Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The volumes, in litres, of juice in large and small bottles have the distributions N(5.10, 0.0102) and N(2.51, 0.0036) respectively.
It is known that 8% of adults in a certain town own a Chantor car. After an advertising campaign, a car dealer wishes to investigate whether this p...
A curve with equation y = f(x) is such that f'(x) = 2x^3 - x3. 1 - x3. It is given that f(8) = 5.
A curve has equation y = x² + 2cx + 4 and a straight line has equation y = 4x + c, where c is a constant. Find the set of values of c for which the...
Find the term independent of x in each of the following expansions.
The first term of a geometric progression and the first term of an arithmetic progression are both equal to a. The third term of the geometric prog...
The functions f and g are defined by f(x) = x² for x ∈ R, g(x) = 2x² - 8x + 14 for x ∈ R.
The circle with equation (x + 1)² + (y − 2)² = 85 and the straight line with equation y = 3x – 20 are shown in the diagram. The line intersects the...
The diagram shows the circle with equation (x – 2)² + y² = 8. The chord AB of the circle intersects the positive y-axis at A and is parallel to the...
Functions f, g and h are defined as follows: f : x ↦ x - 4x^(1/2) + 1 for x ≥ 0, g : x ↦ mx² + n for x ≥ −2, where m and n are constants, h : x ↦ x...
The diagram shows a circle with centre A of radius 5cm and a circle with centre B of radius 8cm. The circles touch at the point C so that ACB is a ...
It is given that a curve has equation y = k(3x – k)⁻¹ + 3x, where k is a constant.
Solve the equation |5x-2| = |4x + 9|.
A curve has equation y = 7 + 4 ln(2x + 5). Find the equation of the tangent to the curve at the point (−2, 7), giving your answer in the form y = m...
The variables x and y satisfy the equation y = 3^(2a)a^x, where a is a constant. The graph of ln y against x is a straight line with gradient 0.239.
The polynomial p(x) is defined by p(x) = 4x³ + 16x² + 9x – 15.
A curve has equation e^(2xy) – e^y = 100.
On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z + 2 - 3i| ≤ 2 and arg z ...
The variables x and y satisfy the equation xⁿy² = C, where n and C are constants. The graph of ln y against ln x is a straight line passing through...
The parametric equations of a curve are x = 1 – cos θ, y = cos θ – ½ cos 2θ.
The angles α and β lie between 0° and 180° and are such that tan(α + β) = 2 and tan α = 3 tan β.
Find the complex numbers w which satisfy the equation w² + 2iw* = 1 and are such that Re w ≤ 0. Give your answers in the form x + iy, where x and y...
The variables x and y satisfy the differential equation (x + 1)(3x + 1) dy/dx = y, and it is given that y = 1 when x = 1.
The points A and B have position vectors 2i + j + k and i – 2j + 2k respectively. The line l has vector equation r = i + 2j – 3k + μ(i – 3j – 2k).
The diagram shows the curve y = sin x cos 2x for 0 ≤ x ≤ ½π, and its maximum point M. [Figure 11.1]
A crane is used to raise a block of mass 600kg vertically upwards at a constant speed through a height of 15 m. There is a resistance to the motion...
A particle P is projected vertically upwards from horizontal ground with speed ums¯¹. P reaches a maximum height of 20m above the ground.
A car of mass mkg is towing a trailer of mass 300kg down a straight hill inclined at 3° to the horizontal at a constant speed. There are resistance...
The total mass of a cyclist and her bicycle is 70kg. The cyclist is riding with constant power of 180 W up a straight hill inclined at an angle a t...
Four coplanar forces act at a point. The magnitudes of the forces are 10N, FN, GN and 2FN. The directions of the forces are as shown in the diagram.
A cyclist starts from rest at a fixed point O and moves in a straight line, before coming to rest k seconds later. The acceleration of the cyclist ...
A bead, A, of mass 0.1kg is threaded on a long straight rigid wire which is inclined at sin¯¹(7/25) to the horizontal. A is released from rest and ...
A fair red spinner has edges numbered 1, 2, 2, 3. A fair blue spinner has edges numbered -3, -2, -1, -1. Each spinner is spun once and the number o...
In a certain country, the probability of more than 10cm of rain on any particular day is 0.18, independently of the weather on any other day.
At a summer camp an arithmetic test is taken by 250 children. The times taken, to the nearest minute, to complete the test were recorded. The resul...
The weights of male leopards in a particular region are normally distributed with mean 55kg and standard deviation 6kg.
A group of 12 people consists of 3 boys, 4 girls and 5 adults.
A factory produces chocolates in three flavours: lemon, orange and strawberry in the ratio 3 : 5 : 7 respectively. Nell checks the chocolates on th...
The lengths, in millimetres, of a random sample of 12 rods made by a certain machine are as follows. 200 201 198 202 200 199 199 201 197 202 200 199
Harry has a five-sided spinner with sectors coloured blue, green, red, yellow and black. Harry thinks the spinner may be biased. He plans to carry ...
A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, p, of all households i...
In the past the time, in minutes, taken by students to complete a certain challenge had mean 25.5 and standard deviation 5.2. A new challenge is de...
The heights of buildings in a large city are normally distributed with mean 18.3m and standard deviation 2.5 m.
In a game a ball is rolled down a slope and along a track until it stops. The distance, in metres, travelled by the ball is modelled by the random ...