Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The diagram shows the curves y = cos x and y = k/(1 + x), where k is a constant, for 0 ≤ x ≤ ½π. The curves touch at the point where x = p. [Figure...
With respect to the origin O, the points A and B have position vectors given by OA = 6i + 2j and OB = 2i + 2j + 3k. The midpoint of OA is M. The po...
Solve the inequality |2x - 1 | > 3|x + 2|.
Find the exact value of ∫₀¹ (2-x)e⁻²ˣ dx.
The equation of a curve is y = x tan⁻¹(½x).
By first expressing the equation tan θ tan(θ + 45°) = 2 cot 2θ as a quadratic equation in tan θ, solve the equation for 0° < θ < 90°.
Let f(x) = 2 / ((2x−1)²(2x + 1)).
Relative to the origin O, the points A, B and D have position vectors given by OA = i + 2j + k, OB = 2i + 5j + 3k and OD = 3i + 2k. A fourth point ...
A tank containing water is in the form of a hemisphere. The axis is vertical, the lowest point is A and the radius is r, as shown in the diagram. T...
Three coplanar forces of magnitudes 100 N, 50 N and 50 N act at a point A, as shown in the diagram. The value of cos α is 3/5. [The diagram]
A car of mass 1800 kg is towing a trailer of mass 400 kg along a straight horizontal road. The car and trailer are connected by a light rigid tow-b...
A particle P is projected vertically upwards with speed 5 m s⁻¹ from a point A which is 2.8 m above horizontal ground.
The diagram shows a ring of mass 0.1 kg threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and t...
A child of mass 35 kg is swinging on a rope. The child is modelled as a particle P and the rope is modelled as a light inextensible string of lengt...
A particle moves in a straight line AB. The velocity v m s⁻¹ of the particle t s after leaving A is given by v = k(t² – 10t + 21), where k is a con...
A particle P of mass 0.3 kg, lying on a smooth plane inclined at 30° to the horizontal, is released from rest. P slides down the plane for a distan...
Particles P of mass mkg and Q of mass 0.2 kg are free to move on a smooth horizontal plane. P is projected at a speed of 2ms⁻¹ towards Q which is s...
A minibus of mass 4000 kg is travelling along a straight horizontal road. The resistance to motion is 900 N.
Four coplanar forces of magnitudes 40 N, 20 N, 50 N and F N act at a point in the directions shown in the diagram. The four forces are in equilibrium.
A car starts from rest and moves in a straight line with constant acceleration am s⁻² for a distance of 50 m. The car then travels with constant ve...
A block B of mass 4 kg is pushed up a line of greatest slope of a smooth plane inclined at 30° to the horizontal by a force applied to B, acting in...
A particle travels in a straight line PQ. The velocity of the particle t s after leaving P is v m s⁻¹, where v = 4.5 + 4t – 0.5t².
Two particles A and B, of masses 3m kg and 2m kg respectively, are attached to the ends of a light inextensible string. The string passes over a fi...
The score when two fair six-sided dice are thrown is the sum of the two numbers on the upper faces.
A company produces small boxes of sweets that contain 5 jellies and 3 chocolates. Jemeel chooses 3 sweets at random from a box.
In a music competition, there are 8 pianists, 4 guitarists and 6 violinists. 7 of these musicians will be selected to go through to the final. How ...
On Mondays, Rani cooks her evening meal. She has a pizza, a burger or a curry with probabilities 0.35, 0.44, 0.21 respectively. When she cooks a pi...
The lengths of female snakes of a particular species are normally distributed with mean 54 cm and standard deviation 6.1 cm.
The numbers of chocolate bars sold per day in a cinema over a period of 100 days are summarised in the following table. Number of chocolate bars so...
For n values of the variable x, it is given that Σ(x – 50) = 144 and Σx = 944. Find the value of n.
A total of 500 students were asked which one of four colleges they attended and whether they preferred soccer or hockey. The numbers of students in...
Two machines, A and B, produce metal rods of a certain type. The lengths, in metres, of 19 rods produced by machine A and 19 rods produced by machi...
Trees in the Redian forest are classified as tall, medium or short, according to their height. The heights can be modelled by a normal distribution...
A fair three-sided spinner has sides numbered 1, 2, 3. A fair five-sided spinner has sides numbered 1, 1, 2, 2, 3. Both spinners are spun once. For...
Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged...
On any given day, the probability that Moena messages her friend Pasha is 0.72.
Juan goes to college each day by any one of car or bus or walking. The probability that he goes by car is 0.2, the probability that he goes by bus ...
In a certain large college, 22% of students own a car.
In a certain town, the time, X hours, for which people watch television in a week has a normal distribution with mean 15.8 hours and standard devia...
A fair four-sided spinner has edges numbered 1, 2, 2, 3. A fair three-sided spinner has edges numbered -2, -1, 1. Each spinner is spun and the numb...
A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The random variable X denotes the number of throws required to obtain ...
The annual salaries, in thousands of dollars, for 11 employees at each of two companies A and B are shown below. Company A 30 32 35 41 41 42 47 49 ...
The lengths, X centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows. 5.2 4.8 5.5 6.1 4.8 3.9 4.4 It is now giv...
In the past the yield of a certain crop, in tonnes per hectare, had mean 0.56 and standard deviation 0.08. Following the introduction of a new fert...