| Exam Board | OCR MEI |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2005 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Forces as vectors |
| Difficulty | Moderate -0.8 This is a straightforward equilibrium problem requiring students to sum three force vectors to zero and solve for three unknowns, followed by a routine magnitude calculation. The concepts are basic (equilibrium means sum of forces = 0) and the arithmetic is simple, making this easier than average for A-level. |
| Spec | 3.03n Equilibrium in 2D: particle under forces |
| Answer | Marks |
|---|---|
| \(\begin{pmatrix} x \\ -7 \end{pmatrix} + \begin{pmatrix} 4 \\ y \end{pmatrix} + \begin{pmatrix} 5 \\ 4 \end{pmatrix} + \begin{pmatrix} 0 \\ -7 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}\) | M1 |
| Equating components gives: \(x = -9\), \(y = 3\), \(z = 12\) | A1 |
| A1 | |
| A1 | [Allow SC 2/4 if 9, -3, -12 obtained] |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| We need \(\sqrt{5^2 + 4^2 + (-7)^2}\) | M1 | |
| \(= \sqrt{90}\) or 9.48683... so 9.49 (3 s. f.) | A1 | Any reasonable accuracy |
| Total: 2 |
**Part (i)**
| $\begin{pmatrix} x \\ -7 \end{pmatrix} + \begin{pmatrix} 4 \\ y \end{pmatrix} + \begin{pmatrix} 5 \\ 4 \end{pmatrix} + \begin{pmatrix} 0 \\ -7 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$ | M1 | |
| Equating components gives: $x = -9$, $y = 3$, $z = 12$ | A1 | |
| | A1 | |
| | A1 | [Allow SC 2/4 if 9, -3, -12 obtained] |
| | | **Total: 4** |
**Part (ii)**
| We need $\sqrt{5^2 + 4^2 + (-7)^2}$ | M1 | |
| $= \sqrt{90}$ or 9.48683... so 9.49 (3 s. f.) | A1 | Any reasonable accuracy |
| | | **Total: 2** |
---3 A particle is in equilibrium when acted on by the forces $\left( \begin{array} { r } x \\ - 7 \\ z \end{array} \right) , \left( \begin{array} { r } 4 \\ y \\ - 5 \end{array} \right)$ and $\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)$, where the units are newtons.\\
(i) Find the values of $x , y$ and $z$.\\
(ii) Calculate the magnitude of $\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)$.
\hfill \mbox{\textit{OCR MEI M1 2005 Q3 [6]}}