Paper 1: Pure Mathematics

(a) Express (3√2 - 1)/(√2 + 1) in the form a + b√2, where a and b are integers. (b) Solve 8^x = 4^x+1.

The points A and B have coordinates (-1, 4) and (3, -2). (a) Find the equation of the line AB in the form ax + by + c = 0. (b) Find the equation of the perpendicular bisector of AB.

Find the set of values of x for which x² - x - 6 > 0 and 2x + 1 < 9.

p(x) = 2x³ + x² - 13x + 6 (a) Use the factor theorem to show that (x - 2) is a factor of p(x). (b) Factorise p(x) completely. (c) Hence solve p(x) = 0 and sketch y = p(x), showing where it meets the axes.

The infinite series Σ_r=0^∞ (x/3 - 1 )^r is convergent. (a) Find the set of possible values of x. (b) Find the sum to infinity in terms of x, simplifying your answer. (c) Find the sum to infinity when x = 4.

(a) Show that cos(x + 60°) + cos(x - 60°) ≡ cos x. (b) Solve 2sin xtan x = 3 for 0° ≤ x < 360°.

(a) Solve 2log₃ x - log₃(x + 6) = 1. (b) Solve 5^2x+1 = 3^x, giving your answer correct to 3 significant figures.

(a) Given y = sin² 3x, show that dy/dx = 3sin 6x. (b) The curve C has equation y = ln(x² + 1) - x. Show that C has exactly one stationary point, find its coordinates, and show that it is a point of inflection.

Find (a) ∫ tan² x dx (b) ∫₀^π/4 sec² xtan³ x dx (c) ∫ 1/(xln x) dx (d) ∫₁² 1/((2x + 1)²) dx

Relative to the origin O, a = 2 i - j + 2 k and b = i + 2 j - 2 k are the position vectors of A and B. (a) Find a unit vector in the direction of a. (b) Find angle AOB. (c) OABC is a parallelogram. Find the position vector of C. (d) Find |AC⃗|.

The number of fish, P, in a lake t years after stocking is modelled by P = 2000/(1 + 9e^-0.4t). (a) Find the initial number of fish. (b) State the long-term number of fish predicted by the model. (c) Find the time taken for the population to reach 1000. (d) Show that dP/dt = 0.4P (1 - P/2000 ). (e) Hence find the greatest rate of increase of the population.

A curve has parametric equations x = t + 1/t, y = t - 1/t, for t > 0. (a) Find a Cartesian equation of the curve. (b) Find dy/dx in terms of t. (c) Find the exact coordinates of the point where the gradient of the curve is 2. (d) Find the equation of the tangent at this point.

(a) Show that x³ - 3x - 1 = 0 has a root α between 1 and 2. (b) Use the iteration x_n+1 = ³√(3x_n + 1) with x₀ = 2 to find x₁, x₂, x₃ to 4 decimal places. (c) Show that α = 1.879 correct to 3 decimal places.