Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
Anton believes that 10% of students at his college are left-handed. Aliya believes that this is an under-estimate. She plans to carry out a hypothe...
Batteries of type A are known to have a mean life of 150 hours. It is required to test whether a new type of battery, type B, has a shorter mean li...
Each box of Seeds & Raisins contains S grams of seeds and R grams of raisins. The weight of a box, when empty, is B grams. S, R and B are independe...
The number of clients who arrive at an information desk has a Poisson distribution with mean 2.2 per 5-minute period.
A random sample of 5 values of a variable X is given below. 2 3 3 5 a
The random variables X and W have probability density functions f and g defined as follows: f(x) = { p(a²-x²) for 0 ≤ x ≤ a, 0 otherwise, g(w) = {...
The equation of a curve is such that dy/dx = 4/(x-3)³ for x > 3. The curve passes through the point (4, 5). Find the equation of the curve.
The coefficient of x⁴ in the expansion of (x + a)⁶ is p and the coefficient of x² in the expansion of (ax + 3)⁴ is q. It is given that p + q = 276....
Solve the equation 8x⁶ + 215x³ – 27 = 0.
The diagram shows the curve with equation y = 10x^(1/2) - (5/2)x^(3/2) for x > 0. The curve meets the x-axis at the points (0, 0) and (4, 0). [Figu...
The diagram shows a sector OAB of a circle with centre O. Angle AOB = θ radians and OP = AP = x. [Figure 6.1]
The diagram shows the graph of y = f(x) where the function f is defined by f(x) = 3 + 2 sin(1/4 x) for 0 ≤ x ≤ 2π. [Figure 8.1]
The second term of a geometric progression is 16 and the sum to infinity is 100.
The equation of a circle is (x − a)² + (y − 3)² = 20. The line y = (1/2)x + 6 is a tangent to the circle at the point P.
The equation of a curve is y = k√(4x + 1) − x + 5, where k is a positive constant.
Use logarithms to solve the equation 12x = 32x+1. Give your answer correct to 3 significant figures.
A curve has equation y = (2 + 3 ln x) / (1 + 2x) Find the equation of the tangent to the curve at the point (1, ⅔). Give your answer in the form ax...
It is given that ∫[from 0 to a] (3e^(2x) - 1) dx = 12, where a is a positive constant.
The polynomial p(x) is defined by p(x) = 2x³ + 3x² + kx − 30, where k is a constant. It is given that (x – 3) is a factor of p(x).
The diagram shows the curve with parametric equations x = 4e^(2t), y = 5e^(-t) cos 2t, for -¼π ≤ t ≤ ¼π. The curve has a maximum point M.
Show that ∫[from ¼π to ½π] (4 cos² 2x + 1/cos² x) dx = ¾√3 + ⅙π - 1.
The variables x and y satisfy the equation y = Ae^(A-B)x, where A and B are constants. The graph of ln y against x is a straight line passing throu...
The diagram shows part of the curve y = 6/(2x + 3). The shaded region is bounded by the curve and the lines x = 6 and y = 2.
The diagram shows the graph of y = 3 – e^(-½x). It is given that the two roots of 3 – e^(-½x) = |5x – 4| are denoted by α and β, where α < β.
The diagram shows the curve with equation y = e^(–½x)(x² – 5x + 4). The curve crosses the x-axis at the points A and B, and has a maximum at the po...
A curve has parametric equations x = (2t + 3)/(t + 2), y = t² + at + 1, where a is a constant. It is given that, at the point P on the curve, the g...
The diagram shows the graph of y = 3 – e⁻½ˣ. It is given that the two roots of 3 – e⁻½ˣ = |5x – 4| are denoted by α and β, where α < β.
The diagram shows the curve with equation y = e⁻½ˣ(x² – 5x + 4). The curve crosses the x-axis at the points A and B, and has a maximum at the point C.
Solve the equation 3e²ˣ - 4e⁻²ˣ = 5. Give the answer correct to 3 decimal places.
Find the coefficient of x³ in the binomial expansion of (3 + x)√1 + 4x.
The equation of a curve is x²y – ay² = 4a³, where a is a non-zero constant.
Relative to the origin O, the points A, B and C have position vectors given by OA = (2 3 3), OB = (4 1 2) and OC = (3 -2 -4). The quadrilateral ABC...
The variables x and y satisfy the differential equation cos 2x dy/dx = 4 tan 2x / sin² 3y, where 0 ≤ x < π/4. It is given that y = 0 when x = π/6. ...
Let f(x) = (3-3x²) / ((2x + 1)(x + 2)²).
The constant a is such that ∫₀ᵃ xe⁻²ˣ dx = 1/8.
The polynomial x³ + 5x² + 31x + 75 is denoted by p(x).
Solve the inequality |5x-3| < 2|3x – 7|.
Solve the equation ln(2x² – 3) = 2 ln x – ln 2, giving your answer in an exact form.
Solve the equation 2 cos x – cos (x/2) = 1 for 0 ≤ x ≤ 2π.
The complex number 2 + yi is denoted by a, where y is a real number and y < 0. It is given that f(a) = a³ – a² – 2a.
The equation cot (x/2) = 3x has one root in the interval 0 < x < π, denoted by α.