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.
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.
Express 2 cos θ + (√5) sin θ in the form Rcos(θ – α) where R > 0 and 0° < α < 90°, giving the value of α correct to 2 decimal places.
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.
By sketching a suitable pair of graphs, show that the equation x³ = 11-2x has exactly one real root.
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...
Use the factor theorem to show that (x + 2) is a factor of the expression 6x³ + 13x² - 33x – 70 and hence factorise the expression completely.
Find ∫ (2 cos θ – 3)(cos θ + 1) dθ.
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...
By sketching a suitable pair of graphs, show that the equation x³ = 11-2x has exactly one real root.
Use the factor theorem to show that (x + 2) is a factor of the expression 6x³ + 13x² - 33x – 70 and hence factorise the expression completely.
Find ∫ (2 cos θ – 3)(cos θ + 1) dθ.
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.
By first expanding 2 sin(x – 30°), express 2 sin(x – 30°) – cosx in the form R sin(x – α), where R > 0 and 0° < α < 90°. Give the exact value of R ...
Express 1 / (x(2x + 3)) in partial fractions.
The diagram shows the curve y = sin x cos² 2x for 0 ≤ x ≤ ¼π and its maximum point M. [Figure X.X]
Solve the equation ln(x² + 1) = 1 + 2lnx, giving your answer correct to 3 significant figures.
Solve the inequality |x-3| < 3x −4.
Express the equation cot θ – 2 tan θ = sin 2θ in the form a cos⁴ θ + b cos² θ + c = 0, where a, b and c are constants to be determined.
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.
Prove that if y = 1/cos θ then dy/dθ = sec θ tan θ.
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.