| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Perpendicularity conditions |
| Difficulty | Moderate -0.8 This is a straightforward application of the scalar product formula requiring direct recall and basic calculation. Part (i) involves computing a·b, finding magnitudes, and using the cosine formula—all standard steps. Part (ii) is even simpler, setting the scalar product to zero and solving for k. Both parts are routine textbook exercises with no problem-solving insight required, making this easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication4.04c Scalar product: calculate and use for angles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{pmatrix}1\\2\\-1\end{pmatrix}\cdot\begin{pmatrix}3\\-1\\2\end{pmatrix}=3-2-2=-1\) | M1 | |
| \(n_1 n_2=\sqrt{6}\sqrt{14}\cos\theta \Rightarrow \cos\theta=\frac{-1}{\sqrt{84}}\approx -0.1091\) | M1, A1 | |
| \(\Rightarrow \theta=96.3°\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{a}\cdot\mathbf{b}=0 \Rightarrow 3-2-k=0 \Rightarrow k=1\) | M1, A1 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{pmatrix}1\\2\\-1\end{pmatrix}\cdot\begin{pmatrix}3\\-1\\2\end{pmatrix}=3-2-2=-1$ | M1 | |
| $n_1 n_2=\sqrt{6}\sqrt{14}\cos\theta \Rightarrow \cos\theta=\frac{-1}{\sqrt{84}}\approx -0.1091$ | M1, A1 | |
| $\Rightarrow \theta=96.3°$ | A1 | |
**Total: 4 marks**
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{a}\cdot\mathbf{b}=0 \Rightarrow 3-2-k=0 \Rightarrow k=1$ | M1, A1 | |
**Total: 2 marks**
---4 You are given that $\mathbf { a } = \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right)$ and $\mathbf { b } = \left( \begin{array} { c } 3 \\ - 1 \\ k \end{array} \right)$.\\
(i) Find the angle between $\mathbf { a }$ and $\mathbf { b }$ when $k = 2$.\\
(ii) Find the value of $k$ such that $\mathbf { a }$ and $\mathbf { b }$ are perpendicular.
\hfill \mbox{\textit{OCR MEI C4 Q4 [6]}}