Cambridge & Pearson Edexcel Past Paper Questions

Express 5 cos θ – 2 sin θ in the form Rcos(θ + α), where R > 0 and 0 < α < ½π. Give the value of α correct to 4 decimal places.

Solve the inequality |3x-2| < |x + 5|.

A curve has equation y = 3 ln(2x + 9) – 2 lnx.

Find the quotient when x⁴ – 2x³ + 8x² – 12x + 13 is divided by x² + 6 and show that the remainder is 1.

Solve the equation 2ln(2x) – ln(x + 3) = 4 ln 2.

A curve has equation y³ sin 2x + 4y = 8. Find the equation of the tangent to the curve at the point where it crosses the y-axis.

It is given that ∫ (1 + e^(x/2))² dx from 0 to a = 10, where a is a positive constant.

Show that 2 cosec² 2x(1 – cos 2x) = sec²x.

A curve has equation y³ sin 2x + 4y = 8.

It is given that ∫₀ᵃ (1 + e½ˣ)² dx = 10, where a is a positive constant.

Show that 2 cosec²2x(1 – cos 2x) = sec²x.

Showing all necessary working, solve the equation ln(x⁴ – 4) = 4 ln x – In 4, giving your answer correct to 2 decimal places.

Given that sin(x – 60°) = 3 cos(x – 45°), find the exact value of tan x.

A curve has equation y = e³ˣ / tan(x/2). Find the x-coordinates of the stationary points of the curve in the interval 0 < x < π. Give your answers ...

The polynomial x⁴ + 2x³ + ax + b, where a and b are constants, is divisible by x² − x + 1. Find the values of a and b.

Let I = ∫(from 1/4 to 3/4) √((x)/(1-x)) dx.

In a certain chemical reaction the amount, x grams, of a substance is decreasing. The differential equation relating x and t, the time in seconds s...

Showing all working and without using a calculator, solve the equation z² + (2√6)z + 8 = 0, giving your answers in the form x + iy, where x and y a...

The positive constant a is such that ∫(from 0 to a) x e^(-x/2) dx = 2.

Let f(x) = (12x² + 4x - 1) / ((x−1)(3x + 2)).

The point P has position vector 3i – 2j + k. The line l has equation r = 4i + 2j + 5k + μ(i + 2j + 3k).

Showing all necessary working, solve the equation 3|2x – 1| = 2x, giving your answers correct to 3 significant figures.

Showing all necessary working, solve the equation cot θ + cot(θ + 45°) = 2, for 0° < θ < 180°.

In the diagram, the tangent to a curve at the point P with coordinates (x, y) meets the x-axis at T. The point N is the foot of the perpendicular f...

Show that (2 sin x - sin 2x) / (1 - cos 2x) = sin x / (1 + cos x)

The equation of a curve is x²(x + 3y) – y³ = 3.

The diagram shows a triangle ABC in which AB = AC = a and angle BAC = θ radians. Semicircles are drawn outside the triangle with AB and AC as diame...

Throughout this question the use of a calculator is not permitted. The complex numbers −3√3 + i and √3 + 2i are denoted by u and v respectively.

The diagram shows the curve y = (x + 1)e^(-x/3) and its maximum point M. [Figure X.X]

Let f(x) = (x - 4x²) / ((3-x)(2 + x²))

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

Expand 4 / √(4-3x) in ascending powers of x, up to and including the term in x², simplifying the coefficients.

Showing all necessary working, solve the equation 5^(2x) = 5^x + 5. Give your answer correct to 3 decimal places.

Showing all necessary working, find the value of ∫(from 0 to π/3) x cos 3x dx, giving your answer in terms of π.

The curve with equation y = (ln x)/(3+x) has a stationary point at x = p.

5 By first expanding (cos²x + sin²x)³, or otherwise, show that cos⁶x + sin⁶x = 1 - (3/4)sin²2x.

6 Express 1/(4-y²) in partial fractions.

The equation of a curve is 2x³ – y³ – 3xy² = 2a³, where a is a non-zero constant.

9 Find the complex number z satisfying the equation 3z - iz* = 1 + 5i, where z* denotes the complex conjugate of z.

The points A and B have position vectors 2i + j + 3k and 4i + j + k respectively. The line l has equation r = 4i + 6j + μ(i + 2j – 2k).

A particle P is projected vertically upwards with speed 24 m s¯¹ from a point 5 m above ground level. Find the time from projection until P reaches...

The diagram shows three coplanar forces acting at the point O. The magnitudes of the forces are 6N, 8 N and 10 N. The angle between the 6 N force a...

A particle P of mass 8 kg is on a smooth plane inclined at an angle of 30° to the horizontal. A force of magnitude 100 N, making an angle of θ° wit...

A particle P moves in a straight line starting from a point O. At time t s after leaving O, the displacement s m from O is given by s = t³ – 4t² + ...

A sprinter runs a race of 200 m. His total time for running the race is 20 s. He starts from rest and accelerates uniformly for 6 s, reaching a spe...

A car has mass 1250 kg. The car is moving along a straight level road at a constant speed of 36 m s¯¹ and is subject to a constant resistance of ma...

The diagram shows a triangular block with sloping faces inclined to the horizontal at 45° and 30°. Particle A of mass 0.8 kg lies on the face incli...

The diagram shows the velocity-time graph for a train which travels from rest at one station to rest at the next station. The graph consists of thr...

A small ball is projected vertically downwards with speed 5 m s⁻¹ from a point A at a height of 7.2 m above horizontal ground. The ball hits the gr...

Coplanar forces of magnitudes 8 N, 12 N and 18 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the sin...