Cambridge & Pearson Edexcel Past Paper Questions
Browse 35,704 questions from Cambridge O-Level & A-Level and Pearson Edexcel IGCSE & IAL exams. Click any question to practice.
A competition is taking place between two choirs, the Notes and the Classics. There is a large audience for the competition. * 30% of the audienc...
The coefficient of x³ in the expansion of (1 + kx)(1 – 2x)⁵ is 20. Find the value of the constant k.
The first, second and third terms of a geometric progression are 2p + 6, −2p and p + 2 respectively, where p is positive. Find the sum to infinity ...
The equation of a curve is y = 2x² + m(2x + 1), where m is a constant, and the equation of a line is y = 6x + 4. Show that, for all values of m, th...
The sum, Sₙ, of the first n terms of an arithmetic progression is given by Sₙ = n² + 4n. The kth term in the progression is greater than 200. Find ...
Functions f and g are defined by f(x) = 4x – 2, for x ∈ R, g(x) = 4/(x+1), for x ∈ R, x ≠ −1.
Prove the identity ( 1 COS X – tan x )( 1 sin x + 1 ) ≡ 1 tan x
The point (4, 7) lies on the curve y = f(x) and it is given that f'(x) = 6x⁻½ – 4x⁻³/₂.
In the diagram, ABC is an isosceles triangle with AB = BC = r cm and angle BAC = θ radians. The point D lies on AC and ABD is a sector of a circle ...
A circle has centre at the point B (5, 1). The point A (−1, –2) lies on the circle.
The diagram shows part of the curve y = 2/(3-2x)² - x and its minimum point M, which lies on the x-axis. [Figure 10.1]
A curve has equation y = 3 cos 2x + 2 for 0 ≤ x ≤ π.
Given that ln(2x + 1) – ln(x − 3) = 2, find x in terms of e.
The polynomial p(x) is defined by p(x) = x³ + ax² + bx + 16, where a and b are constants. It is given that (x + 2) is a factor of p(x) and that the...
The diagram shows the curve y = 2 + e⁻²ˣ. The curve crosses the y-axis at the point A, and the point B on the curve has x-coordinate 1. The shaded ...
Solve the equation |2x – 5| = |x + 6|.
The sequence of values given by the iterative formula xn+1 = (6+8xn) / (8+xn²) with initial value x₁ = 2 converges to α.
It is given that 3 sin 2θ = cos θ where θ is an angle such that 0° < θ < 90°.
A curve is defined by the parametric equations x = 3t - 2 sin t, y = 5t + 4 cos t, where 0 ≤ t ≤ 2π. At each of the points P and Q on the curve, th...
A curve has equation y = f(x) where f(x) = (4x³ + 8x - 4) / (2x - 1)
Solve the equation 7 cot θ = 3 cosec θ for 0° < θ < 90°.
Given that 2³ˣ⁺² + 8 2³ˣ - 7 to 4 significant figures. = 5, find the value of 2³ˣ and hence, using logarithms, find the value of x correct
Sketch, on a single diagram, the graphs of y = |x - a constant. 1 a and y = 2x – a, – a, where a is a positive
The diagram shows the curve with equation y = x-2 x² + 8 and the lines x = 14 and y = 0. [Figure 4.1]
The equation of a curve is 2e²ˣy - y³ + 4 = 0.
Find ∫ (8/(4x+1) + 8/cos²(4x+1)) dx.
A curve has equation y = f(x) where f(x) = x⁴ – 5x³ + 6x² + 5x – 15. As shown in the diagram, the curve crosses the x-axis at the points A and B wi...
Given that ln(2x + 1) – ln(x − 3) = 2,
The sequence of values given by the iterative formula xₙ₊₁ = (6+8xₙ)/(8+xₙ²) with initial value x₁ = 2 converges to α.
A curve is defined by the parametric equations x = 3t – 2 sin t, y = 5t + 4 cos t, where 0 ≤ t ≤ 2π. At each of the points P and Q on the curve, th...
A curve has equation y = f(x) where f(x) = (4x³ + 8x – 4) / (2x – 1).
Solve the inequality 2 – 5x > 2|x-3|.
The parametric equations of a curve are x = 3 - cos 2θ, y = 2θ + sin 2θ, for 0 < θ < ½π. Show that dy/dx = cot θ.
Solve the equation log₁₀(2x + 1) = 2log₁₀(x + 1) – 1. Give your answers correct to 3 decimal places.
By sketching a suitable pair of graphs, show that the equation cosec x = 1 + e¯½ˣ has exactly two roots in the interval 0 < x < π.
Express √6 cos θ + 3 sin θ in the form R cos(θ – α), where R > 0 and 0° < α < 90°. State the exact value of R and give α correct to 2 decimal places.
Verify that –1 + √5i is a root of the equation 2x³ + x² + 6x – 18 = 0.
The coordinates (x, y) of a general point of a curve satisfy the differential equation x dy/dx = (1 - 2x²)y, for x > 0. It is given that y = 1 when...
Let f(x) = (8 + 5x + 12x²)/((1-x)(2 + 3x)²).
The diagram shows the curve y = (2 – x)e¯½ˣ, and its minimum point M. [Figure 10.1]
Two lines have equations r = i + 2j + k + λ(ai + 2j – k) and r = 2i + j – k + μ(2i – j + k), where a is a constant.
Solve the equation ln(1 + e¯3x) = 2. Give the answer correct to 3 decimal places.
Expand √1 + 6x in ascending powers of x, up to and including the term in x³, simplifying the coefficients.
The variables x and y satisfy the relation 2ʸ = 3¹⁻²ˣ.
Show that the equation tan(θ + 60°) = 2 cot θ can be written in the form tan²θ + 3√3 tan θ – 2 = 0.
The diagram shows the curve with parametric equations x = tan θ, y = cos² θ, for −π < θ < π. [Figure X.X]
The complex number u is defined by u = 7+ i 1-i
The variables x and t satisfy the differential equation e³ᵗ dx/dt = cos² 2x, for t ≥ 0. It is given that x = 0 when t = 0.
With respect to the origin O, the position vectors of the points A, B, C and D are given by OA = (2, 1, 5), OB = (4, -1, 1), OC = (1, 1, 2) and OD ...
Let f(x) = 7x + 18 (3x + 2)(x² + 4)