Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The equation of a curve is 3x² + 4xy + 3y² = 5.
(a) The variables x and y satisfy the differential equation (dy/dx) = (4 + 9y²)/e^(2x+1). It is given that y = 0 when x = 1.
Let f(x) = (2x² + 17x - 17) / ((1 + 2x)(2-x)²).
The diagram shows the curve y = (x + 5)√3 – 2x and its maximum point M. [Figure 10.1]
The points A and B have position vectors i + 2j – 2k and 2i – j + k respectively. The line l has equation r = i − j + 3k + μ(2i – 3j + 4k).
Solve the equation ln(x + 5) = 5 + lnx. Give your answer correct to 3 decimal places.
Find the quotient and remainder when 2x⁴ – 27 is divided by x² + x + 3.
The parametric equations of a curve are x = (cos θ) / (2 - sin θ), y = θ + 2 cos θ. Show that dy/dx = (2 − sin θ)².
The diagram shows the part of the curve y = x² cos 3x for 0 ≤ x ≤ π/6, and its maximum point M, where x = a. [Figure 5.1]
The variables x and y satisfy the differential equation dy/dx = (y² + 4)/(x(y + 4)) for x > 0. It is given that x = 4 when y = 2√3. Solve the diffe...
The lines l and m have equations l: r = ai + 3j + bk + λ(ci – 2j + 4k), m: r = i +2j + 3k + μ(2i – 3j + k). Relative to the origin O, the position ...
Let f(x) = (21 – 8x – 2x²) / ((1 + 2x)(3−x)²).
The complex number z is defined by z = (5a - 2i) / (3 + ai), where a is an integer. It is given that arg z = –π/4.
Two particles P and Q, of masses mkg and 0.3kg respectively, are at rest on a smooth horizontal plane. P is projected at a speed of 5ms¯¹ directly ...
A particle P of mass 0.4 kg is projected vertically upwards from horizontal ground with speed 10ms-1.
A particle moves in a straight line starting from rest. The displacement s m of the particle from a fixed point O on the line at time t s is given ...
The velocity of a particle at time t s after leaving a fixed point O is vms¯¹. The diagram shows a velocity-time graph which models the motion of t...
Four coplanar forces act at a point. The magnitudes of the forces are F N, 10 N, 50 N and 40 N. The directions of the forces are as shown in the di...
Two particles P and Q, of masses 0.2kg and 0.1 kg respectively, are attached to the ends of a light inextensible string. The string passes over a f...
A car of mass 1200kg is travelling along a straight horizontal road. The power of the car's engine is constant and is equal to 16kW. There is a con...
A particle of mass 1.6kg is dropped from a height of 9 m above horizontal ground. The speed of the particle at the instant before hitting the groun...
Two particles A and B, of masses 3.2kg and 2.4kg respectively, lie on a smooth horizontal table. A moves towards B with a speed of vms¯¹ and collid...
Coplanar forces of magnitudes 30N, 15N, 33N and PN act at a point in the directions shown in the diagram, where tan α = 4/3. The system is in equil...
An athlete of mass 84kg is running along a straight road.
A particle of mass 0.6kg is placed on a rough plane which is inclined at an angle of 35° to the horizontal. The particle is kept in equilibrium by ...
A particle P starts at rest and moves in a straight line from a point O. At time ts after leaving O, the velocity of P, vms¯¹, is given by v = bt +...
Two particles P and Q, of masses 2kg and 0.25 kg respectively, are connected by a light inextensible string that passes over a fixed smooth pulley....
Two particles P and Q, of masses 0.1kg and 0.4kg respectively, are free to move on a smooth horizontal plane. Particle P is projected with speed 4m...
A car of mass 1500kg is towing a trailer of mass mkg along a straight horizontal road. The car and the trailer are connected by a tow-bar which is ...
A smooth ring R of mass 0.2kg is threaded on a light string ARB. The ends of the string are attached to fixed points A and B with A vertically abov...
A lorry of mass 15000kg moves on a straight horizontal road in the direction from A to B. It passes A and B with speeds 20ms⁻¹ and 25ms⁻¹ respectiv...
A particle starts from rest from a point O and moves in a straight line. The acceleration of the particle at time ts after leaving O is a m s¯², wh...
An elevator is pulled vertically upwards by a cable. The elevator accelerates at 0.4ms⁻² for 5 s, then travels at constant speed for 25 s. The elev...
The diagram shows the vertical cross-section XYZ of a rough slide. The section YZ is a straight line of length 2 m inclined at an angle of α to the...
A summary of 50 values of x gives Σ(x – q) = 700, Σ(x − q)² = 14235, where q is a constant.
A different committee of 6 people is to be chosen from 6 men and 8 women. Three of the 6 men are brothers.
A mathematical puzzle is given to a large number of students. The times taken to complete the puzzle are normally distributed with mean 14.6 minute...
The populations of 150 villages in the UK, to the nearest hundred, are summarised in the table. Population | 100 – 800 | 900 – 1200 | 1300 – 2000 |...
Eli has four fair 4-sided dice with sides labelled 1, 2, 3, 4. He throws all four dice at the same time. The random variable X denotes the number o...
A children's wildlife magazine is published every Monday. For the next 12 weeks it will include a model animal as a free gift. There are five diffe...
The random variable X takes the values –2, 2 and 3. It is given that P(X = x) = k(x² – 1), where k is a constant.
A sports event is taking place for 4 days, beginning on Sunday. The probability that it will rain on Sunday is 0.4. On any subsequent day, the prob...
The following back-to-back stem-and-leaf diagram represents the monthly salaries, in dollars, of 27 employees at each of two companies, A and B. Co...
A fair 5-sided spinner has sides labelled 1, 2, 3, 4, 5. The spinner is spun repeatedly until a 2 is obtained on the side on which the spinner land...
The lengths of Western bluebirds are normally distributed with mean 16.5cm and standard deviation 0.6 cm. A random sample of 150 of these birds is ...
In a group of 25 people there are 6 swimmers, 8 cyclists and 11 runners. Each person competes in only one of these sports. A team of 7 people is se...
Two fair coins are thrown at the same time repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variab...