| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Line intersection verification |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring standard techniques: finding a direction vector and vector equation for Lā, then checking if lines intersect by solving simultaneous equations. While it involves multiple steps, the methods are routine for C4 level with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\mathbf{r} = (2\mathbf{i}-3\mathbf{j}+5\mathbf{k}) + t(3\mathbf{i}-\mathbf{j}+5\mathbf{k})\) or equivalent | M1 | For either point + t(diff between vectors); Completely correct including \(\mathbf{r}\); For point + (\(s\) or \(t\)) direction vector |
| \((s,t) = (\frac{3}{5},\frac{5}{5})\) or \((\frac{22}{11},\frac{11}{11})\) or similar | M1 A1 2 | For solving any relevant pair of eqns; For both parameters correct |
| (ii) \(L(2)(\mathbf{r}) = 3\mathbf{i}+2\mathbf{j}-9\mathbf{k}+s(4\mathbf{i}-4\mathbf{j}+5\mathbf{k})\) or \((s\mathbf{r} = \mathbf{a} + \mathbf{lb})\) | M1 | For point + (\(s\) or \(t\)) direction vector; For 2/3 eqns with 2 different parameters |
| \(2+3t=3+4s,\,-3-t=2-4s,\,5+5t=ā9+5s\) | M1 | For solving any relevant pair of eqns; For both parameters correct |
| \(t=3\) or \(s=2\) | A1 A1 3 | For substituting relevant pair of eqns into new 3rd eqn and solving to find 'a' |
| Basic check other eqn & interp \(\sqrt{\_}\) | M1 | For substituting (\(t,s\)) into eqn of \(AB\) or \(OT\) and produce \(3\mathbf{i}+6\mathbf{j}-3\mathbf{k}\) |
(i) $\mathbf{r} = (2\mathbf{i}-3\mathbf{j}+5\mathbf{k}) + t(3\mathbf{i}-\mathbf{j}+5\mathbf{k})$ or equivalent | M1 | For either point + t(diff between vectors); Completely correct including $\mathbf{r}$; For point + ($s$ or $t$) direction vector
$(s,t) = (\frac{3}{5},\frac{5}{5})$ or $(\frac{22}{11},\frac{11}{11})$ or similar | M1 A1 2 | For solving any relevant pair of eqns; For both parameters correct
(ii) $L(2)(\mathbf{r}) = 3\mathbf{i}+2\mathbf{j}-9\mathbf{k}+s(4\mathbf{i}-4\mathbf{j}+5\mathbf{k})$ or $(s\mathbf{r} = \mathbf{a} + \mathbf{lb})$ | M1 | For point + ($s$ or $t$) direction vector; For 2/3 eqns with 2 different parameters
$2+3t=3+4s,\,-3-t=2-4s,\,5+5t=ā9+5s$ | M1 | For solving any relevant pair of eqns; For both parameters correct
$t=3$ or $s=2$ | A1 A1 3 | For substituting relevant pair of eqns into new 3rd eqn and solving to find 'a'
Basic check other eqn & interp $\sqrt{\_}$ | M1 | For substituting ($t,s$) into eqn of $AB$ or $OT$ and produce $3\mathbf{i}+6\mathbf{j}-3\mathbf{k}$ | B1 5 |3 The line $L _ { 1 }$ passes through the points $( 2 , - 3,1 )$ and $( - 1 , - 2 , - 4 )$. The line $L _ { 2 }$ passes through the point $( 3,2 , - 9 )$ and is parallel to the vector $4 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }$.\\
(i) Find an equation for $L _ { 1 }$ in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$.\\
(ii) Prove that $L _ { 1 }$ and $L _ { 2 }$ are skew.
\hfill \mbox{\textit{OCR C4 Q3 [7]}}