Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The function f is such that f'(x) = 5 – 2x² and (3, 5) is a point on the curve y = f(x). Find f(x).
In the diagram, AYB is a semicircle with AB as diameter and OAXB is a sector of a circle with centre O and radius r. Angle AOB = 2θ radians. Find a...
Variables u, x and y are such that u = 2x(y – x) and x + 3y = 12. Express u in terms of x and hence find the stationary value of u.
A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height, h m, of ...
The point C lies on the perpendicular bisector of the line joining the points A (4, 6) and B (10, 2). C also lies on the line parallel to AB throug...
Relative to an origin O, the position vectors of points A and B are given by OA = 2i + 4j + 4k and OB = 3i + j + 4k.
The equation of a curve is y = 4 / (2x – 1).
The function f is defined by f : x → 2x² – 6x + 5 for x ∈ R.
Functions f and g are defined by f:x ↦ 10 – 3x, x ∈ R, g:x ↦ 10 / (3 – 2x), x ∈ R, x ≠ 3/2. Solve the equation ff(x) = gf(2).
A curve is such that dy/dx = 8 / (5 – 2x)². Given that the curve passes through (2, 7), find the equation of the curve.
Relative to an origin O, the position vectors of points A and B are given by OA = 2i – 5j – 2k and OB = 4i – 4j + 2k. The point C is such that AB =...
Find the term that is independent of x in the expansion of
In the diagram, triangle ABC is right-angled at C and M is the mid-point of BC. It is given that angle ABC = 1/3 π radians and angle BAM = θ radian...
The diagram shows a circle with radius r cm and centre O. The line PT is the tangent to the circle at P and angle POT = α radians. The line OT meet...
Three points have coordinates A (0, 7), B (8, 3) and C (3k, k). Find the value of the constant k for which
A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.
The diagram shows the part of the curve y = 8/x + 2x for x > 0, and the minimum point M. [Figure 10.1]
The function f is defined by f : x → 6x − x² – 5 for x ∈ R.
The point A has coordinates (−2, 6). The equation of the perpendicular bisector of the line AB is 2y = 3x + 5.
The diagram shows a circle with radius r cm and centre O. Points A and B lie on the circle and ABCD is a rectangle. Angle AOB = 2θ radians and AD =...
A curve has equation y = 3 + 12 / (2 - x).
The diagram shows the straight line x + y = 5 intersecting the curve y = 4/x at the points A (1, 4) and B (4, 1). Find, showing all necessary worki...
Relative to an origin O, the position vectors of three points A, B and C are given by OA = 3i + pj – 2pk, OB = 6i + (p + 4)j + 3k and OC = (p − 1)i...
The equation of a curve is y = 8√x − 2x.
The function f is defined by f(x) = 3 tan(1/2 x) - 2, for -1/2 π ≤ x ≤ 1/2 π.
Given that 5ˣ = 3⁴ʸ, use logarithms to show that y = mx and find the value of the constant m correct to 3 significant figures.
Solve the inequality |4 – x| ≤ |3 – 2x|.
Given that ∫₀ᵃ 4e^(2x+3) dx = 835, find the value of the constant a correct to 3 significant figures.
The sequence of values given by the iterative formula xn+1 = (2x_n² + x_n + 9) / (x_n + 1)² with x₁ = 2, converges to α.
The diagram shows the curve y = tan 2x for 0 ≤ x ≤ (1/6)π. The shaded region is bounded by the curve and the lines x = (1/6)π and y = 0. [Figure 6]
The parametric equations of a curve are x = t³ + 6t + 1, y = t⁴ – 2t³ + 4t² – 12t + 5.
The diagram shows the curve with equation y = 3x² ln((1/2)x). The curve crosses the x-axis at the point P and has a minimum point M. [Figure 8]
Solve the equation |x + a| = |2x – 5a|, giving x in terms of the positive constant a.
Use logarithms to solve the equation 3ˣ⁺⁴ = 5²ˣ, giving your answer correct to 3 significant figures.
Find the equation of the tangent to the curve y = e⁴ˣ / (2x + 3) at the point on the curve for which x = 0. Give your answer in the form ax + by + ...
The variables x and y satisfy the equation y = K/a²ˣ, where K and a are constants. The graph of ln y against x is a straight line passing through t...
The diagram shows the curve with parametric equations x = 2 - cos 2t, y = 2 sin³ t + 3 cos³t + 1 for 0 ≤ t ≤ ½π. The end-points of the curve are (1...