Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
In an orchestra, there are 11 violinists, 5 cellists and 4 double bass players. A small group of 6 musicians is to be selected from these 20.
At the Nonland Business College, all students sit an accountancy examination at the end of their first year of study. On average, 80% of the studen...
The daily rainfall, x mm, in a certain village is recorded on 250 consecutive days. The results are summarised in the following cumulative frequenc...
In a group of students, the numbers of boys and girls studying Art, Music and Drama are given in the following table. Each of these 160 students is...
The following back-to-back stem-and-leaf diagram shows the reaction times in seconds in an experiment involving two groups of people, A and B. ...
Jake attempts the crossword puzzle in his daily newspaper every day. The probability that he will complete the puzzle on any given day is 0.75, ind...
The Quivers Archery club has 12 Junior members and 20 Senior members. For the Junior members, the mean age is 15.5 years and the standard deviation...
A fair red spinner has 4 sides, numbered 1, 2, 3, 4. A fair blue spinner has 3 sides, numbered 1, 2, 3. When a spinner is spun, the score is the nu...
A group consists of 5 men and 2 women. Find the number of different ways that the group can stand in a line if the women are not next to each other.
A fair 6-sided die has the numbers -1, -1, 0, 0, 1, 2 on its faces. A fair 3-sided spinner has edges numbered -1, 0, 1. The die is thrown and the s...
A box contains 3 red balls and 5 blue balls. One ball is taken at random from the box and not replaced. A yellow ball is then put into the box. A s...
Out of a class of 8 boys and 4 girls, a group of 7 people is chosen at random.
The weights of apples sold by a store can be modelled by a normal distribution with mean 120 grams and standard deviation 24 grams. Apples weighing...
The lifetimes, in hours, of a particular type of light bulb are normally distributed with mean 2000 hours and standard deviation σ hours. The proba...
The heights, in cm, of the 11 members of the Anvils athletics team and the 11 members of the Brecons swimming team are shown below. Anvils 173 158 ...
The coefficient of x² in the expansion of (4 + ax)(1 + x/2)^6 is 3. Find the value of the constant a.
The point M is the mid-point of the line joining the points (3, 7) and (−1, 1). Find the equation of the line through M which is parallel to the li...
A curve is such that dy/dx = k/√x, where k is a constant. The points P (1, -1) and Q (4, 4) lie on the curve. Find the equation of the curve.
The diagram shows a circle with centre O and radius r cm. Points A and B lie on the circle and angle AOB = 2θ radians. The tangents to the circle a...
The diagram shows a solid cone which has a slant height of 15 cm and a vertical height of h cm. [Figure 5]
The diagram shows a three-dimensional shape OABCDEFG. The base OABC and the upper surface DEFG are identical horizontal rectangles. The parallelogr...
Functions f and g are defined by f(x) = 2x² + 8x + 1 for x ∈ R, g(x) = 2x - k for x ∈ R, where k is a constant.
The diagram shows part of the curve y = 1 - 4/(2x + 1)². The curve intersects the x-axis at A. The normal to the curve at A intersects the y-axis a...
Find the exact value of ∫₀¹ (2e^(2x) – 1)² dx. Show all necessary working.
A curve has equation y = (3 + 2 ln x) / (1 + ln x). Find the exact gradient of the curve at the point for which y = 4.
The polynomial p(x) is defined by p(x) = ax³ + ax² – 15x – 18, where a is a constant. It is given that (x – 2) is a factor of p(x).
It is given that ∫₀^a (3x² + 4 cos 2x – sin x) dx = 2, where a is a constant.
The equation of a curve is x² – 4xy – 2y² = 1.
The polynomial f(x) is defined by f(x) = x⁴ - 3x³ + 5x² – 6x + 11. Find the quotient and remainder when f(x) is divided by (x² + 2).
The variables x and y satisfy the equation y = kxᵃ, where k and a are constants. The graph of ln y against ln x is a straight line passing through ...
The sequence x₁, x₂, x₃, ... defined by x₁ = 1, xn+1 = xn / ln(2xn) converges to the value α.
Find the exact coordinates of the stationary point of the curve with equation y = e^(-½x) (2x + 5).
The parametric equations of a curve are x = 3 sin 2θ, y = 1 + 2 tan 2θ, for 0 ≤ θ < π/4.
Find the exact value of ∫₀² (2e²ˣ – 1)² dx. Show all necessary working.
A curve has equation y = (3 + 2 lnx) / (1 + ln x). Find the exact gradient of the curve at the point for which y = 4.
It is given that ∫₀ᵃ (3x² + 4 cos 2x - sin x) dx = 2, where a is a constant.
Given that ln(1 + e2y) = x, express y in terms of x.
Solve the inequality |2x-3| > 4|x + 1|.
The parametric equations of a curve are x = 2t + sin 2t, y = ln(1 − cos 2t). dy Show that = cosec 2t. dx