Paper 3: Pure Mathematics and Mechanics

(a) Show that, for small θ measured in radians, (1 - cos 2θ)/(θtan θ) ≈ 2. (b) Show that sec θ ≈ 1 + 1/2θ² for small θ.

A sequence is defined by u₁ = 2 and u_n+1 = 1/(1 - u_n). (a) Find u₂, u₃ and u₄. (b) Describe the behaviour of the sequence. (c) Find Σ_n=1³⁰ u_n.

The curve C has equation y = e^xsin x, for 0 ≤ x ≤ 2π. (a) Find the exact x-coordinates of the stationary points of C. (b) Determine the nature of each stationary point.

In triangle OAB, OA⃗ = a and OB⃗ = b. The point X lies on OA with OX : XA = 2 : 1, and Y is the midpoint of AB. (a) Find XY⃗ in terms of a and b. (b) The line XY meets the line OB extended at Z, where OZ⃗ = λ b. Find λ.

The height, h metres, of a tree t years after planting is modelled by dh/dt = (8 - h)/10. When planted, the tree is 1 m tall. (a) Solve the differential equation to find h in terms of t. (b) Find the time taken for the tree to reach 6 m. (c) State the height the model predicts the tree will approach, and give one criticism of the model.

(a) Use the substitution x = 2sin θ to find the exact value of ∫₀¹ 1/(√(4 - x²)) dx. (b) Use the substitution u = √x to find the exact value of ∫₀⁴ e^√x dx.

A particle moves in a straight line. Its acceleration at time t seconds is (6t - 4) m s^-2. When t = 0 its velocity is 2 m s^-1 and it is at the origin. Find its velocity and displacement when t = 3.

A particle of mass 5 kg hangs in equilibrium from two light strings. One string makes an angle of 30° with the horizontal and the other makes an angle of 60° with the horizontal, on the opposite side. Find the tension in each string.

Particle A, of mass 4 kg, lies on a rough plane inclined at 30° to the horizontal. The coefficient of friction is 0.25. A is attached to a light inextensible string that passes over a smooth pulley at the top of the plane to particle B, of mass 3 kg, which hangs freely. The system is released from rest and B moves downwards. Find the acceleration of the system and the tension in the string.

A non-uniform rod AB of length 5 m and mass 8 kg hangs horizontally from two vertical strings attached at A and B. The tension in the string at A is 31.36 N and the tension at B is 47.04 N. (a) Find the distance of the centre of mass of the rod from A. (b) A particle of mass m kg is attached at A so that the two tensions become equal. Find m. (c) Instead, a particle of mass 2 kg is attached to the rod. The string at B breaks if its tension exceeds 60 N. Find the range of distances from A at which the particle can be placed without the string at B breaking.

A ball is thrown from a point 1.5 m above horizontal ground with speed 18 m s^-1 at an angle of 35° above the horizontal. A wall 3 m high stands 20 m horizontally from the point of projection. (a) Show that the ball passes over the wall and find by how much it clears it. (b) Find the greatest height of the ball above the ground. (c) Find the horizontal distance from the point of projection to where the ball lands. (d) Find the speed and direction of the ball just before it lands.