| Exam Board | OCR MEI |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Geometric properties using vectors |
| Difficulty | Moderate -0.8 This is a straightforward vectors question requiring only basic operations: calculating magnitudes using Pythagoras, vector addition/subtraction, and recognizing parallel vectors through scalar multiples. Part (iii) involves simple geometry observation (perpendicular vectors). All techniques are routine for M1 level with no problem-solving insight needed, making it easier than average but not trivial due to multiple parts. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \( | \mathbf{p} | = \sqrt{8^2 + 1^2}\) |
| \( | \mathbf{p} | = \sqrt{65}\) |
| \( | \mathbf{q} | = \sqrt{4^2 + (-7)^2} = \sqrt{65}\); they are equal |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\mathbf{p} + \mathbf{q} = 12\mathbf{i} - 6\mathbf{j}\) | M1 | |
| \(\mathbf{p} + \mathbf{q} = 6(2\mathbf{i} - \mathbf{j})\), so \(\mathbf{p} + \mathbf{q}\) is parallel to \(2\mathbf{i} - \mathbf{j}\) | E1 | Accept argument based on gradients being equal. "Parallel" may be implied |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Diagram with \(\mathbf{p} + \mathbf{q}\) drawn correctly | B1 | One mark for each of \(\mathbf{p}+\mathbf{q}\) and \(\mathbf{p}-\mathbf{q}\) drawn correctly |
| Diagram with \(\mathbf{p} - \mathbf{q}\) drawn correctly | B1 | SC1 if arrows missing or incorrect from otherwise correct vectors |
| The angle is \(90°\) | B1 | cao |
## Question 3:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $|\mathbf{p}| = \sqrt{8^2 + 1^2}$ | M1 | For applying Pythagoras' theorem |
| $|\mathbf{p}| = \sqrt{65}$ | A1 | |
| $|\mathbf{q}| = \sqrt{4^2 + (-7)^2} = \sqrt{65}$; they are equal | A1 | Condone no explicit statement that they are equal |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{p} + \mathbf{q} = 12\mathbf{i} - 6\mathbf{j}$ | M1 | |
| $\mathbf{p} + \mathbf{q} = 6(2\mathbf{i} - \mathbf{j})$, so $\mathbf{p} + \mathbf{q}$ is parallel to $2\mathbf{i} - \mathbf{j}$ | E1 | Accept argument based on gradients being equal. "Parallel" may be implied |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Diagram with $\mathbf{p} + \mathbf{q}$ drawn correctly | B1 | One mark for each of $\mathbf{p}+\mathbf{q}$ and $\mathbf{p}-\mathbf{q}$ drawn correctly |
| Diagram with $\mathbf{p} - \mathbf{q}$ drawn correctly | B1 | SC1 if arrows missing or incorrect from otherwise correct vectors |
| The angle is $90°$ | B1 | cao |
---3 The vectors $\mathbf { p }$ and $\mathbf { q }$ are given by
$$\mathbf { p } = 8 \mathbf { i } + \mathbf { j } \text { and } \mathbf { q } = 4 \mathbf { i } - 7 \mathbf { j } .$$
(i) Show that $\mathbf { p }$ and $\mathbf { q }$ are equal in magnitude.\\
(ii) Show that $\mathbf { p } + \mathbf { q }$ is parallel to $2 \mathbf { i } - \mathbf { j }$.\\
(iii) Draw $\mathbf { p } + \mathbf { q }$ and $\mathbf { p } - \mathbf { q }$ on the grid.
Write down the angle between these two vectors.
\hfill \mbox{\textit{OCR MEI M1 Q3 [8]}}