Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
A geometric progression has a first term of 3 and a second term of 2.4. For this progression, find
DO NOT USE A CALCULATOR IN THIS QUESTION. In this question lengths are in centimetres. [Figure 8.1: A diagram showing a triangle ABC with angle A =...
In the diagram OP = 2b, OS = 3a, SR = b and PQ = a. The lines OR and QS intersect at X. [Figure 9.1: A diagram showing vectors OP, OS, SR, PQ. Line...
The number, b, of bacteria in a sample is given by b = P+Qe^(2t), where P and Q are constants and t is time in weeks. Initially there are 500 bacte...
The diagram shows the graph of the cubic function y = f(x). The intercepts of the curve with the axes are all integers. [Figure 1.1]
Solve the equation `cot(2x+π/3)-√3 = 0`, where `-π < x < π` radians. Give your answers in terms of π.
Find the possible values of the constant c for which the line `y = c` is a tangent to the curve `y = 5 sin(x/3) + 4`.
DO NOT USE A CALCULATOR IN THIS QUESTION. The polynomial `p(x) = 10x³ + ax² -10x+b`, where a and b are integers, is divisible by `2x+1`. When `p(x)...
[Figure 7.1] The diagram shows triangle OAC, where `OA = a`, `OB = b` and `OC = c`. The point B lies on the line AC such that `AB:BC = m:n`, where ...
The diagram shows a circle, centre O, radius 12cm, and a rectangle ABCD. The diagonals AC and BD intersect at O. The sides AB and AD of the rectang...
The diagram shows the graph of the curve `y = 1/(x+2)² + 3/(x+2)` for `x > -2`. The points A and B lie on the curve such that the x-coordinates of ...
(a) The diagram shows the velocity-time graph for a particle P, travelling in a straight line with velocity vms⁻¹ at a time t seconds. P accelerate...
[Figure showing a Cartesian coordinate plane with x from -6 to 6 and y from 0 to 20.]
It is given that d²y/dx² = e^(2x) + 1/((x+1)²) for x > -1.
Variables x and y are such that when √y is plotted against log₂(x+1), where x > -1, a straight line is obtained which passes through (2,10.4) and (...
The diagram shows a circle, centre O, radius 10cm. The points A and B lie on the circumference of the circle. The tangent at A and the tangent at B...
A function f(x) is such that f(x) = ln(2x+3)+ln4, for x > a, where a is a constant.
The first three terms of an arithmetic progression are lgx, lgx³, lgx⁹, where x > 0.
A particle P moves in a straight line such that, t seconds after passing through a fixed point O, its displacement, s metres, is given by s = (2t+1...
The first three terms, in descending powers of x, of the expansion of (ax+3)(1-b/x)⁵ can be written as 32x⁵-160x⁴+cx³, where a, b and c are constan...
Solve the following simultaneous equations. x+5y=-4 3y-xy = 6
Solve the equation 4e^(2x-3) = 7e^(5-x).
In this question a and b are constants. The normal to the curve y = ax⁻³ + 3x - 2 at the point where x = 1 has equation y = -1/4 x + b. Find the va...
Solve the equation log₃(11x-8) = 1 + 2/logₓ3 given that x > 1.
DO NOT USE A CALCULATOR IN THIS QUESTION. Find the x-coordinates of the points of intersection of the curves y = x³-2x²-4x-16 and y = 7x³-7x²-17x-4.
A 4-digit code is to be formed using 4 different numbers selected from 2, 3, 4, 5, 6, 7, 8 and 9. Find how many possible codes there are if the cod...
In this question all lengths are in centimetres. The volume of a cylinder with radius r and height h is πr²h and its curved surface area is 2πrh. T...
In this question all lengths are in centimetres. The diagram shows triangle ABC which has area 2√5/3 cm². Angle A is acute. [Figure 9.1]
The coordinates of points A and B are (-5,6) and (4, – 6) respectively. The point C lies on the line AB, between A and B, such that AC/CB = 1/2.
[Figure 1.0] The diagram shows the graph of the cubic polynomial y = f(x).
The function g is defined by g(x) = 5 sin(3x/4) - 2 for all values of x.