Coplanar forces of magnitudes 16N, 12N, 24N and 8N act at a point in the directions shown in the diagram. [Figure 2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

1. [M1] With correct number of relevant terms. Allow sin/cos mix. Allow sign errors. Do not allow 'forces to the left = forces to the right' e.g. 12cos30-8sin30=16sin30 unless subsequently 'corrected'.
2. [A1] Fy = ±(30-4√3) [=23.07...].
3. [A1] Fx = ±(6√3-12) [=-1.607...].
4. [M1] Attempt to find F. Must have correct number of relevant terms. (Forces must have or not have components as required). All forces resolved/not resolved as appropriate, but allow consistent sin/cos muddle. Allow use of their θ provided correctly derived from equations with the correct number of relevant terms.
5. [M1] Attempt to find θ. Must have correct number of relevant terms. (Forces must have or not have components as required). All forces resolved/not resolved as appropriate, but allow consistent sin/cos muddle. Allow upside down so tan⁻¹ (30-4√3)/(6√3-12). Allow use of their F provided correctly derived from equations with the correct number of relevant terms. Note: watch for use of sin⁻¹ (6√3-12)/(30-4√3) or sin⁻¹ (1.607)/(23.07) which leads to correct answer of angle 4.0° scores M0A0.
6. [A1] [23.1277...] Both correct Allow 4.0° but not simply 4° . [3.986...] Allow answers about the direction such as ‘Above the west', 'north of west' etc, or clockwise 183.99 from x axis, or resultant sketch with angle indicated. If not specified in working please check original diagram to see if direction specified there instead. Allow a bearing of 274.0°. Allow explanation of direction that could be drawn uniquely. Or e.g. 86.0° to left of the y-axis or 176.0° from the positive x-axis.