| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Line intersection: show lines are skew |
| Difficulty | Standard +0.3 This is a standard C4 vectors question testing routine techniques: finding a direction vector from two points, writing a vector equation, and proving lines are skew by showing they're not parallel and don't intersect. Part (i) is straightforward recall, while part (ii) requires systematic checking but follows a well-practiced method with no novel insight needed. Slightly above average due to the algebraic work in part (ii), but still a textbook exercise. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\mathbf{r} = (2\mathbf{i}-3\mathbf{j}+\mathbf{k})\) or \(\mathbf{r} = \mathbf{i}-2\mathbf{j}-4\mathbf{k}) + t(3\mathbf{i}+5\mathbf{k})\) | M1, A1 2 | For (either point) + t(diff betw vectors); Completely correct including \(\mathbf{r} =\). AEF |
| (ii) \(L(2)(\mathbf{r}) = 3\mathbf{i}+2\mathbf{j}-3\mathbf{k}+s(4\mathbf{i} - 4\mathbf{j} + 5\mathbf{k})\) and \(L(1) \& L(2)\) must be of form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\) | M1 | For point + (s or t) direction vector |
| \(2+3t=3+4s\), \(-3-t=2-4s\), \(1+5t=-9+5s\) | M1 | For 2/3 eqns with 2 different parameters |
| (s,t) = \((+/-3,2)\) or \((+/-1,+1)\) or \((+/- 4,2)\) or \((+/-4,2)\) or \((0,1)\) or \((-/+8,-7)\) | M1, A1 | For solving any relevant pair of eqns; For both parameters correct |
| Basic check other eqn & interp ∨ | B1 5 |
(i) $\mathbf{r} = (2\mathbf{i}-3\mathbf{j}+\mathbf{k})$ or $\mathbf{r} = \mathbf{i}-2\mathbf{j}-4\mathbf{k}) + t(3\mathbf{i}+5\mathbf{k})$ | M1, A1 2 | For (either point) + t(diff betw vectors); Completely correct including $\mathbf{r} =$. **AEF**
(ii) $L(2)(\mathbf{r}) = 3\mathbf{i}+2\mathbf{j}-3\mathbf{k}+s(4\mathbf{i} - 4\mathbf{j} + 5\mathbf{k})$ and $L(1) \& L(2)$ must be of form $\mathbf{r} = \mathbf{a} + t\mathbf{b}$ | M1 | For point + (s or t) direction vector
$2+3t=3+4s$, $-3-t=2-4s$, $1+5t=-9+5s$ | M1 | For 2/3 eqns with 2 different parameters
(s,t) = $(+/-3,2)$ or $(+/-1,+1)$ or $(+/- 4,2)$ or $(+/-4,2)$ or $(0,1)$ or $(-/+8,-7)$ | M1, A1 | For solving any relevant pair of eqns; For both parameters correct
Basic check other eqn & interp ∨ | B1 5 | | **7**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 $\mathbf{4i} - \mathbf{4j} + \mathbf{5k}$.
\begin{enumerate}[label=(\roman*)]
\item Find an equation for $L_1$ in the form $\mathbf{r} = \mathbf{a} + t\mathbf{b}$. [2]
\item Prove that $L_1$ and $L_2$ are skew. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR C4 2005 Q3 [7]}}