| Exam Board | OCR MEI |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Parallel or perpendicular vectors condition |
| Difficulty | Moderate -0.8 This is a straightforward vectors question requiring only basic operations: calculating magnitudes using Pythagoras (both give √65), adding vectors componentwise to show p+q = 12i-6j = 6(2i-j), and recognizing perpendicular vectors. All parts are routine recall and simple verification with no problem-solving or geometric insight needed. |
| 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 | 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 | 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 | Mark | Guidance |
| \(\mathbf{p}+\mathbf{q}\) drawn correctly | B1 | SC1 if arrows missing or incorrect from otherwise correct vectors |
| \(\mathbf{p}-\mathbf{q}\) drawn correctly | B1 | |
| The angle is \(90°\) | B1 | Cao |
# Question 5:
## Part (i)
| Answer | 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 | 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 | Mark | Guidance |
|--------|------|----------|
| $\mathbf{p}+\mathbf{q}$ drawn correctly | B1 | SC1 if arrows missing or incorrect from otherwise correct vectors |
| $\mathbf{p}-\mathbf{q}$ drawn correctly | B1 | |
| The angle is $90°$ | B1 | Cao |
---5 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 2012 Q5 [8]}}