Paper 2: Pure Mathematics 2

The function f is defined by f(x) = 3/(x - 1) + 2, x > 1. (a) State the range of f. (b) Find f^-1(x), stating its domain. (c) Find ff(4).

The points P, Q and R have coordinates (1, 2, 3), (4, -2, 3) and (1, 2, -9). (a) Find |PQ⃗| and |PR⃗|. (b) Show that angle QPR is a right angle and find the area of triangle PQR.

f(x) = e^x - 4x (a) Show that the equation f(x) = 0 has a root α in the interval [0, 1]. (b) Using x_n+1 = 1/4 e^x_n with x₀ = 0.3, find x₁, x₂, x₃ to 5 decimal places. (c) Show that α = 0.357 to 3 decimal places.

(a) Express 5cos x - 12sin x in the form Rcos(x + α), where R > 0 and 0 < α < π/2. Give α to 4 decimal places. (b) Hence solve 5cos x - 12sin x = 6.5 for 0 ≤ x < 2π.

(a) Express (x + 11)/((x + 3)(x - 1)²) in partial fractions. (b) Hence find the exact value of ∫₂³ (x + 11)/((x + 3)(x - 1)²) dx, giving your answer in the form a + bln 3/5.

(a) Find ∫ x² e^x dx. (b) Hence find the exact value of ∫₀¹ x² e^x dx.

(a) Find the particular solution of dy/dx = (y + 1)cos x, given that y = 2 when x = 0. Give your answer in the form y = f(x). (b) Find the exact maximum and minimum values of y.

A curve has parametric equations x = 3cos t, y = 2sin t, for 0 ≤ t < 2π. (a) Find dy/dx in terms of t. (b) Find the equation of the tangent to the curve at the point where t = π/4, in the form ax + by = c√2. (c) Find a Cartesian equation of the curve.

Jo saves money each month. She saves £50 in the first month and increases the amount by £5 each month after that. (a) Find the total amount she has saved after 24 months. (b) Find the month in which her total savings first exceed £5000.

(a) Prove that (1 - cos 2x)/(sin 2x) ≡ tan x. (b) Prove that csc 2x + cot 2x ≡ cot x. (c) Hence solve csc 2x + cot 2x = 2 for 0° < x < 360°, giving your answers to 1 decimal place.

Water is poured into an inverted cone at a constant rate of 20 cm³ s^-1. When the depth of water is h cm, the radius of the surface is h/2 cm. (a) Show that the volume of water is V = (π h³)/12. (b) Find the rate at which the depth is increasing when h = 10. (c) The cone is initially empty. Find the time taken for the depth to reach 10 cm.

(a) Use the substitution u = x + 1 to find the exact value of ∫₀³ x√(x + 1) dx. (b) Use the trapezium rule with 3 strips to estimate the same integral. (c) Find the percentage error in your estimate and explain why the trapezium rule gives an overestimate here.