Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The time taken, in minutes, by a ferry to cross a lake has a normal distribution with mean 85 and standard deviation 6.8.
Megan sends messages to her friends in one of 3 different ways: text, email or social media. For each message, the probability that she uses text i...
Mr and Mrs Keene and their 5 children all go to watch a football match, together with their friends Mr and Mrs Uzuma and their 2 children. Find the...
On average, 34% of the people who go to a particular theatre are men.
A fair five-sided spinner has sides numbered 1, 1, 1, 2, 3. A fair three-sided spinner has sides numbered 1, 2, 3. Both spinners are spun once and ...
The times in minutes taken by 13 pupils at each of two schools in a cross-country race are recorded in the table below. Thaters School 38 43 48 52 ...
The following parts relate to binomial expansions.
The following parts relate to trigonometric equations.
A weather balloon in the shape of a sphere is being inflated by a pump. The volume of the balloon is increasing at a constant rate of 600 cm³ per s...
The nth term of an arithmetic progression is (1/2)(3n – 15). Find the value of n for which the sum of the first n terms is 84.
The function f is defined for x ∈ R by f: x ↦ a - 2x, where a is a constant.
The equation of a curve is y = 2x² + kx + k − 1, where k is a constant.
In the diagram, OAB is a sector of a circle with centre O and radius 2r, and angle AOB = π/6 radians. The point C is the midpoint of OA. [Figure 7.1]
The diagram shows part of the curve y = 6/x. The points (1, 6) and (3, 2) lie on the curve. The shaded region is bounded by the curve and the lines...
Functions f and g are such that f(x) = 2 - 3 sin 2x for 0 ≤ x ≤ π, g(x) = -2f(x) for 0 ≤ x ≤ π. The diagram below shows the graph of y = f(x). [Fig...
The equation of a curve is y = 54x – (2x – 7)³.
The equation of a circle with centre C is x² + y² – 8x + 4y − 5 = 0.
Given that 2^y = 9^3x, use logarithms to show that y = kx and find the value of k correct to 3 significant figures.
Find the exact coordinates of the stationary point on the curve with equation y = 5xex^(1/2)x.
The equation of a curve is cos 3x + 5 sin y = 3. Find the gradient of the curve at the point (1/3 π, 1/6 π).
The variables x and y satisfy the equation y = Ax^(-2p), where A and p are constants. The graph of ln y against ln x is a straight line passing thr...
The polynomial p(x) is defined by p(x) = 6x³ + ax² - 4x – 3, where a is a constant. It is given that (x + 3) is a factor of p(x).
It is given that ∫[from 0 to a] (4/(2x+1) + 8x) dx = 10, where a is a positive constant.
Given that 2ʸ = 9³ˣ, use logarithms to show that y = kx and find the value of k correct to 3 significant figures.
Find the exact coordinates of the stationary point on the curve with equation y = 5xe½ˣ.
The equation of a curve is cos 3x + 5 sin y = 3. Find the gradient of the curve at the point (¹⁄₆π, ¹⁄₆π).
The variables x and y satisfy the equation y = Ax⁻²ᵖ, where A and p are constants. The graph of ln y against ln x is a straight line passing throug...
It is given that ∫₀ᵃ ( 4/(2x + 1) + 8x) dx = 10, where a is a positive constant.
Find the set of values of x for which 2(3^(1−2x)) < 5^x. Give your answer in a simplified exact form.
Express the equation tan(θ+ 60°) = 2 + tan(60° – θ) as a quadratic equation in tan θ, and hence solve the equation for 0° ≤ θ ≤ 180°.
The curve with equation y = e^(2x)(sin x + 3 cos x) has a stationary point in the interval 0 ≤ x ≤ π.
The diagram shows a circle with centre O and radius r. The tangents to the circle at the points A and B meet at T, and angle AOB is 2x radians. The...
Let f(x) = cos x / (1 + sin x)
A certain curve is such that its gradient at a point (x, y) is proportional to y / (x√x). The curve passes through the points with coordinates (1, ...
With respect to the origin O, the vertices of a triangle ABC have position vectors OA = 2i + 5k, OB = 3i + 2j + 3k and OC = i + j + k.
The complex number u is defined by u = 3i / (a + 2i), where a is real.
Find the quotient and remainder when 6x⁴ + x³ − x² + 5x – 6 is divided by 2x² − x + 1.
The variables x and y satisfy the equation y² = Aekx, where A and k are constants. The graph of In y against x is a straight line passing through t...
Find the exact value of ∫₄¹ x^(3/2) ln x dx.
A curve has equation y = cos x sin 2x. Find the x-coordinate of the stationary point in the interval 0 < x < ½π, giving your answer correct to 3 si...
The diagram shows the curve y = x / (1 + 3x⁴) for x ≥ 0, and its maximum point M. [Figure 6.1]
The variables x and y satisfy the differential equation dy/dx = (y - 1) / ((x + 1)(x + 3)). It is given that y = 2 when x = 0. Solve the differenti...