Cambridge & Pearson Edexcel Past Paper Questions

The diagram shows curves with equations y = 2x¹/² + 13x⁻¹/² and y = 3x⁻¹/² + 12. The curves intersect at points A and B. [Diagram provided]

The equation of a curve is y = f(x), where f(x) = (4x – 3)³/² - 20x.

The coordinates of points A, B and C are (6, 4), (p, 7) and (14, 18) respectively, where p is a constant. The line AB is perpendicular to the line BC.

It is given that θ is an acute angle in degrees such that sin θ = 3/5. Find the exact value of sin(θ + 60°).

A curve has equation y = 3 tan (1/2)x cos 2x. Find the gradient of the curve at the point for which x = 5/6 π.

Find ∫₄¹⁰ 4/(2x-5) dx, giving your answer in the form ln a, where a is an integer.

Sketch, on the same diagram, the graphs of y = |3x – 5| and y = 2x + 7.

The polynomial p(x) is defined by p(x) = 6x³ + ax² + bx – 20, where a and b are constants. It is given that (x + 2) is a factor of p(x) and that th...

Show that cosec θ(3 sin 2θ + 4 sin³ θ) = 4 + 6 cos θ – 4 cos² θ.

The curve with equation e²ˣ – 18x + y³ + y = 11 has a stationary point at (p, q).

When the polynomial ax³ + 4ax² -7x - 5 is divided by (x + 2), the remainder is 33.

Solve the equation sec θcos(θ – 60°) = 4 for −180° < θ < 180°.

The diagram shows the curve with equation y = 6e⁻¹/²x. The points on the curve with x-coordinates 0 and 2 are denoted by A and B respectively. The ...

Sketch, on the same diagram, the graphs of y = |3 – x| and y = 9 – 2x.

Find the quotient when 6x³ – 5x² – 24x – 4 is divided by (2x + 1), and show that the remainder is 6.

The diagram shows the curve with parametric equations x = 3 ln(2t – 3), y = 4t ln t. The curve crosses the y-axis at the point A. At the point B, t...

Prove that sin 2x(cot x + 3 tan x) = 4 – 2 cos 2x.

Find ∫ from 4 to 10 of (4 / (2x-5)) dx, giving your answer in the form ln a, where a is an integer.

The curve with equation e^(2x) – 18x + y³ + y = 11 has a stationary point at (p, q).

Find the exact coordinates of the points on the curve y = x² / (1-3x) at which the gradient of the tangent is equal to 8.

The variables x and y are related by the equation y = ab^x, where a and b are constants. The diagram shows the result of plotting In y against x fo...

The complex number u is defined by u = (3 + 2i) / (a - 5i), where a is real.

Given that sin(x + π/3) – sin(x – π/3) = cos(x + π/3) – cos(x – π/3), find the exact value of tan x.

The parametric equations of a curve are x = √t + 3, y = ln t, for t > 0.

The variables x and θ satisfy the differential equation (x / tan θ) (dx / dθ) = x² + 3. It is given that x = 1 when θ = 0. Solve the differential e...

By sketching a suitable pair of graphs, show that the equation √x = e^x - 3 has only one root.

The diagram shows the curve y = xe^(-x²/4), for x ≥ 0, and its maximum point M. [Figure 9]

Let f(x) = (24x + 13) / ((1 - 2x)(2 + x)²)

In the diagram, OABCDEFG is a cuboid in which OA = 3 units, OC = 2 units and OD = 2 units. Unit vectors i, j and k are parallel to OA, OD and OC re...

1 (a) Sketch the graph of y = |4x − 2|. (b) Solve the inequality 1 + 3x < |4x − 2|.

2 The parametric equations of a curve are x = (ln t)², y = e^(2-t²), for t > 0. Find the gradient of the curve at the point where t = e, simplifyin...

3 The polynomial 2x³ + ax² – 11x + b is denoted by p(x). It is given that p(x) is divisible by (2x – 1) and that when p(x) is divided by (x + 1) th...

4 (a) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z − 4 − 3i| ≤ 2 and ...

5 Find the exact value of ∫(from 0 to 6) [x(x + 1) / (x² + 4)] dx.

6 (a) By sketching a suitable pair of graphs, show that the equation cot x = 2 − cos x has one root in the interval 0 < x < ½π. (b) Show by calcula...

7 (a) By expressing 3θ as 2θ + θ, prove the identity cos 3θ = 4 cos³ θ – 3 cos θ. (b) Hence solve the equation cos 3θ + cos θ cos 2θ = cos² θ for 0...

8 It is given that (2 + 3ai) / (a + 2i) = λ(2 – i), where a and λ are real constants. (a) Show that 3a² + 4a – 4 = 0. (b) Hence find the possible v...

9 The diagram shows the curve y = sin x cos 2x, for 0 ≤ x ≤ π, and a maximum point M, where x = a. The shaded region between the curve and the x-ax...

10 The equations of the lines l and m are given by l: r = (3i - 2j + k) + λ(3i + j + 2k) and m: r = (6i - 3j + 6k) + μ(-2i + 4j + ck), where c is a...

11 The variables x and y satisfy the differential equation x² dy/dx + y² + y = 0. It is given that x = 1 when y = 1. (a) Solve the differential equ...

Find the set of values of x satisfying the inequality | 2x+1 – 2| < 0.5, giving your answer to 3 significant figures.

The polynomial 2x³ + ax² + bx + 6, where a and b are constants, is denoted by p(x). When p(x) is divided by (x + 2) the remainder is –38 and when p...

Solve the quadratic equation (3 + i)w² – 2w + 3 − i = 0, giving your answers in the form x + iy, where x and y are real.

Find the exact coordinates of the stationary points of the curve y = e^(3x²-1) / (1-x²).

Show that the equation cot² θ + 2 cos 2θ = 4 can be written in the form 4 sin⁴ θ + 3 sin² θ − 1 = 0.

The equation of a curve is x³ + y² + 3x² + 3y = 4.

The variables x and y satisfy the differential equation e^(4x) dy/dx = cos² 3y. It is given that y = 0 when x = 2. Solve the differential equation,...

Let f(x) = (17x² − 7x + 16) / ((2 + 3x²)(2−x)).

The diagram shows the curve y = x cos 2x, for x ≥ 0. [Figure X.X]

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