Paper 1: Pure Mathematics 1

(a) Express 2x² - 12x + 23 in the form a(x + b)² + c, where a, b, c are integers. (b) Write down the coordinates of the minimum point of the curve y = 2x² - 12x + 23.

Show that ∫₁⁹ (x - 3)/√x dx = 16/3.

(a) A sequence is defined by u₁ = 2, u_n+1 = 3u_n - k, where k is a constant. Given that u₃ = 8, find the value of k. (b) Find the value of Σ_r=1¹⁰ (2^r + 3).

The equation kx² + 4x + (k - 3) = 0, where k is a constant, has two distinct real roots. Find the set of possible values of k.

Solve, for 0 ≤ x < 180°, 2cos²(2x - 10°) = 3sin(2x - 10°).

The mass, m grams, of a radioactive substance t years after it was first measured is modelled by m = 50e^-kt. The half-life of the substance is 8 years. (a) Show that k = (ln 2)/8. (b) Find the mass after 20 years. (c) Find the time taken for the mass to fall to 1 gram. (d) Find the rate at which the mass is decreasing when t = 0.

(a) Prove, from first principles, that the derivative of x³ is 3x². (b) The curve C has equation y = x³ - 3x + 1. Find the equation of the tangent to C at the point where x = 2, in the form y = mx + c.

The curve C has equation y = x(x - 2)(x - 4). (a) Sketch C, showing where it meets the coordinate axes. (b) Find the total area of the regions bounded by C and the x-axis between x = 0 and x = 4.

Two variables x and y satisfy y = ab^x, where a and b are constants. A graph of log₁₀ y against x is a straight line with gradient 0.3 and intercept 1.2 on the vertical axis. (a) Find the values of a and b, giving your answers to 3 significant figures. (b) Use the model to estimate y when x = 5.

The circle C has equation x² + y² + 4x - 6y - 12 = 0. (a) Find the centre and radius of C. (b) The line y = 2x + 2 meets C at the points P and Q. Find the coordinates of P and Q. (c) Find the exact length of PQ.

(a) Prove by contradiction that there are infinitely many prime numbers. (b) Prove that, for all integers n, (2n + 1)² - (2n - 1)² is a multiple of 8. (c) Show that the statement x² ≥ x for all real x is false.

(a) Find the binomial expansion of √(4 - x), in ascending powers of x, up to and including the term in x². Give each coefficient as a simplified fraction. (b) State the range of values of x for which the expansion is valid. (c) Use your expansion with x = 0.04 to find an estimate for √3.96, giving your answer to 6 decimal places.

An open box is made from a square sheet of card of side 30 cm by cutting a square of side x cm from each corner and folding up the sides. (a) Show that the volume of the box is V = x(30 - 2x)² cm³ and state the possible values of x. (b) Use calculus to find the value of x that maximises V. (c) Justify that this value gives a maximum. (d) Find the maximum volume.