Advanced level 2025 Centre Regional Mock Pure mathematics with mechanics 3

Advanced level 2025 Centre Regional Mock Pure mathematics with mechanics 3

Advanced level 2025 Centre Regional Mock Pure mathematics with mechanics 3

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1. A particle of mass 2 kg has a position vector given by $\mathbf{r}=[(cos2t+1)\mathbf{i}+(sin2t-1)\mathbf{j}]$ m.

a) Find the magnitude of its acceleration when $t=\pi$. b) Find the magnitude of the force acting on the particle when $t=\pi$. c) For $0<t<\pi$, determine the time when the interval is parallel to the vector $\mathbf{i}-\mathbf{j}$.

2. A car of mass 1500 kg with its engine working at a constant rate of 240 kW, has a constant resistance $R$ which is linearly related with the speed, $v$.

On the level road, the car attains a maximum speed of 120 m/s, and on an inclined slope with angle $\theta$, the car attains a steady speed of 80 m/s.

a) Determine the resistance $R$ as a function of the linear speed $v$. b) Show that when $v=50m/s$, the resistance is 1650 N.

3. A particle starts from rest and moves such that its acceleration $a$ is given by a=s1​, where $s$ is its displacement in meters from the initial position.

Determine the velocity of the particle when it has traveled a distance of 4 meters.

4. A uniform beam PQ of length 2 m and mass $m$ kg is free to rotate in a vertical plane about a hinge at P. It is supported in a horizontal position by a string attached to the beam at the point R, a distance of 23​ m from P, and to a point S vertically above P. The string is inclined at an angle of 60∘ to the horizontal.

a) Show that the magnitude of the tension $T$ in the string is 853​mg​. b) Find, in terms of $m$, the magnitude of the reaction at the hinge P. c) Show that the angle, to the nearest degree, between the reaction at the hinge and the beam PQ is 56°.