Cambridge & Pearson Edexcel Past Paper Questions

The polynomial p(x) is defined by p(x) = ax³ + ax² – 15x – 18, where a is a constant. It is given that (x – 2) is a factor of p(x).

It is given that ∫₀^a (3x² + 4 cos 2x – sin x) dx = 2, where a is a constant.

Showing all necessary working, solve the equation sec α cosec α = 7 for 0° < α < 90°.

The equation of a curve is x² – 4xy – 2y² = 1.

The polynomial f(x) is defined by f(x) = x⁴ - 3x³ + 5x² – 6x + 11. Find the quotient and remainder when f(x) is divided by (x² + 2).

Solve the equation |4x + 5| = |x − 7|.

The variables x and y satisfy the equation y = kxᵃ, where k and a are constants. The graph of ln y against ln x is a straight line passing through ...

The sequence x₁, x₂, x₃, ... defined by x₁ = 1, xn+1 = xn / ln(2xn) converges to the value α.

Find the exact coordinates of the stationary point of the curve with equation y = e^(-½x) (2x + 5).

Show that ∫(from 2 to 18) (3/(2x)) dx = ln 27.

The parametric equations of a curve are x = 3 sin 2θ, y = 1 + 2 tan 2θ, for 0 ≤ θ < π/4.

Express 0.5 cos θ – 1.2 sin θ in the form R cos(θ + α), where R > 0 and 0° < α < 90°, giving the value of α correct to 2 decimal places.

Find the exact value of ∫₀² (2e²ˣ – 1)² dx. Show all necessary working.

A curve has equation y = (3 + 2 lnx) / (1 + ln x). Find the exact gradient of the curve at the point for which y = 4.

It is given that ∫₀ᵃ (3x² + 4 cos 2x - sin x) dx = 2, where a is a constant.

Given that ln(1 + e2y) = x, express y in terms of x.

Solve the inequality |2x-3| > 4|x + 1|.

The parametric equations of a curve are x = 2t + sin 2t, y = ln(1 − cos 2t). dy Show that = cosec 2t. dx

The number of insects in a population t weeks after the start of observations is denoted by N. The population is decreasing at a rate proportional ...

The curve with equation y = e^2x ln(x – 1) has a stationary point when x = p. (i) Show that p satisfies the equation x = 1 + exp exp(2(x-1)) where...

COS X (i) By differentiating -, show that if y sin x 10 dy = - cosec²x. [2] = cot x then dy dx (ii) Show that 2 π ∫ x cosec²x dx = 4(π + 1n 4). [6] π

Two lines l and m have equations r = ai + 2j + 3k + λ(i – 2j + 3k) and r = 2i + j + 2k + μ(2i – j + k) respectively, where a is a constant. It is g...

Let f(x) = x² + x + 6 x²(x + 2) (i) Express f(x) in partial fractions. (ii) Hence, showing full working, show that the exact value of S 4 f(x) dx...

(i) By first expanding cos(2x + x), show that cos 3x = 4 cos³ x − 3 cos x. 3 [4] (ii) Hence solve the equation cos 3x + 3 cos x + 1 = 0, for 0 ≤ x...

10 (a) The complex number u is given by u = −3 – (2√10)i. Showing all necessary working and without using a calculator, find the square roots of u....

Solve the equation 5 ln(4 – 3ˣ) = 6. Show all necessary working and give the answer correct to 3 decimal places.

The curve with equation y = e⁻²ˣ / (1-x²) has a stationary point in the interval −1 < x < 1. Find dy/dx and hence find the x-coordinate of this sta...

The polynomial x⁴ + 3x³ + ax + b, where a and b are constants, is denoted by p(x). When p(x) is divided by x² + x − 1 the remainder is 2x + 3. Find...

The context for the parts is: Express (√6) sin x + cos x in the form R sin(x + α), where R > 0 and 0° < α < 90°. Hence solve an equation.

The equation of a curve is 2x²y – xy² = a³, where a is a positive constant. Show that there is only one point on the curve at which the tangent is ...

The variables x and θ satisfy the differential equation sin(θ/2) dx/dθ = (x + 2) cos(θ/2) for 0 < θ < π. It is given that x = 1 when θ = π/3. Solve...

The context for the parts involves complex numbers.

Let f(x) = (2x² + x + 8) / ((2x−1)(x² + 2)).

It is given that ∫(from 0 to a) x cos(x/3) dx = 3, where the constant a is such that 0 < a < (3/2)π.

The line l has equation r = i + 3j – 2k + λ(i – 2j + 3k). The plane p has equation 2x + y − 3z = 5.

Solve the inequality 2|x + 2| > |3x – 1|.

The polynomial 6x³ + ax² + bx − 2, where a and b are constants, is denoted by p(x). It is given that (2x + 1) is a factor of p(x) and that when p(x...

Showing all necessary working, solve the equation (3^(2x) + 3^(-x)) / (3^(2x) - 3^(-x)) = 4. Give your answer correct to 3 decimal places.

By first expanding tan(2x + x), show that the equation tan 3x = 3 cot x can be written in the form tan⁴x - 12 tan²x + 3 = 0.

By sketching a suitable pair of graphs, show that the equation ln(x + 2) = 4e⁻ˣ has exactly one real root.

Throughout this question the use of a calculator is not permitted. The complex number with modulus 1 and argument 1/3π is denoted by w. The complex...

The plane m has equation x + 4y – 8z = 2. The plane n is parallel to m and passes through the point P with coordinates (5, 2, −2).

The diagram shows the graph of y = sec x for 0 ≤ x < 1/2π.

The variables x and t satisfy the differential equation 5 dx/dt = (20−x)(40 – x). It is given that x = 10 when t = 0.

The diagram shows the graph of y = e^(cosx) sin³x for 0 ≤ x ≤ π, and its maximum point M. The shaded region R is bounded by the curve and the x-axis.

A crane is lifting a load of 1250 kg vertically at a constant speed Vm s⁻¹. Given that the power of the crane is a constant 20 kW, find the value o...

The total mass of a cyclist and her bicycle is 75 kg. The cyclist ascends a straight hill of length 0.7 km inclined at 1.5° to the horizontal. Her ...

A block of mass 3 kg is at rest on a rough plane inclined at 60° to the horizontal. A force of magnitude 15 N acting up a line of greatest slope of...

Two blocks A and B of masses 4 kg and 5 kg respectively are joined by a light inextensible string. The blocks rest on a smooth plane inclined at an...

A small ring P is threaded on a fixed smooth horizontal rod AB. Three horizontal forces of magnitudes 4.5 N, 7.5 N and F N act on P [Figure X.X].