Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
A random variable X has probability density function given by f(x) = { (9-x²) 0 0 ≤ x ≤ 3, otherwise.
The random variable T denotes the time, in seconds, for 100 m races run by Tania. T is normally distributed with mean µ and variance σ². A random s...
The masses, in grams, of apples from a certain farm have mean µ and standard deviation 5.2. The farmer says that the value of µ is 64.6. A quality ...
The mass, in kilograms, of a block of cheese sold in a supermarket is denoted by the random variable M. The masses of a random sample of 40 blocks ...
Andy and Jessica are doing a survey about musical preferences. They plan to choose a representative sample of six students from the 256 students at...
The probability that a certain spinner lands on red on any spin is p. The spinner is spun 140 times and it lands on red 35 times.
A certain kind of firework is supposed to last for 30 seconds, on average, after it is lit. An inspector suspects that the fireworks actually last ...
In a certain large document, typing errors occur at random and at a constant mean rate of 0.2 per page.
A machine is supposed to produce random digits. Bob thinks that the machine is not fair and that the probability of it producing the digit 0 is les...
(a) The probability density function of the random variable X is given by f(x) = { kx(4-x) 0 ≤ x ≤ 2, { 0 otherwise, where k is a consta...
A random variable X has probability density function given by f(x) = { (1/18)(9-x²) for 0 ≤ x ≤ 3, 0 otherwise. The median of X is m.
(a) The proportion of people having a particular medical condition is 1 in 100000. A random sample of 2500 people is obtained. The number of people...
Points A and B have coordinates (5, 2) and (10, −1) respectively.
The first, second and third terms of an arithmetic progression are a, 2a and a² respectively, where a is a positive constant. Find the sum of the f...
A geometric progression is such that the third term is 1764 and the sum of the second and third terms is 3444. Find the 50th term.
The graph with equation y = f(x) is transformed to the graph with equation y = g(x) by a stretch in the x-direction with factor 0.5, followed by a ...
The equation of a curve is y = 4x² + 20x + 6.
The equation of a curve is such that dy/dx = 3x^(1/2) – 3x^(-1/2). The curve passes through the point (3, 5).
Functions f and g are defined by f(x) = x + 1/x for x > 0, g(x) = ax + 1 for x ∈ R, where a is a constant.
The diagram shows a cross-section RASB of the body of an aircraft. The cross-section consists of a sector OARB of a circle of radius 2.5m, with cen...
Solve the inequality |2x – 5| > x.
Use logarithms to solve the equation 14e⁻²ˣ = 5ˣ⁺¹, giving your answer correct to 3 significant figures.
It is given that sec θ = √17 where 0 < θ < ½π. Find the exact value of tan(θ + ¼π).
A curve has equation 4e²ˣy + y² = 21. Find the gradient of the curve at the point (0, -7).
The polynomial p(x) is defined by p(x) = 12x³ – 9x² + 8x – 4.
The diagram shows the curve with equation y = (2lnx)/(3x+1). The curve crosses the x-axis at the point A and has a maximum point B. The shaded regi...
The expression f(θ) is defined by f(θ) = 12 sin θ cos θ + 16 cos² θ.
Solve the equation sec θ = 5 cosec θ for 0° < θ < 360°.
The solutions of the equation |4x − 1| = |x + 3| are x = p and x = q, where p < q. Find the exact values of p and q, and hence determine the exact ...
The variables x and y satisfy the equation y = Axk, where A and k are constants. The graph of In y against In x is a straight line passing through ...
The polynomial p(x) is defined by p(x) = ax³ + 23x² - ax – 8, where a is a constant. It is given that (2x + 1) is a factor of p(x).
The curve with equation y = xln(4x + 1) – 3x has one stationary point P.
The diagram shows the curves y = 6 / (3x + 2) and y = 3e⁻ˣ – 3 for values of x between 0 and 4. The shaded region is bounded by the two curves and ...
The diagram shows the curve with parametric equations x = 3 cos 2θ, y = 4 sin θ, for π < θ < 3π/2. Points P and Q lie on the curve. The gradient of...
A curve has equation 4e^(2xy) + y² = 21. Find the gradient of the curve at the point (0, -7).
The diagram shows the curve with equation y = (2 ln x) / (3x + 1). The curve crosses the x-axis at the point A and has a maximum point B. The shade...
On a sketch of an Argand diagram shade the region whose points represent complex numbers z satisfying the inequalities |z| ≤ 3, Rez ≥ −2 and π < ar...
Solve the equation 2^(3x−1) = 5(3^−x). Give your answer in the form (ln a)/(ln b), where a and b are integers.
Solve the equation tan(x + 45°) = 2 cotx for 0° < x < 180°.
The complex numbers u and w are defined by u = 2e^(πi/4) and w = 3e^(πi/3).
The equation of a curve is y = x/cos²x, for 0 ≤ x < π/2. At the point where x = a, the tangent to the curve has gradient equal to 12.
In a certain chemical reaction the amount, x grams, of a substance is increasing. The differential equation satisfied by x and t, the time in secon...