Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
Solve the inequality |2x – 1| < 3|x + 1|.
The variables x and y satisfy the equation x = A(3⁻ʸ), where A is a constant.
Using integration by parts, find the exact value of ∫₀² tan⁻¹(½x)dx.
The complex number u is given by u = 10 – 4√6i. Find the two square roots of u, giving your answers in the form a + ib, where a and b are real and ...
A curve is such that the gradient at a general point with coordinates (x, y) is proportional to y / √(x + 1). The curve passes through the points w...
The equation of a curve is y = e⁻⁵ˣ tan²x for −π/2 < x < π/2. Find the x-coordinates of the stationary points of the curve. Give your answers corre...
Let f(x) = (14 - 3x + 2x²) / ((2 + x)(3 + x²))
The diagram shows a trapezium ABCD in which AD = BC = r and AB = 2r. The acute angles BAD and ABC are both equal to x radians. Circular arcs of rad...
With respect to the origin O, the points A and B have position vectors given by OA = 2i – j and OB = j – 2k.
Expand (1 + 3x)⁻² in ascending powers of x, up to and including the term in x³, simplifying the coefficients.
Solve the equation 4ˣ = 3 + 4⁻ˣ. Give your answer correct to 3 decimal places.
The parametric equations of a curve are x = t + ln(t + 2), y = (t − 1)e⁻²ᵗ, where t > -2.
Let f(x) = (15-6x) / ((1+2x)(4-x))
For the curve shown in the diagram, the normal to the curve at the point P with coordinates (x, y) meets the x-axis at N. The point M is the foot o...
The diagram shows the curve y = (ln x) / x³ and its maximum point M. [Figure X.X]
The quadrilateral ABCD is a trapezium in which AB and DC are parallel. With respect to the origin O, the position vectors of A, B and C are given b...
A winch operates by means of a force applied by a rope. The winch is used to pull a load of mass 50 kg up a line of greatest slope of a plane incli...
Two particles A and B have masses mkg and 0.1 kg respectively, where m > 0.1. The particles are attached to the ends of a light inextensible string...
Three particles P, Q and R, of masses 0.1 kg, 0.2 kg and 0.5 kg respectively, are at rest in a straight line on a smooth horizontal plane. Particle...
Two cyclists, Isabella and Maria, are having a race. They both travel along a straight road with constant acceleration, starting from rest at point...
A particle moving in a straight line starts from rest at a point A and comes instantaneously to rest at a point B. The acceleration of the particle...
Three coplanar forces of magnitudes 10N, 25 N and 20 N act at a point O in the directions shown in the diagram.
A slide in a playground descends at a constant angle of 30° for 2.5 m. It then has a horizontal section in the same vertical plane as the sloping s...
A particle of mass 0.6 kg is projected with a speed of 4 m s⁻¹ down a line of greatest slope of a smooth plane inclined at 10° to the horizontal.
Coplanar forces of magnitudes 34 N, 30N and 26 N act at a point in the directions shown in the diagram.
A ring of mass 0.3 kg is threaded on a horizontal rough rod. The coefficient of friction between the ring and the rod is 0.8. A force of magnitude ...
A particle of mass 12 kg is stationary on a rough plane inclined at an angle of 25° to the horizontal. A pulling force of magnitude P N acts at an ...
A car of mass 1250 kg is pulling a caravan of mass 800 kg along a straight road. The resistances to the motion of the car and caravan are 440 N and...
A particle A is projected vertically upwards from level ground with an initial speed of 30 m s⁻¹. At the same instant a particle B is released from...
A particle P moving in a straight line starts from rest at a point O and comes to rest 16 s later. At time t s after leaving O, the acceleration a ...
Particles P of mass 0.4 kg and Q of mass 0.5 kg are free to move on a smooth horizontal plane. P and Q are moving directly towards each other with ...
A cyclist is travelling along a straight horizontal road. She is working at a constant rate of 150 W. At an instant when her speed is 4 m s⁻¹, her ...
Four coplanar forces act at a point. The magnitudes of the forces are 20 N, 30 N, 40 N and F N. The directions of the forces are as shown in the di...
A particle is projected vertically upwards with speed u m s⁻¹ from a point on horizontal ground. After 2 seconds, the height of the particle above ...
A car of mass 1400 kg is towing a trailer of mass 500 kg down a straight hill inclined at an angle of 5° to the horizontal. The car and trailer are...
A particle moves in a straight line and passes through the point A at time t = 0. The velocity of the particle at time t s after leaving A is v m s...
A particle P of mass 0.3 kg rests on a rough plane inclined at an angle θ to the horizontal, where sin θ = 7/25. A horizontal force of magnitude 4 ...
A bag contains 12 marbles, each of a different size. 8 of the marbles are red and 4 of the marbles are blue.
A company produces a particular type of metal rod. The lengths of these rods are normally distributed with mean 25.2 cm and standard deviation 0.4 ...
To gain a place at a science college, students first have to pass a written test and then a practical test. Each student is allowed a maximum of tw...
The times taken by 200 players to solve a computer puzzle are summarised in the following table. Time (t seconds) 0 ≤ t < 10 10 ≤ t < 20 20 ≤ t < 4...
In Questa, 60% of the adults travel to work by car.
Sharma knows that she has 3 tins of carrots, 2 tins of peas and 2 tins of sweetcorn in her cupboard. All the tins are the same shape and size, but ...
An ordinary fair die is thrown repeatedly until a 5 is obtained. The number of throws taken is denoted by the random variable X.
The weights of bags of sugar are normally distributed with mean 1.04 kg and standard deviation σ kg. In a random sample of 2000 bags of sugar, 72 w...