Paper 1: Pure Mathematics

Find the value of log₃ 27 + log₃ 1/9. Circle your answer.

The function f is defined by f(x) = 2 - e^-x for all real x. Which of the following is the range of f?

Solve the inequality |2x - 5| ≤ 3.

(a) Find the first three terms, in ascending powers of x, of the binomial expansion of (1 - 3x)^-1/2. (b) State the range of values of x for which the expansion is valid.

Prove by contradiction that √3 is irrational.

(a) Show that (sin 2θ)/(1 + cos 2θ) ≡ tan θ. (b) Hence solve (sin 2θ)/(1 + cos 2θ) = √3 for 0 ≤ θ < 2π.

The functions f and g are defined by f(x) = (2x + 1)/(x - 3), x ≠ 3 and g(x) = x² + 1. (a) Find f^-1(x) and state its domain. (b) Find the exact value of fg(2).

A curve has equation y = x e^-2x. (a) Show that dy/dx = e^-2x(1 - 2x). (b) Find the exact coordinates of the stationary point and determine its nature. (c) Find the equation of the tangent to the curve at the origin.

(a) Use the substitution u = 1 + x² to show that ∫₀¹ 4x/((1 + x²)²) dx = 1. (b) Use integration by parts to find the exact value of ∫₁^e x² ln x dx.

A curve is defined by the parametric equations x = t², y = 2t - t³, for t ≠ 0. (a) Find dy/dx in terms of t. (b) Find the exact coordinates of the points where the tangent to the curve is horizontal.

The temperature T °C of a cup of tea t minutes after it is made is modelled by T = 20 + 70e^-kt. After 10 minutes the temperature is 60°C. (a) Show that k = 1/10 ln 7/4. (b) Find the time taken for the tea to cool to 30°C. (c) Find the rate at which the tea is cooling when t = 10.

f(x) = x³ - 2x - 5 (a) Show that the equation f(x) = 0 has a root α between x = 2 and x = 3. (b) Using x₀ = 2, apply the Newton–Raphson method twice to find x₂, giving your answer to 4 decimal places. (c) Explain why the Newton–Raphson method would fail with x₀ = √(2/3).

(a) A geometric series has sum to infinity 8. The sum of its first two terms is 6. Find the two possible values of the common ratio and the corresponding first terms. (b) Find the sum of all the integers from 1 to 200 inclusive that are NOT divisible by 3.

A circle C has centre (2, -1) and passes through the point (5, 3). (a) Find an equation for C. (b) Show that the point (1, 6) lies outside C. (c) The line y = x + k is a tangent to C. Find the two possible exact values of k.

Water drains from a tank. The volume of water, V litres, after t minutes is modelled by dV/dt = -k√V, where k is a positive constant. Initially V = 100. After 10 minutes, V = 64. (a) Solve the differential equation to find V in terms of t. (b) Find the time taken for the tank to empty. (c) Give one reason why the model may not be accurate as the tank nearly empties.

(a) Express 3sin x + 4cos x in the form Rsin(x + α), where R > 0 and 0 < α < π/2. Give α to 4 decimal places. (b) Find the maximum value of 3sin x + 4cos x and the smallest positive value of x at which it occurs. (c) Solve 3sin x + 4cos x = 2.5 for 0 ≤ x < 2π. Give your answers to 3 significant figures.