Cambridge & Pearson Edexcel Past Paper Questions

Sketch, on the same diagram, the graphs of y = 3x and y = |x − 3|.

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

The curve with equation y = xe²ˣ + 5e⁻ˣ has a minimum point M.

The diagram shows the curve with parametric equations x = ln(2t + 3), y = (2t-3)/(2t+3). [Figure 5.1] The curve crosses the y-axis at the point A a...

The polynomials f(x) and g(x) are defined by f(x) = 4x³ + ax² + 8x + 15 and g(x) = x² + bx + 18, where a and b are constants.

By first expanding cos(2θ + θ), show that cos 3θ = 4 cos³ θ – 3 cos θ.

The polynomial p(x) is defined by p(x) = ax³ + bx – 10, where a and b are constants. It is given that (x + 2) is a factor of p(x) and that the rema...

Sketch, on the same diagram, the graphs of y = x + 3 and y = |2x – 1|.

The curve with equation y = 5x – 2 tan 2x has exactly one stationary point in the interval 0 ≤ x < ¼π. Find the coordinates of this stationary poin...

Given that ∫ (from a to a+14) (1/3x) dx = ln 2, find the value of the positive constant a.

A curve has equation x² + 4x cos 3y = 6. Find the exact value of the gradient of the normal to the curve at the point (√2, π/12).

By sketching a suitable pair of graphs on the same diagram, show that the equation ln x = 2e^(–x) has exactly one root.

Prove that 4 sin x sin(x + π/6) = √3 – √3 cos 2x + sin 2x.

Find the exact value of ∫₂₋₁(4e²ˣ - 2e⁻ˣ) dx.

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

The curve with equation y = xe²ˣ + 5e⁻ˣ has a minimum point M.

The diagram shows the curve with parametric equations x = ln(2t + 3), y = (2t-3)/(2t+3). The curve crosses the y-axis at the point A and the x-axis...

The polynomials f(x) and g(x) are defined by f(x) = 4x³ + ax² + 8x + 15 and g(x) = x² + bx + 18, where a and b are constants.

By first expanding cos(2θ + θ), show that cos 3θ = 4 cos³ θ – 3 cos θ.

Solve the equation 4|5ˣ – 1| = 5ˣ, giving your answers correct to 3 decimal places.

Express 5 sin x – 3 cos x in the form Rsin(x – α), where R > 0 and 0 < α < ½π. Give the exact value of R and give α correct to 2 decimal places.

The curve with equation y = xe¹⁻²ˣ has one stationary point.

Using the substitution u = √x, find the exact value of ∫₈₃ (1 / (x+1)√x) dx.

Show that the equation cot 2θ + cot θ = 2 can be expressed as a quadratic equation in tan θ.

When (a + bx)√1 + 4x, where a and b are constants, is expanded in ascending powers of x, the coefficients of x and x² are 3 and –6 respectively. Fi...

Given that y = ln(lnx), show that `dy/dx = 1/(x lnx)`.

The constant a is such that ∫ᵃ₁ (lnx / √x) dx = 6. [exp(x) is an alternative notation for eˣ.]

Two lines l and m have equations r = 3i + 2j + 5k + s(4i – j + 3k) and r = i − j − 2k + t(−i + 2j + 2k) respectively.

The complex number 1 + 2i is denoted by u. The polynomial 2x³ + ax² + 4x + b, where a and b are real constants, is denoted by p(x). It is given tha...

Find the value of x for which 3(2¹⁻ˣ) = 7ˣ. Give your answer in the form ln a / ln b, where a and b are integers.

Solve the inequality |3x – a| > 2|x + 2a|, where a is a positive constant.

Given the complex numbers u = a + ib and w = c + id, where a, b, c and d are real, prove that (u + w)* = u* + w*.

Express 4x² - 13x + 13 / (2x - 1)(x - 3) in partial fractions.

Using the expansions of sin(3x + 2x) and sin(3x - 2x), show that ½(sin 5x + sin x) = sin 3x cos 2x.

The variables x and y satisfy the differential equation e²ˣ dy/dx = 4xy², and it is given that y = 1 when x = 0. Solve the differential equation, o...

By first expanding (cos² θ + sin²θ)², show that cos⁴ θ + sin⁴θ = 1 − ½ sin² 2θ.

The equation of a curve is ye²ˣ – y²eˣ = 2.

With respect to the origin O, the position vectors of the points A and B are given by OA = (1, 2, -1) and OB = (0, 3, 1).

The equation of a curve is y = √tan x, for 0 ≤ x < ½π. [Figure X.X]

Find the quotient and remainder when 2x⁴ + 1 is divided by x² − x + 2.

Sketch the graph of y = |2x – 3|.

Solve the equation 4ˣ⁻² = 4ˣ – 4², giving your answer correct to 3 decimal places.

Find the exact value of ∫ from π/2 to π of x sin(½x) dx.

Solve the equation sin θ = 3 cos 2θ + 2, for 0° ≤ θ ≤ 360°.

By first expanding cos(x – 60°), show that the expression 2 cos(x - 60°) + cos x can be written in the form R cos(x – α), where R > 0 and 0° < α < ...

The equation of a curve is ln(x + y) = x – 2y.

In the diagram, OABCD is a pyramid with vertex D. The horizontal base OABC is a square of side 4 units. The edge OD is vertical and OD = 4 units. T...

Let f(x) = 1 / ((9-x)√x) Find the x-coordinate of the stationary point of the curve with equation y = f(x).

A large plantation of area 20 km² is becoming infected with a plant disease. At time t years the area infected is x km² and the rate of increase of...

The complex number –√3 + i is denoted by u.