Cambridge Past Paper Questions
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The position vectors of points A and B, relative to an origin O, are given by OA = (6, -2, -6) and OB = (3, k, -3), where k is a constant.
The curve C₁ has equation y = x² – 4x + 7. The curve C₂ has equation y² = 4x + k, where k is a constant. The tangent to C₁ at the point where x = 3...
(a) In an arithmetic progression, the sum of the first ten terms is equal to the sum of the next five terms. The first term is a. (b) The sum to in...
The diagram shows part of the curve y = √(4x + 1) + 9/√(4x + 1) and the minimum point M. [Figure 11.1]
Show that ln(x³ – 4x) – ln(x² – 2x) = ln(x + 2).
Find the equation of the normal to the curve x² ln y + 2x + 5y = 11 at the point (3, 1).
The polynomial p(x) is defined by p(x) = 5x³ + ax² + bx – 16, where a and b are constants. It is given that (x – 2) is a factor of p(x) and that th...
The diagram shows the curve with equation y = (8 + x³)/(2 - 5x). The maximum point is denoted by M.
The polynomial p(x) is defined by p(x) = 4x³ + (k + 1)x² – mx + 3k, where k and m are constants. Given that (x + 1) is a factor of p(x), express m ...
Find the exact coordinates of the stationary point of the curve with equation y = 3x / ln x
The diagram shows the curve with parametric equations x = 3t - 6e⁻²ᵗ, y = 4t eᵗ, for 0 ≤ t ≤ 2. At the point P on the curve, the y-coordinate is 1....
Find the exact coordinates of the stationary point of the curve with equation y = 3x / ln x.
The diagram shows the curve with parametric equations x = 3t - 6e⁻²ᵗ, y = 4teᵗ, for 0 ≤ t ≤ 2. At the point P on the curve, the y-coordinate is 1. ...
Use the trapezium rule with 3 intervals to estimate the value of 3 ∫ |2ˣ – 4|dx. 0
Showing all necessary working, solve the equation ln(2x – 3) = 2lnx – ln(x – 1). Give your answer correct to 2 decimal places.
Find the gradient of the curve x³ + 3xy² – y³ = 1 at the point with coordinates (1, 3).
By first expressing the equation cot θ – cot(θ + 45°) = 3 as a quadratic equation in tan θ, solve the equation for 0° < θ < 180°.
The diagram shows the curves y = 4 cos ½x and y = 1/(4-x) for 0 ≤ x < 4. When x = a, the tangents to the curves are perpendicular.
Let f(x) = (16-17x) / ((2 + x)(3 - x)²).
The diagram shows a set of rectangular axes Ox, Oy and Oz, and four points A, B, C and D with position vectors **OA** = 3**i**, **OB** = 3**i** + 4...
Throughout this question the use of a calculator is not permitted. The complex number (√3) + i is denoted by u.
Find the coefficient of x³ in the expansion of (3 – x)(1 + 3x)⅓ in ascending powers of x.
Showing all necessary working, solve the equation 9ˣ = 3ˣ + 12. Give your answer correct to 2 decimal places.
Showing all necessary working, solve the equation cot 2θ = 2 tan θ for 0° < θ < 180°.
Find the exact coordinates of the point on the curve y = x / (1 + ln x) at which the gradient of the tangent is equal to ¼.
Throughout this question the use of a calculator is not permitted. It is given that the complex number −1 + (√3)i is a root of the equation kx³ + 5...
In the diagram, A is the mid-point of the semicircle with centre O and radius r. A circular arc with centre A meets the semicircle at B and C. The ...
The variables x and y satisfy the differential equation dy/dx = x e^(x+y). It is given that y = 0 when x = 0.
Let f(x) = (10x + 9) / ((2x + 1)(2x + 3)²).
The points A and B have position vectors i + 2j – k and 3i + j + k respectively. The line l has equation r = 2i + j + k + μ(i + j + 2k).
The diagram shows the curve y = sin 3x cos x for 0 ≤ x ≤ ½π and its minimum point M. The shaded region R is bounded by the curve and the x-axis. [F...
Use logarithms to solve the equation 53−2x = 4(7x), giving your answer correct to 3 decimal places.
Show that ∫₀^(1/4 π) x² cos 2x dx = (1/8)(π² – 8).
Let f(θ) = (1 - cos 2θ + sin 2θ) / (1 + cos 2θ + sin 2θ)
The equation of a curve is y = (1 + e⁻ˣ) / (1 - e⁻ˣ), for x > 0.
The variables x and y satisfy the differential equation (x + 1)y dy/dx = y² + 5. It is given that y = 2 when x = 0. Solve the differential equation...
The diagram shows the curve y = x⁴ – 2x³ – 7x − 6. The curve intersects the x-axis at the points (a, 0) and (b, 0), where a < b. It is given that b...
The curve y = sin(x + (1/3)π) cos x has two stationary points in the interval 0 ≤ x ≤ π.
Throughout this question the use of a calculator is not permitted. The complex number u is defined by u = 4i / (1-(√3)i)
Let f(x) = 2x(5 – x) / ((3+x)(1-x)²)