Cambridge & Pearson Edexcel Past Paper Questions

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.

Solve the equation z² – 6iz – 12 = 0, giving the answers in the form x + iy, where x and y are real and exact.

Relative to the origin O, the points A, B and C have position vectors given by OA = (1, 1, 1), OB = (3, 1, 2) and OC = (5, -2, -2).

The variables x and θ satisfy the differential equation x sin² θ (dx/dθ) = tan² θ – 2 cot θ, for 0 < θ < ½π and x > 0. It is given that x = 2 when ...

The diagram shows part of the curve y = sin √x. This part of the curve intersects the x-axis at the point where x = a. [Figure 8.1]

The diagram shows a semicircle with diameter AB, centre O and radius r. The shaded region is the minor segment on the chord AC and its area is one ...

Let f(x) = (4 – x + x²) / ((1+x)(2 + x²))

Solve the equation ln(2x – 1) = 2ln(x + 1) – lnx. Give your answer correct to 3 decimal places.

Expand √(1 + 2x)/(1 - 2x) in ascending powers of x, up to and including the term in x², simplifying the coefficients.

Find the exact value of ∫(from 0 to π/4) x sec²x dx.

The parametric equations of a curve are x = 2t - tan t, y = ln(sin 2t), for 0 < t < ½π. Show that dy/dx = cot t.

Solve the quadratic equation (1 – 3i)z² – (2 + i)z + i = 0, giving your answers in the form x + iy, where x and y are real.

Show that the equation √5 secx + tanx = 4 can be expressed as Rcos(x + α) = √5, where R > 0 and 0° < α < 90°. Give the exact value of R and the val...

The curve with equation y = x³/(eˣ - 1) has a stationary point at x = p, where p > 0.

With respect to the origin O, the position vectors of the points A, B and C are given by OA = (0, 5, 2), OB = (1, 0, 1) and OC = (4, -3, -2). The m...

A gardener is filling an ornamental pool with water, using a hose that delivers 30 litres of water per minute. Initially the pool is empty. At time...

Let f(x) = (5 - x + 6x²)/((3 - x)(1 + 3x²)).

Coplanar forces of magnitudes PN, QN, 16N and 22N act at a point in the directions shown in the diagram. The forces are in equilibrium.

Small smooth spheres A and B, of equal radii and of masses 6 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially A is moving towa...

A constant resistance of magnitude 1400N acts on a car of mass 1250kg.

A block of mass 8 kg is placed on a rough plane which is inclined at an angle of 18° to the horizontal. The block is pulled up the plane by a light...

A particle P moves on the x-axis from the origin O with an initial velocity of −20ms⁻¹. The acceleration a ms⁻² at time ts after leaving O is given...

Fig. 6.1 shows particles A and B, of masses 4kg and 3kg respectively, attached to the ends of a light inextensible string that passes over a small ...

A cyclist is riding a bicycle along a straight horizontal road AB of length 50m. The cyclist starts from rest at A and reaches a speed of 6ms⁻¹ at ...

A particle P of mass 0.4kg is in limiting equilibrium on a plane inclined at 30° to the horizontal.

A particle of mass 0.3 kg is held at rest by two light inextensible strings. One string is attached at an angle of 60° to a horizontal ceiling. The...

A car of mass 1200kg is travelling along a straight horizontal road AB. There is a constant resistance force of magnitude 500N. When the car passes...

A block A of mass 80kg is connected by a light, inextensible rope to a block B of mass 40kg. The rope joining the two blocks is taut and is paralle...

Three particles A, B and C of masses 0.3 kg, 0.4kg and mkg respectively lie at rest in a straight line on a smooth horizontal plane. The distance b...