Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The masses, in kilograms, of large sacks of flour and small sacks of flour have the independent distributions N(40, 1.5²) and N(12, 0.7²) respectiv...
A fair spinner has five sides numbered 1, 2, 3, 4, 5. The score on one spin is denoted by X. Fiona has another spinner, also with five sides number...
Each week a sports team plays one home match and one away match. In their home matches they score goals at a constant average rate of 2.1 goals per...
The length of time, T minutes, that a passenger has to wait for a bus at a certain bus stop is modelled by the probability density function given b...
The masses, in grams, of plums of a certain type have the distribution N(40.4, 5.2²). The plums are packed in bags, with each bag containing 6 rand...
A shop obtains apples from a certain farm. It has been found that 5% of apples from this farm are Grade A. Following a change in growing conditions...
In the data-entry department of a certain firm, it is known that 0.12% of data items are entered incorrectly, and that these errors occur randomly ...
The score on one spin of a 5-sided spinner is denoted by the random variable X with probability distribution as shown in the table. x | 0 | 1 | 2 |...
The random variable X has the distribution Po(λ).
A random variable X has probability density function given by f(x) = { k/x² 1 ≤ x ≤ a, 0 otherwise, where k and a are positive constants.
A random sample of 100 values of a variable X is taken. These values are summarised below. n = 100 Σχ = 1556 Σχ² = 29004 Calculate unbiased estimat...
Each day at the gym, Sarah completes three runs. The distances, in metres, that she completes in the three runs have the independent distributions ...
The number of customers who visit a particular shop between 9.00 am and 10.00 am has the distribution Po(λ). In the past the value of λ was 5.2. Fo...
The random variable A has the distribution Po(1.5). A₁ and A₂ are independent values of A.
Sunita has a six-sided die with faces marked 1, 2, 3, 4, 5, 6. The probability that the die shows a six on any throw is p. Sunita throws the die 50...
The length, X centimetres, of worms of a certain type is modelled by the probability density function f(x) = { (1/125)(10-x)(x - 5) 5 ≤ x ≤ 10,...
A market researcher is investigating the length of time that customers spend at an information desk. He plans to choose a sample of 50 customers on...
The equation of a curve is y = (x − 3)√x + 1 + 3. The following points lie on the curve. Non-exact values are rounded to 4 decimal places. A (2, k)...
The coefficient of x in the expansion of (4x + 10/x)³ is p. The coefficient of 1/x in the expansion of (2x + k/x)⁵ is q. Given that p = 6q, find th...
The function f is defined by f(x) = 2x² + 3 for x ≥ 0.
Points A and B have coordinates (8, 3) and (p, q) respectively. The equation of the perpendicular bisector of AB is y = −2x + 4. Find the values of...
The point A has coordinates (1, 5) and the line l has gradient −2/3 and passes through A. A circle has centre (5, 11) and radius √52.
The first, second and third terms of an arithmetic progression are a, 3/2a and b respectively, where a and b are positive constants. The first, sec...
The diagram shows part of the curve with equation y² = x – 2 and the lines x = 5 and y = 1. The shaded region enclosed by the curve and the lines i...
The gradient of a curve is given by dy/dx = 6(3x – 5)³ – kx², where k is a constant. The curve has a stationary point at (2, -3.5).
The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm, held together by a thin rope which is wrapped tightly around th...
The solutions of the equation 5|x| = 5 – 2x are x = a and x = b, where a < b. Find the value of |3a – 1| + |7b – 1|.
Solve the equation sin(2θ + 30°) = 5 cos(2θ + 60°) for 0° < θ < 180°.
The diagram shows the curve with equation y = (ln x)² – 2 ln x. The curve crosses the x-axis at the points A and B, and has a minimum point M.
The diagram shows the curve with parametric equations x = 4t + e²ᵗ, y = 6t sin 2t, for 0 ≤ t ≤ 1. The point P on the curve has parameter p and y-co...
The diagram shows the curve with equation y = (lnx)² – 2 lnx. The curve crosses the x-axis at the points A and B, and has a minimum point M. [Figur...
The diagram shows the curve with parametric equations x = 4t + e^(2t), y = 6t sin 2t, for 0 ≤ t ≤ 1. The point P on the curve has parameter p and y...
Solve the inequality 2|3x – 1| < |x + 1|.
Find the real root of the equation (2eˣ + e⁻ˣ) / (2 + eˣ) = 3, giving your answer correct to 3 decimal places. Your working should show clearly tha...
Given that cos(x – 30°) = 2 sin(x + 30°), show that tan x = (2-√3) / (1-2√3)
Prove that (1 - cos 2θ) / (1 + cos 2θ) = tan² θ.
The parametric equations of a curve are x = ln(2 + 3t), y = t / (2 + 3t)
The diagram shows the curve y = (tan⁻¹ x) / √x and its maximum point M where x = a. [Figure 7.1]
With respect to the origin O, the points A and B have position vectors given by OA = (1, 2, 1) and OB = (3, 1, -2). The line l has equation r = (2,...
The equation of a curve is y = x⁻² ln x for x > 0. The curve has one stationary point.
The variables x and t satisfy the differential equation dx/dt = x²(1 + 2x), and x = 1 when t = 0. Using partial fractions, solve the differential e...