OCR MEI C4 — Question 4 5 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVectors 3D & Lines
TypeShow lines intersect and find intersection point
DifficultyStandard +0.3 This is a straightforward line intersection problem requiring students to equate components and solve simultaneous equations. While it involves 3D vectors, the method is standard and the question explicitly tells students the lines meet (removing the need to verify consistency), making it slightly easier than average.
Spec1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation4.04a Line equations: 2D and 3D, cartesian and vector forms
4 Show that the straight lines with equations \(\mathbf { r } = \begin{array} { r r r } 2 & + \lambda & 0 \\ 4 & & 1 \end{array}\) and \(\mathbf { r } = \quad + \mu \quad\) meet.
Find their point of intersection.
Question 4:
AnswerMarks Guidance
\(\begin{pmatrix}4+3\lambda\\2\\4+\lambda\end{pmatrix} = \begin{pmatrix}-1-\mu\\4+\mu\\9+3\mu\end{pmatrix}\)
\(4+3\lambda = -1-\mu \quad (1)\)M1 equating components
\(2 = 4+\mu \quad (2)\)
\(4+\lambda = 9+3\mu \quad (3)\)
\((2)\Rightarrow \mu = -2\)B1 \(\mu = -2\)
\((1)\Rightarrow 4+3\lambda = -1+2 \Rightarrow \lambda = -1\)A1 \(\lambda = -1\)
\((3)\Rightarrow 4+(-1) = 9+3\times(-2)\), so consistentA1 checking third component
Point of intersection is \((1, 2, 3)\)A1 dependent on all previous marks being obtained
[5] 
## Question 4:

$\begin{pmatrix}4+3\lambda\\2\\4+\lambda\end{pmatrix} = \begin{pmatrix}-1-\mu\\4+\mu\\9+3\mu\end{pmatrix}$ | |

$4+3\lambda = -1-\mu \quad (1)$ | M1 | equating components
$2 = 4+\mu \quad (2)$ | |
$4+\lambda = 9+3\mu \quad (3)$ | |
$(2)\Rightarrow \mu = -2$ | B1 | $\mu = -2$
$(1)\Rightarrow 4+3\lambda = -1+2 \Rightarrow \lambda = -1$ | A1 | $\lambda = -1$
$(3)\Rightarrow 4+(-1) = 9+3\times(-2)$, so consistent | A1 | checking third component
Point of intersection is $(1, 2, 3)$ | A1 | dependent on all previous marks being obtained
| [5] |

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4 Show that the straight lines with equations $\mathbf { r } = \begin{array} { r r r } 2 & + \lambda & 0 \\ 4 & & 1 \end{array}$ and $\mathbf { r } = \quad + \mu \quad$ meet.\\
Find their point of intersection.

\hfill \mbox{\textit{OCR MEI C4  Q4 [5]}}
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