Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
Solve the inequality |2x + 1| < 3|x-2|.
Expand 1/√(1 + 6x) in ascending powers of x, up to and including the term in x³, simplifying the coefficients.
It is given that x = ln(1 – y) – ln y, where 0 < y < 1.
The parametric equations of a curve are x = ln cos θ, y = 3θ - tan θ, where 0 ≤ θ < ½π.
The diagram shows a semicircle with centre O, radius r and diameter AB. The point P on its circumference is such that the area of the minor segment...
The plane with equation 2x + 2y – z = 5 is denoted by m. Relative to the origin O, the points A and B have coordinates (3, 4, 0) and (−1, 0, 2) res...
Throughout this question the use of a calculator is not permitted. The complex numbers u and w are defined by u = −1 + 7i and w = 3 + 4i.
The diagram shows the curve y = sin x cos² 2x for 0 ≤ x ≤ ¼π and its maximum point M. [Figure X.X]
The parametric equations of a curve are x = t² + 1, y = 4t + ln(2t − 1).
In a certain chemical process a substance A reacts with and reduces a substance B. The masses of A and B at time t after the start of the process a...
Throughout this question the use of a calculator is not permitted. The complex number 2 – i is denoted by u.
Let f(x) = (5x² - 7x + 4) / ((3x + 2)(x² + 5)).
Relative to the origin O, the point A has position vector given by OA = i + 2j + 4k. The line l has equation r = 9i – j + 8k + μ(3i – j + 2k).
The diagram shows the curve y = x² cos 2x for 0 ≤ x ≤ ¼π. The curve has a maximum point at M where x = p. [Figure 10.1]
Prove the identity cot x - tan x / cot x + tan x = cos 2x.
Expand (3 + 2x)⁻³ in ascending powers of x up to and including the term in x², simplifying the coefficients.
Using the substitution u = eˣ, solve the equation 4e⁻ˣ = 3eˣ + 4. Give your answer correct to 3 significant figures.
Find the exact value of ∫(from 0 to π/2) θ sin(½θ) dθ.
A curve has equation y = ¾ ln(1 + 3 cos²x) for 0 ≤ x ≤ ½π.
The equation cot x = 1 - x has one root in the interval 0 < x < π, denoted by α.
The diagram shows a sketch of the curve y = e^(½x) / x for x > 0, and its minimum point M. [Figure 7.1]
In a certain chemical reaction, a compound A is formed from a compound B. The masses of A and B at time t after the start of the reaction are x and...
Let f(x) = (3x²-4) / (x²(3x + 2)).
The points A and B have position vectors given by OA = i − 2j + 2k and OB = 3i + j + k. The line l has equation r = 2i + j + mk + µ(i – 2j – 4k), w...
Throughout this question the use of a calculator is not permitted.
A particle of mass 0.6 kg is dropped from a height of 8 m above the ground. The speed of the particle at the instant before hitting the ground is 1...
A particle of mass 0.8 kg is projected with a speed of 12 m s⁻¹ up a line of greatest slope of a rough plane inclined at an angle of 10° to the hor...
Two light inextensible strings are attached to a particle of weight 25 N. The strings pass over two smooth fixed pulleys and have particles of weig...
A car of mass 800 kg is moving up a hill inclined at θ° to the horizontal, where sin θ = 0.15. The initial speed of the car is 8 m s⁻¹. Twelve seco...
A particle P moves in a straight line ABCD with constant deceleration. The velocities of P at A, B and C are 20 m s⁻¹, 12 m s⁻¹ and 6 m s⁻¹ respect...
A particle P moves in a straight line passing through a point O. At time t s, the velocity of P, v m s⁻¹, is given by v = qt + rt², where q and r a...
As shown in the diagram, a particle A of mass 0.8 kg lies on a plane inclined at an angle of 30° to the horizontal and a particle B of mass 1.2 kg ...
The diagram shows a wire ABCD consisting of a straight part AB of length 5 m and a part BCD in the shape of a semicircle of radius 6 m and centre O...
A particle A moves in a straight line with constant speed 10 m s⁻¹. Two seconds after A passes a point O on the line, a particle B passes through O...
A car of mass 1200 kg is moving on a straight road against a constant force of 850 N resisting the motion.
The diagram shows a fixed block with a horizontal top surface and a surface which is inclined at an angle of θ° to the horizontal, where sin θ = 3/...
A man pushes a wheelbarrow of mass 25 kg along a horizontal road with a constant force of magnitude 35 N at an angle of 20° below the horizontal. T...
The four coplanar forces shown in the diagram are in equilibrium. Find the values of P and θ. [Figure X.X]
A train travels between two stations, A and B. The train starts from rest at A and accelerates at a constant rate for T s until it reaches a speed ...
A particle P moves in a straight line starting from a point O. At time t s after leaving O, the velocity, v m s⁻¹, of P is given by v = (2t – 5)³.
A particle is projected vertically upwards from a point O with a speed of 12 m s⁻¹. Two seconds later a second particle is projected vertically upw...