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.
Prove the identity (sin θ – cos θ) / (sin θ + cos θ) = (tan θ – 1) / (tan θ + 1).
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...
The first, second and last terms in an arithmetic progression are 56, 53 and –22 respectively. Find the sum of all the terms in the progression.
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...
Prove the identity (1 + cos θ)/(1 – cos θ) – (1 – cos θ)/(1 + cos θ) = 4/(sin θ tan θ)
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.
Find the coefficient of x in the expansion of (2x - 1/x)^5.
The point A has coordinates (−2, 6). The equation of the perpendicular bisector of the line AB is 2y = 3x + 5.
Prove the identity (1/cos θ - tan θ)² = (1 - sin θ) / (1 + sin θ)
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...
The first two terms of an arithmetic progression are 16 and 24. Find the least number of terms of the progression which must be taken for their sum...
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.
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.