Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The number of accidents on a certain road has a Poisson distribution with mean 0.4 per 50-day period.
Bottles of Lanta contain approximately 300 ml of juice. The volume of juice, in millilitres, in a bottle is 300 + X, where X is a random variable w...
The volumes, in millilitres, of large and small cups of tea are modelled by the distributions N(200, 30) and N(110, 20) respectively.
A national survey shows that 95% of year 12 students use social media. Arvin suspects that the percentage of year 12 students at his college who us...
By using a suitable substitution, solve the equation 4 (2x-3)2 - - 3 = 0.
Solve the equation tan θ + 2 sin θ = 3 for 0° < θ < 180°. tan θ-2 sin θ
A line has equation y = 3x + k and a curve has equation y = x² + kx + 6, where k is a constant. Find the set of values of k for which the line and ...
In the diagram, the graph of y = f(x) is shown with solid lines. The graph shown with broken lines is a transformation of y = f(x). [Figure 5.1]
A curve is such that dy 6 = and A (1, -3) lies on the curve. A point is moving along the curve dx (3x-2)3 and at A the y-coordinate of the point is...
Functions f and g are defined as follows: f : x → x² + 2x + 3 for x ≤ −1, g: x → 2x + 1 for x ≥ −1.
The points A (7, 1), B (7, 9) and C (1, 9) are on the circumference of a circle.
The first term of a progression is cos θ, where 0 < θ < ½π.
The diagram shows a sector ABC which is part of a circle of radius a. The points D and E lie on AB and AC respectively and are such that AD = AE = ...
The diagram shows the curve with equation y = 9(x^(½) - 4x^(-½)). The curve crosses the x-axis at the point A. [Figure 11.1]
Solve the equation sec² θ cot θ = 8 for 0 < θ < π.
The parametric equations of a curve are x = e2t cos 4t, y = 3 sin 2t. Find the gradient of the curve at the point for which t = 0.
The diagram shows part of the curve with equation y = 5x / (4x³ + 1). The shaded region is bounded by the curve and the lines x = 1, x = 3 and y = ...
The polynomial p(x) is defined by p(x) = x² + ax + b, where a and b are constants. It is given that (x + 2) is a factor of p(x) and that the remain...
Solve the equation ln(x³ – 3) = 3 ln x – ln 3. Give your answer correct to 3 significant figures.
The polynomial ax³ + 5x² – 4x + b, where a and b are constants, is denoted by p(x). It is given that (x + 2) is a factor of p(x) and that when p(x)...
By first expressing the equation tan(x + 45°) = 2 cotx + 1 as a quadratic equation in tanx, solve the equation for 0° < x < 180°.
The variables x and y satisfy the differential equation (1-cosx) dy/dx = y sinx. It is given that y = 4 when x = π.
Express √7 sin x + 2 cos x in the form R sin(x + α), where R > 0 and 0° < α < 90°. State the exact value of R and give α correct to 2 decimal place...
Let f(x) = 5a / ((2x – a)(3a – x)), where a is a positive constant.
Two lines have equations r = (1, 3, 2) + s(2, -1, 3) and r = (2, 1, 4) + t(-1, 1, 4)
The complex numbers u and v are defined by u = −4 + 2i and v = 3 + i.
Let f(x) = (e^(2x) + 1) / (e^(2x) - 1), for x > 0.
The diagram shows the curve y = sin 2x cos²x for 0 ≤ x ≤ ½π, and its maximum point M. [Figure 10.1]
Two particles P and Q of masses 0.2 kg and 0.3 kg respectively are free to move in a horizontal straight line on a smooth horizontal plane. P is pr...
A car of mass 1400 kg is travelling at constant speed up a straight hill inclined at α to the horizontal, where sin α = 0.1. There is a constant re...
A particle Q of mass 0.2 kg is held in equilibrium by two light inextensible strings PQ and QR. P is a fixed point on a vertical wall and R is a fi...
An elevator moves vertically, supported by a cable. The diagram shows a velocity-time graph which models the motion of the elevator. The graph cons...
A block of mass 5 kg is being pulled along a rough horizontal floor by a force of magnitude X N acting at 30° above the horizontal (see diagram). T...
A particle moves in a straight line. It starts from rest from a fixed point O on the line. Its velocity at time t s after leaving O is v m s⁻¹, whe...
Two particles P and Q of masses 0.5 kg and mkg respectively are attached to the ends of a light inextensible string. The string passes over a fixed...
A fair spinner with 5 sides numbered 1, 2, 3, 4, 5 is spun repeatedly. The score on each spin is the number on the side on which the spinner lands.
Georgie has a red scarf, a blue scarf and a yellow scarf. Each day she wears exactly one of these scarves. The probabilities for the three colours ...
The time spent by shoppers in a large shopping centre has a normal distribution with mean 96 minutes and standard deviation 18 minutes.
The random variable X takes the values 1, 2, 3, 4 only. The probability that X takes the value x is kx(5-x), where k is a constant.
A driver records the distance travelled in each of 150 journeys. These distances, correct to the nearest km, are summarised in the following table....
There are 400 students at a school in a certain country. Each student was asked whether they preferred swimming, cycling or running and the results...
A construction company notes the time, t days, that it takes to build each house of a certain design. The results for a random sample of 60 such ho...
The diagram shows the graph of the probability density function, f, of a random variable X. [Figure 2.1]
An architect wishes to investigate whether the buildings in a certain city are higher, on average, than buildings in other cities. He takes a large...
On average, 1 in 400 microchips made at a certain factory are faulty. The number of faulty microchips in a random sample of 1000 is denoted by X.