Definite integration

Model 2 is a simplified model for the sign. In this model, the lamina is in the shape of a trapezium with vertices at the same points O, A, B and C as in Model 1 (see diagram).

The centre of mass of the lamina in Model 2 is at the point with coordinates ( .

Find the value of and the value of

The sign is placed on a rough plane inclined at 50° to the horizontal. The line OA rests on the plane with O below A. The line OA lies along a line of greatest slope of the plane.

You may assume that all forces acting on the sign lie in the vertical plane containing OA and the plane is sufficiently rough to prevent sliding.

You are given that and from Model 1 are both within 0.04 m of the actual coordinates of the centre of mass of the sign.

Determine whether the sign will topple. Justify your answer.

Use your answer to part (b) to explain whether Model 2 would have predicted the same

result as in part (c)(i).

In this question you must show detailed reasoning.

The region bounded by the x‑axis, the y‑axis, the line x = 4 and the curve with equation

The centre of mass of the lamina is at the point G ) (see diagram).

Determine the value of . Give your answer correct to 3 significant figures.

P is the point on the curved edge of the lamina where x = 3. The lamina is freely suspended from P and hangs in equilibrium in a vertical plane.

Determine the acute angle that the longest straight edge of the lamina makes with the vertical.

The diagram below shows the curve with polar equation r = sin 3θ for 0 ⩽ θ ⩽ .

Find the values of θ at the pole.

Find the polar coordinates of the point on the curve where r takes its maximum value.

In this question you must show detailed reasoning.

Find the exact area enclosed by the curve.

Given that sin 3θ = 3 sin θ - 4 sin³θ, find a cartesian equation for the curve.

Find giving your answer correct to 3 significant figures.

Using the modelling assumptions, determine the value of P giving your answer correct to

3 significant figures.