Paper 2: Pure and Mechanics

Find d/dx(tan 3x).

For small θ measured in radians, which is the best approximation for cos θ?

(a) Express (5x + 4)/((x + 2)(x - 1)) in partial fractions. (b) Hence find ∫ (5x + 4)/((x + 2)(x - 1)) dx.

A curve has equation x² + xy + y² = 7. Find the equation of the normal to the curve at the point (1, 2), giving your answer in the form ax + by + c = 0 where a, b, c are integers.

(a) Show that Σ_r=1ⁿ (3r - 1) = (n(3n + 1))/2. (b) Find the smallest value of n for which Σ_r=1ⁿ (3r - 1) > 1000.

Solve the equation 3sec² x - 4tan x - 4 = 0 for 0° ≤ x < 360°. Give your answers to 1 decimal place.

(a) The curve y = x² - 4x + 5 and the line y = x + 1 intersect at two points. Find the exact area of the region enclosed between them. (b) Use the trapezium rule with 4 strips to estimate ∫₀² √(1 + x³) dx, and state whether your answer is an overestimate or an underestimate, giving a reason.

Points A and B have position vectors a = 2i + j - 3k and b = 5i - j + 3k relative to the origin O. (a) Find |AB⃗|. (b) Find the position vector of the midpoint of AB. (c) Show that angle AOB is a right angle. (d) Find the exact area of triangle OAB.

(a) Solve 3^2x - 10 × 3^x + 9 = 0. (b) Solve log₂(x + 3) + log₂(x - 3) = 4.

A particle moves in a straight line with constant velocity. Which statement must be true?

A particle's velocity is v = 3t² - 2t m s⁻¹. What is its acceleration when t = 2?

A car travelling at 25 m s^-1 brakes with constant deceleration and stops in a distance of 80 m. Find the deceleration and the time taken to stop.

Particle A, of mass 3 kg, lies on a smooth horizontal table. It is connected by a light inextensible string passing over a smooth pulley at the edge of the table to particle B, of mass 2 kg, which hangs freely. The system is released from rest. (a) Find the acceleration of the particles and the tension in the string. (b) Find the speed of B after it has fallen 0.5 m (A has not reached the pulley).

A block of mass 5 kg is released from rest on a rough plane inclined at 30° to the horizontal. The coefficient of friction is 0.2. (a) Find the acceleration of the block. (b) Find the distance it slides in the first 3 seconds. (c) Find the least value of the coefficient of friction for which the block would remain at rest.

A uniform plank AB has length 4 m and mass 12 kg. It rests horizontally on two supports, C and D, where AC = 0.5 m and DB = 1 m. A box of mass 20 kg is placed on the plank. (a) The box is placed at the midpoint of AB. Find the reaction at each support. (b) The box is moved towards B. Find the greatest distance from A at which the box can be placed without the plank tilting. Model the box as a particle.

A ball is projected from the top of a vertical cliff 20 m above the sea, with speed 15 m s^-1 at 40° above the horizontal. (a) Find the time taken for the ball to reach the sea. (b) Find the horizontal distance from the foot of the cliff to where the ball lands. (c) Find the speed of the ball as it hits the sea. (d) State one modelling assumption you have made.

A particle P of mass 0.5 kg moves in a horizontal plane. At time t seconds its velocity is v = (3t² - 4) i + 2t j m s⁻¹. When t = 0, P is at the point with position vector 2 i - j m. (a) Find the magnitude of the acceleration of P when t = 2. (b) Find the position vector of P when t = 2. (c) Find the time at which P is moving parallel to j. (d) Find the magnitude of the resultant force on P when t = 1.