| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Parallel and perpendicular lines |
| Difficulty | Standard +0.3 This is a standard C4 vectors question testing routine techniques: (i) angle between lines using dot product formula, (ii) parallel lines requiring direction vectors to be scalar multiples, (iii) intersection requiring solving simultaneous equations. All three parts use well-practiced methods with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(-6\mathbf{i} + 8\mathbf{j} - 2\mathbf{k}\) and \(\mathbf{i} + 3\mathbf{j} + 2\mathbf{k}\) only | M1 | of any two vectors (\(-6 + 24 - 4 = 14\)) |
| Correct method for scalar product | M1 | of any vector (\(\sqrt{36 + 64 + 4} = \sqrt{104}\) or \(\sqrt{1 + 9 + 4} = \sqrt{14}\)) |
| Correct method for magnitude | M1 | |
| 68 or 68.5 (68.47546); 1.2(0) (1.1951222) rad | A1 | [N.B. 61 (60.562) will probably have been generated by 5i - |
| (ii) Indication that relevant vectors are parallel | M1 | \(-6\mathbf{i} + 8\mathbf{j} - 2\mathbf{k}\) & \(3\mathbf{i} + c\mathbf{j} + \mathbf{k}\) with some indic of method of attack |
| \(c = -4\) | A1 | eg \(-6\mathbf{i} + 8\mathbf{j} - 2\mathbf{k} = \lambda(3\mathbf{i} + c\mathbf{j} + \mathbf{k})\) |
| \(c = -4\) WW \(\to\) B2 | ||
| (iii) Produce 2/3 equations containing \(t, u\) (& \(c\)) | M1 | eg \(3 + t = 2 + 3u, -8 + 3t = 1 + cu\) and \(2t = 3 + u\) |
| Solve the 2 equations not containing 'c' | M1 | |
| \(t = 2, u = 1\) | A1 | |
| Subst their \((t,u)\) into equation containing \(c\) | M1 | |
| \(c = -3\) | A1 | |
| Alternative method for final 4 marks | ||
| Solve two equations, one with 'c', for \(t\) and \(u\) in terms of \(c\), and substitute into third equation | (M2) | |
| \(c = -3\) | (A2) |
(i) Use $-6\mathbf{i} + 8\mathbf{j} - 2\mathbf{k}$ and $\mathbf{i} + 3\mathbf{j} + 2\mathbf{k}$ only | M1 | of any two vectors ($-6 + 24 - 4 = 14$)
Correct method for scalar product | M1 | of any vector ($\sqrt{36 + 64 + 4} = \sqrt{104}$ or $\sqrt{1 + 9 + 4} = \sqrt{14}$)
Correct method for magnitude | M1 |
68 or 68.5 (68.47546); 1.2(0) (1.1951222) rad | A1 | [N.B. 61 (60.562) will probably have been generated by 5i - | - 2k and 3i - 8j] | 4
(ii) Indication that relevant vectors are parallel | M1 | $-6\mathbf{i} + 8\mathbf{j} - 2\mathbf{k}$ & $3\mathbf{i} + c\mathbf{j} + \mathbf{k}$ with some indic of method of attack
$c = -4$ | A1 | eg $-6\mathbf{i} + 8\mathbf{j} - 2\mathbf{k} = \lambda(3\mathbf{i} + c\mathbf{j} + \mathbf{k})$ | 2
| | | $c = -4$ WW $\to$ B2
(iii) Produce 2/3 equations containing $t, u$ (& $c$) | M1 | eg $3 + t = 2 + 3u, -8 + 3t = 1 + cu$ and $2t = 3 + u$
Solve the 2 equations not containing 'c' | M1 |
$t = 2, u = 1$ | A1 |
Subst their $(t,u)$ into equation containing $c$ | M1 |
$c = -3$ | A1 | | 5
Alternative method for final 4 marks | |
Solve two equations, one with 'c', for $t$ and $u$ in terms of $c$, and substitute into third equation | (M2) |
$c = -3$ | (A2) |
---9 Lines $L _ { 1 } , L _ { 2 }$ and $L _ { 3 }$ have vector equations
$$\begin{aligned}
& L _ { 1 } : \mathbf { r } = ( 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) + s ( - 6 \mathbf { i } + 8 \mathbf { j } - 2 \mathbf { k } ) , \\
& L _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } - 8 \mathbf { j } ) + t ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) , \\
& L _ { 3 } : \mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + u ( 3 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) .
\end{aligned}$$
(i) Calculate the acute angle between $L _ { 1 }$ and $L _ { 2 }$.\\
(ii) Given that $L _ { 1 }$ and $L _ { 3 }$ are parallel, find the value of $c$.\\
(iii) Given instead that $L _ { 2 }$ and $L _ { 3 }$ intersect, find the value of $c$.
4
\hfill \mbox{\textit{OCR C4 2007 Q9 [11]}}