Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The random variable X takes the values −1, 1, 2, 3 only. The probability that X takes the value x is kx², where k is a constant.
The weights, in kg, of the 11 members of the Dolphins swimming team and the 11 members of the Sharks swimming team are shown below. Dolphins 62 75...
The results of a survey by a large supermarket show that 35% of its customers shop online.
Find the number of different arrangements that can be made of all 9 letters in the word CAMERAMAN in each of the following cases.
The function f is defined by f(x) = 1 3x + 2 + x² for x < -1.
The graph of y = f(x) is transformed to the graph of y = 1 + f(x).
The diagram shows part of the curve with equation y = x² + 1. The shaded region enclosed by the curve, the y-axis and the line y = 5 is rotated thr...
A curve has equation y = x² – 2x – 3. A point is moving along the curve in such a way that at P the y-coordinate is increasing at 4 units per secon...
Solve the equation tan θ + 3 sin θ + 2 = 2 tan θ - 3 sin θ + 1 for 0° ≤ θ ≤ 90°.
The coefficient of 1/x in the expansion of (2x + a/x²)⁵ is 720.
The diagram shows a sector AOB which is part of a circle with centre O and radius 6cm and with angle AOB = 0.8 radians. The point C on OB is such t...
A woman's basic salary for her first year with a particular company is $30000 and at the end of the year she also gets a bonus of $600.
The function f is defined by f(x) = 2x² + 12x + 11 for x ≤ −4.
The gradient of a curve at the point (x, y) is given by dy/dx = 2(x + 3)¹/² - x. The curve has a stationary point at (a, 14), where a is a positive...
A diameter of a circle C₁ has end-points at (−3, −5) and (7, 3).
Solve the equation 2 sin(θ + 30°) + 5 cos θ = 2 sin θ for 0° < θ < 90°.
It is given that ∫⁽³ᵃ⁾ₐ (2/(2x-5)) dx = ln(7/2). Find the value of the positive constant a.
A curve has equation 3x² - y² – 4 ln(2y + 3) = 26. Find the equation of the tangent to the curve at the point (3, −1).
A curve has equation y = x³e⁻⁰.²ˣ where x ≥ 0. At the point P on the curve, the gradient of the curve is 15.
The diagram shows part of the curve with equation y = 4 sin²x + 8 sin x + 3, where x is measured in radians. The curve crosses the x-axis at the po...
Solve the equation In 3 + ln(2x + 5) = 2ln(x + 2). Give your answer in a simplified exact form.
Find ∫(π/3 to π/6) x sec²x dx. Give your answer in a simplified exact form.
The variables x and y satisfy the differential equation dy/dx = (1+4y²) / e^(x²). It is given that y = 0 when x = 1.
The equation of a curve is x³ + 3xy² – y³ = 5.
In the diagram, OABCDEFG is a cuboid in which OA = 2 units, OC = 3 units and OD = 2 units. Unit vectors i, j and k are parallel to OA, OC and OD re...
Let f(x) = (2 + 11x - 10x²) / ((1 + 2x)(1-2x)(2 + x)).
A lorry of mass 16000 kg is travelling along a straight horizontal road. The engine of the lorry is working at constant power. The work done by the...
A particle P of mass 0.4 kg is on a rough horizontal floor. The coefficient of friction between P and the floor is μ. A force of magnitude 3 N is a...
The diagram shows the vertical cross-section of a surface. A, B and C are three points on the cross-section. The level of B is h m above the level ...
A cyclist travels along a straight road with constant acceleration. He passes through points A, B and C. The cyclist takes 2 seconds to travel alon...
Coplanar forces, of magnitudes FN, 3 N, 6 N and 4 N, act at a point P, as shown in the diagram. [Figure 5.1]
On a straight horizontal test track, driverless vehicles (with no passengers) are being tested. A car of mass 1600 kg is towing a trailer of mass 7...
A particle moves in a straight line through the point O. The displacement of the particle from O at time t s is s m, where s = t² - 3t + 2 for 0 ≤ ...
The 40 members of a club include Ranuf and Saed. All 40 members will travel to a concert. 35 members will travel in a coach and the other 5 will tr...
An ordinary fair die is thrown repeatedly until a 1 or a 6 is obtained.
The weights of apples of a certain variety are normally distributed with mean 82 grams. 22% of these apples have a weight greater than 87 grams.
Richard has 3 blue candles, 2 red candles and 6 green candles. The candles are identical apart from their colours. He arranges the 11 candles in a ...
In Greenton, 70% of the adults own a car. A random sample of 8 adults from Greenton is chosen.
Box A contains 7 red balls and 1 blue ball. Box B contains 9 red balls and 5 blue balls. A ball is chosen at random from box A and placed in box B....
Helen measures the lengths of 150 fish of a certain species in a large pond. These lengths, correct to the nearest centimetre, are summarised in th...
The booklets produced by a certain publisher contain, on average, 1 incorrect letter per 30 000 letters, and these errors occur randomly. A randoml...
Lengths of a certain species of lizard are known to be normally distributed with standard deviation 3.2 cm. A naturalist measures the lengths of a ...
In the past, the mean time taken by Freda for a particular daily journey was 39.2 minutes. Following the introduction of a one-way system, Freda wi...