Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The masses, in kilograms, of small and large bags of wheat have the independent distributions N(16.0,0.4) and N(51.0,0.9) respectively. Find the pr...
The times, T minutes, taken by a random sample of 75 students to complete a test were noted. The results were summarised by Σt = 230 and Σt² = 930....
A random variable X has probability density function f defined by f(x) = { a/x² - 18/x³, 2 ≤ x ≤ 3 { 0, otherwise, where a i...
The lengths, in centimetres, of worms of a certain kind are normally distributed with mean μ and standard deviation 2.3. An article in a magazine s...
The numbers of customers arriving at service desks A and B during a 10-minute period have the independent distributions Po(1.8) and Po(2.1) respect...
The number of accidents per year on a certain road has the distribution Po(λ). In the past the value of λ was 3.3. Recently, a new speed limit was ...
A random variable X has the distribution B(4500000, 1000000). Use a Poisson distribution to calculate an estimate of P(X≥ 4).
The lengths of a random sample of 50 roads in a certain region were measured. Using the results, a 95% confidence interval for the mean length, in ...
A factory owner models the number of employees who use the factory canteen on any day by the distribution B(25, p). In the past the value of p was ...
A population is normally distributed with mean 35 and standard deviation 8.1 . A random sample of size 140 is chosen from this population and the s...
A machine puts sweets into bags at random. The numbers of lemon and orange sweets in a bag have the independent distributions Po(3.7) and Po(2.6) r...
The time, X hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by f(x) = { 2/...
The heights of one-year-old trees of a certain variety are known to have mean 2.3 m. A scientist believes that, on average, trees of this age and v...
A random variable X has probability density function f defined by f(x) = { a/x² - 18/x³ for 2 ≤ x ≤ 3, 0 otherwise, where a is a constant.
Find the term independent of x in the expansion of (2x² - 3/x)⁶
The equation of a curve is such that dy/dx = kx³ + 2/x², where k is a constant. The curve passes through the point S (2, 20) and the gradient of th...
The equation of a curve is y = 4x^(1/2) - x. The curve has a maximum point when x = a and crosses the x-axis at the point with coordinates (b, 0), ...
The coordinates of the points P and Q are (1, 1) and (7, 11) respectively. The line segment PQ forms a diameter of a circle.
The first three terms of a geometric progression are a, b and c respectively, where a, b and c are positive constants. The first three terms of an ...
The function f is defined by f(x) = 4/(3x-6)² + 1/(3x-6)³ for x > 2. The function g is defined by g(x) = 4x-3 for x > a.
The diagram shows a circle with centre A and radius r passing through points B, C and D. A larger circle of radius s has centre C and passes throug...
Find ∫ 6 sin²x dx.
Solve the equation e²ˣ (e²ˣ – 8) = 48.
Solve the equation cotθ tan(θ+45°) = 7 for 0° < θ < 90°.
The diagram shows the curve with equation y = 8e⁻¹/²ˣ - 1. The curve meets the axes at the points A and B. The shaded region is bounded by the curv...
A curve has parametric equations x = tan θ, y = sin θ – 2 sin³θ, for 0 < θ < ½π.
The polynomial p(x) is defined by p(x) = 2x⁴ + kx³ + kx² + 17x + 18, where k is a constant. It is given that (x+2) is a factor of p(x).
The equation of a curve is y = 4e¹⁻²ˣ √(3x-1).
Solve the equation ln(3x+5) – ln(x - 2) = 4. Give your answer in an exact form.
Solve the equation 2tan²θ+3 secθ = 18 for −180° < θ < 180°.
The polynomial p(x) is defined by 4 p(x) = x² - 10x³ +20x² - 30x+40.
The diagram shows the curve with equation y = 4 cos2x+8 sinx for 0 ≤ x ≤ π. The maximum points on the curve are denoted by A and B, and the shaded ...
A curve has equation 5x²y +4e² - 7x+10 = 0.
The diagram shows the curve with equation y = 8e⁻¹⁄₂ˣ - 1. The curve meets the axes at the points A and B. The shaded region is bounded by the curv...
A curve has parametric equations x = tan θ, y = sin θ-2 sin³θ, for 0 < θ < ½π.
The equation of a curve is y = 4e¹⁻²ˣ √3x-1.
The diagram shows the graph of y = e^(sin 2x) cos 4x for 0 < x < 1/2π, and its maximum point M. [Figure 4.1]
The parametric equations of a curve are x = t² - ln(t+1), y = t/(2t+1)