A particle P of mass mkg is attached to one end of a light inextensible string of length 2.6 m. The other end of the string is attached to a fixed point on a horizontal ceiling, and the string is taut. The particle is held in equilibrium by a force of magnitude 35N, acting in a vertical plane which is perpendicular to the ceiling and contains the string. The force acts in a direction perpendicular to the string (see diagram). The tension in the string is TN and the vertical distance of P from the ceiling is 2.4 m. [Figure 3.1]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [1] Resolving in any direction to form an equation Resolving either vertically/horizontally or parallel/perpendicular to the string. Correct number of terms, allow sign errors, allow sin/cos mix; forces that need resolving should be resolved. M0 for using an angle other than 23 (or better) or 67 (or better).
2. [1] Allow Tcosθ+35sinθ=mg with θ = 23 or better (22.61986495) substituted OR T = mg cosθ with θ = 23 or better (22.61986495) substituted. Allow with their T or m if already found.
3. [1] Allow Tsinθ=35cosθ with θ = 23 or better (22.61986495) substituted OR mg sinθ=35 with θ = 23 or better (22.61986495) substituted.
4. [1] T=84 and m=9.1 Allow T =84.1 and m = 9.11. Allow AWRT 84.0 for T and 9.10 for m.