A-LevelMathematicsMechanics9709/41 · May/June 2023 · Q59 Marks

Four coplanar forces act at a point. The magnitudes of the forces are F N, 10 N, 50 N and 40 N. The directions of the forces are as shown in the diagram. [Figure 5.1]

(a)Given that the forces are in equilibrium, find the value of F and the value of θ.[6]
(b)Given instead that F = 10√2 and θ = 45, find the direction and the exact magnitude the resultant force.[3]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[M1] Resolving either direction. 3 terms; allow sign errors and allow sin/cos mix. Must be an equation with either = 0 or with an attempt to balance forces.
[A1] [F sinθ=50-20√3=15.358...]
[A1] [F cosθ=10]
[M1] Attempt to solve for θ; one missing term in total θ = tan⁻¹1.535898....
[M1] Attempt to solve for F: one missing term in total.
[A1] Both correct (18.327530..., 56.932462...).
[B1] Allow non-exact values for √2 etc. in correct expression.
[B1] Allow non-exact values for √2 etc. in correct expression. Could be implied by correct answer.
[B1] Allow vertically downwards, south, 180°, negative y-direction. Resultant force must be exact and positive (so 20√3-40 is B0).

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.