Cambridge & Pearson Edexcel Past Paper Questions

Show that ln(x³ – 4x) – ln(x² – 2x) = ln(x + 2).

Solve the inequality |3x – 5| < |x + 3|.

Find the equation of the normal to the curve x² ln y + 2x + 5y = 11 at the point (3, 1).

Find ∫ tan² 3x dx.

The polynomial p(x) is defined by p(x) = 5x³ + ax² + bx – 16, where a and b are constants. It is given that (x – 2) is a factor of p(x) and that th...

The diagram shows the curve with equation y = (8 + x³)/(2 - 5x). The maximum point is denoted by M.

Show that 2 cosec 2θ cot θ = cosec² θ.

The polynomial p(x) is defined by p(x) = 4x³ + (k + 1)x² – mx + 3k, where k and m are constants. Given that (x + 1) is a factor of p(x), express m ...

Solve the equation |4 + 2x| = |3 – 5x|.

Find the exact coordinates of the stationary point of the curve with equation y = 3x / ln x

Find the exact value of ∫(from 0 to π/2) (4 sin 2x + 2 cos²x)dx. Show all necessary working.

Find the quotient and remainder when 2x³ + x² – 8x is divided by (2x + 1).

The diagram shows the curve with parametric equations x = 3t - 6e⁻²ᵗ, y = 4t eᵗ, for 0 ≤ t ≤ 2. At the point P on the curve, the y-coordinate is 1....

Express 4 sin θ + 4 cos θ in the form R sin(θ + α), where R > 0 and 0° < α < 90°.

Find the exact coordinates of the stationary point of the curve with equation y = 3x / ln x.

Find the quotient and remainder when 2x³ + x2 – 8x is divided by (2x + 1).

The diagram shows the curve with parametric equations x = 3t - 6e⁻²ᵗ, y = 4teᵗ, for 0 ≤ t ≤ 2. At the point P on the curve, the y-coordinate is 1. ...

Use the trapezium rule with 3 intervals to estimate the value of 3 ∫ |2ˣ – 4|dx. 0

Showing all necessary working, solve the equation ln(2x – 3) = 2lnx – ln(x – 1). Give your answer correct to 2 decimal places.

Find the gradient of the curve x³ + 3xy² – y³ = 1 at the point with coordinates (1, 3).

By first expressing the equation cot θ – cot(θ + 45°) = 3 as a quadratic equation in tan θ, solve the equation for 0° < θ < 180°.

Differentiate 1 with respect to θ. sin²θ

By first expanding sin(2x + x), show that sin 3x = 3 sin x – 4 sin³ x.

The diagram shows the curves y = 4 cos ½x and y = 1/(4-x) for 0 ≤ x < 4. When x = a, the tangents to the curves are perpendicular.

Let f(x) = (16-17x) / ((2 + x)(3 - x)²).

The diagram shows a set of rectangular axes Ox, Oy and Oz, and four points A, B, C and D with position vectors **OA** = 3**i**, **OB** = 3**i** + 4...

Throughout this question the use of a calculator is not permitted. The complex number (√3) + i is denoted by u.

Find the coefficient of x³ in the expansion of (3 – x)(1 + 3x)⅓ in ascending powers of x.

Showing all necessary working, solve the equation 9ˣ = 3ˣ + 12. Give your answer correct to 2 decimal places.

Showing all necessary working, solve the equation cot 2θ = 2 tan θ for 0° < θ < 180°.

Find the exact coordinates of the point on the curve y = x / (1 + ln x) at which the gradient of the tangent is equal to ¼.

Throughout this question the use of a calculator is not permitted. It is given that the complex number −1 + (√3)i is a root of the equation kx³ + 5...

In the diagram, A is the mid-point of the semicircle with centre O and radius r. A circular arc with centre A meets the semicircle at B and C. The ...

The variables x and y satisfy the differential equation dy/dx = x e^(x+y). It is given that y = 0 when x = 0.

Let f(x) = (10x + 9) / ((2x + 1)(2x + 3)²).

The points A and B have position vectors i + 2j – k and 3i + j + k respectively. The line l has equation r = 2i + j + k + μ(i + j + 2k).

The diagram shows the curve y = sin 3x cos x for 0 ≤ x ≤ ½π and its minimum point M. The shaded region R is bounded by the curve and the x-axis. [F...

Use logarithms to solve the equation 53−2x = 4(7x), giving your answer correct to 3 decimal places.

Show that ∫₀^(1/4 π) x² cos 2x dx = (1/8)(π² – 8).

Let f(θ) = (1 - cos 2θ + sin 2θ) / (1 + cos 2θ + sin 2θ)

The equation of a curve is y = (1 + e⁻ˣ) / (1 - e⁻ˣ), for x > 0.

The variables x and y satisfy the differential equation (x + 1)y dy/dx = y² + 5. It is given that y = 2 when x = 0. Solve the differential equation...

The diagram shows the curve y = x⁴ – 2x³ – 7x − 6. The curve intersects the x-axis at the points (a, 0) and (b, 0), where a < b. It is given that b...

The curve y = sin(x + (1/3)π) cos x has two stationary points in the interval 0 ≤ x ≤ π.

Throughout this question the use of a calculator is not permitted. The complex number u is defined by u = 4i / (1-(√3)i)

Let f(x) = 2x(5 – x) / ((3+x)(1-x)²)

The line l has equation r = i + 2j + 3k + μ(2i – j – 2k).

Given that tan α = 12/5 and tan θ = 4/3, show that the coplanar forces shown in the diagram are in equilibrium. [Figure 1.1]

A particle P is projected vertically upwards with speed 25 m s⁻¹ from a point 3 m above horizontal ground.

A lorry has mass 12000 kg.