Cambridge & Pearson Edexcel Past Paper Questions

A certain kind of firework is supposed to last for 30 seconds, on average, after it is lit. An inspector suspects that the fireworks actually last ...

In a certain large document, typing errors occur at random and at a constant mean rate of 0.2 per page.

A machine is supposed to produce random digits. Bob thinks that the machine is not fair and that the probability of it producing the digit 0 is les...

(a) The probability density function of the random variable X is given by f(x) = { kx(4-x) 0 ≤ x ≤ 2, { 0 otherwise, where k is a consta...

A random variable X has probability density function given by f(x) = { (1/18)(9-x²) for 0 ≤ x ≤ 3, 0 otherwise. The median of X is m.

(a) The proportion of people having a particular medical condition is 1 in 100000. A random sample of 2500 people is obtained. The number of people...

Points A and B have coordinates (5, 2) and (10, −1) respectively.

The first, second and third terms of an arithmetic progression are a, 2a and a² respectively, where a is a positive constant. Find the sum of the f...

Find the set of values of k for which the equation 8x² + kx + 2 = 0 has no real roots.

A geometric progression is such that the third term is 1764 and the sum of the second and third terms is 3444. Find the 50th term.

The graph with equation y = f(x) is transformed to the graph with equation y = g(x) by a stretch in the x-direction with factor 0.5, followed by a ...

The equation of a curve is y = 4x² + 20x + 6.

Prove the identity sin θ/(sin θ + cos θ) + cos θ/(sin θ - cos θ) = (tan²θ + 1)/(tan²θ - 1).

The equation of a curve is such that dy/dx = 3x^(1/2) – 3x^(-1/2). The curve passes through the point (3, 5).

Functions f and g are defined by f(x) = x + 1/x for x > 0, g(x) = ax + 1 for x ∈ R, where a is a constant.

The diagram shows a cross-section RASB of the body of an aircraft. The cross-section consists of a sector OARB of a circle of radius 2.5m, with cen...

Find the coordinates of the minimum point of the curve y = 9/4 x² – 12x + 18.

Solve the inequality |2x – 5| > x.

Use logarithms to solve the equation 14e⁻²ˣ = 5ˣ⁺¹, giving your answer correct to 3 significant figures.

It is given that sec θ = √17 where 0 < θ < ½π. Find the exact value of tan(θ + ¼π).

By sketching a suitable pair of graphs on the same diagram, show that the equation e^(x/2) = x⁵ has exactly one real root.

A curve has equation 4e²ˣy + y² = 21. Find the gradient of the curve at the point (0, -7).

The polynomial p(x) is defined by p(x) = 12x³ – 9x² + 8x – 4.

The diagram shows the curve with equation y = (2lnx)/(3x+1). The curve crosses the x-axis at the point A and has a maximum point B. The shaded regi...

The expression f(θ) is defined by f(θ) = 12 sin θ cos θ + 16 cos² θ.

Solve the equation sec θ = 5 cosec θ for 0° < θ < 360°.

The solutions of the equation |4x − 1| = |x + 3| are x = p and x = q, where p < q. Find the exact values of p and q, and hence determine the exact ...

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

The polynomial p(x) is defined by p(x) = ax³ + 23x² - ax – 8, where a is a constant. It is given that (2x + 1) is a factor of p(x).

The curve with equation y = xln(4x + 1) – 3x has one stationary point P.

The diagram shows the curves y = 6 / (3x + 2) and y = 3e⁻ˣ – 3 for values of x between 0 and 4. The shaded region is bounded by the two curves and ...

The diagram shows the curve with parametric equations x = 3 cos 2θ, y = 4 sin θ, for π < θ < 3π/2. Points P and Q lie on the curve. The gradient of...

By sketching a suitable pair of graphs on the same diagram, show that the equation e^(2x) = x⁵ has exactly one real root.

A curve has equation 4e^(2xy) + y² = 21. Find the gradient of the curve at the point (0, -7).

The diagram shows the curve with equation y = (2 ln x) / (3x + 1). The curve crosses the x-axis at the point A and has a maximum point B. The shade...

Sketch the graph of y = |2x + 1|.

On a sketch of an Argand diagram shade the region whose points represent complex numbers z satisfying the inequalities |z| ≤ 3, Rez ≥ −2 and π < ar...

Solve the equation 2^(3x−1) = 5(3^−x). Give your answer in the form (ln a)/(ln b), where a and b are integers.

Solve the equation tan(x + 45°) = 2 cotx for 0° < x < 180°.

The complex numbers u and w are defined by u = 2e^(πi/4) and w = 3e^(πi/3).

Prove the identity cos 4θ + 4 cos 2θ + 3 = 8 cos⁴ θ.

The equation of a curve is y = x/cos²x, for 0 ≤ x < π/2. At the point where x = a, the tangent to the curve has gradient equal to 12.

In a certain chemical reaction the amount, x grams, of a substance is increasing. The differential equation satisfied by x and t, the time in secon...

The diagram shows part of the curve y = (3 - x)e^(−x/3) for x ≥ 0, and its minimum point M. [Figure 9.1]

Let f(x) = (2x² + 7x + 8) / ((1+x)(2 + x)²).

In the diagram, OABCD is a solid figure in which OA = OB = 4 units and OD = 3 units. The edge OD is vertical, DC is parallel to OB and DC = 1 unit....

Solve the equation 2^(3x−1) = 5(3^(1−x)). Give your answer in the form (ln a)/(ln b) where a and b are integers.

The polynomial 2x³ – x² + a, where a is a constant, is denoted by p(x). It is given that (2x + 3) is a factor of p(x).

The equation of a curve is y = sin x sin 2x. The curve has a stationary point in the interval 0 < x < ½π. Find the x-coordinate of this point, givi...

Express 4 cos x − sin x in the form Rcos(x + α), where R > 0 and 0° < α < 90°. State the exact value of R and give α correct to 2 decimal places.