or \((-2.5,1.5)\) if \(t\) found first |
| eg \(1+2\times0.2\neq2+5\times-1.2\) oe isw; NB A0 for \(1+2\times0.2=2+5\times-1.2\) unless clarified by suitable comment | A1 | Correct substitution of correct values in correct equation; or find same parameter from second pair of equations; A1 for correct demonstration of inconsistency; NB clear statement needed if two different values of same parameter found |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2\mathbf{i}-4\mathbf{j}-4\mathbf{k}=-2(-\mathbf{i}+2\mathbf{j}+2\mathbf{k})\) oe | B1 | Allow equivalent in words, but scale factors must be correct; eg direction of \(A\) is \(-\frac{1}{2}\times\) direction of \(C\) |
| eg line \(A\) goes through \((1,4,1)\) but line \(C\) goes through \((1,15,11)\), so they do not coincide so the lines are parallel; eg demonstration of different \(y\) or \(z\) values on each line for (say) \(x=1\), so lines are parallel | B1 | |
| [2] |
# Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1-s=2+t$; $4+2s=8+3t$; $1+2s=2+5t$ | B1 | For all three equations; NB third equation may appear later, or with values already substituted; or M1 for one value (of $s$ or $t$) found from one pair of equations |
| Value of either $s$ or $t$ obtained from valid method | M1 | eqns (i) and (ii): $s=0.2$, $t=-1.2$; A1 for substitution of this value (of $s$ or $t$) in third equation and obtaining the other parameter |
| Correct pair of values | A1 | eqns (i) and (iii): $s=-\frac{4}{7}$, $t=-\frac{3}{7}$; eqns (ii) and (iii): $s=4.25$, $t=1.5$; NB $(0.2,-0.12)$ or $(-\frac{4}{7},-\frac{12}{7})$ or $(4.25,-5.25)$ if $s$ found first; $(-2.5,-1.2)$ or $(\frac{19}{14},-\frac{3}{7})$ or $(-2.5,1.5)$ if $t$ found first |
| eg $1+2\times0.2\neq2+5\times-1.2$ oe isw; NB A0 for $1+2\times0.2=2+5\times-1.2$ unless clarified by suitable comment | A1 | Correct substitution of correct values in correct equation; or find same parameter from second pair of equations; A1 for correct demonstration of inconsistency; NB clear statement needed if two different values of same parameter found |
| **[4]** | | |
# Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2\mathbf{i}-4\mathbf{j}-4\mathbf{k}=-2(-\mathbf{i}+2\mathbf{j}+2\mathbf{k})$ oe | B1 | Allow equivalent in words, but scale factors must be correct; eg direction of $A$ is $-\frac{1}{2}\times$ direction of $C$ |
| eg line $A$ goes through $(1,4,1)$ but line $C$ goes through $(1,15,11)$, so they do not coincide so the lines are parallel; eg demonstration of different $y$ or $z$ values on each line for (say) $x=1$, so lines are parallel | B1 | |
| **[2]** | | |5 The equations of three lines are as follows.
$$\begin{array} { l l }
\text { Line } A : & \mathbf { r } = \mathbf { i } + 4 \mathbf { j } + \mathbf { k } + s ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \\
\text { Line } B : & \mathbf { r } = 2 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } + t ( \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) \\
\text { Line } C : & \mathbf { r } = - \mathbf { i } + 19 \mathbf { j } + 15 \mathbf { k } + u ( 2 \mathbf { i } - 4 \mathbf { j } - 4 \mathbf { k } )
\end{array}$$
(i) Show that lines $A$ and $B$ are skew.\\
(ii) Determine, giving reasons, the geometrical relationship between lines $A$ and $C$.
\hfill \mbox{\textit{OCR C4 2014 Q5 [6]}}