Cambridge & Pearson Edexcel Past Paper Questions

Alethia models the length of time, in minutes, by which her train is late on any day by the random variable X with probability density function giv...

The coefficient of x⁴ in the expansion of (3 + x)⁵ is equal to the coefficient of x² in the expansion of (2x + a/x)⁶. Find the value of the positiv...

The second and third terms of a geometric progression are 10 and 8 respectively. Find the sum to infinity.

The equation of a curve is such that dy/dx = 3(4x - 7)½ − 4x⁻½. It is given that the curve passes through the point (4, 3/2). Find the equation of ...

The first, second and third terms of an arithmetic progression are k, 6k and k + 6 respectively.

The equation of a curve is y = 4x² - kx + (1/2)k² and the equation of a line is y = x - a, where k and a are constants.

The diagram shows the curve with equation y = 5x^(1/2) and the line with equation y = 2x + 2. Find the exact area of the shaded region which is bou...

The diagram shows a sector OBAC of a circle with centre O and radius 10cm. The point P lies on OC and BP is perpendicular to OC. Angle AOC = π/6 an...

The equation of a circle is x² + y² + ax + by − 12 = 0. The points A (1, 1) and B (2, −6) lie on the circle.

The equation of a curve is y = 3x + 1 − 4(3x + 1)½ for x > -1/3.

Functions f and g are defined as follows: f(x) = (2x + 1)/(2x - 1) for x ≠ 1/2, g(x) = x² + 4 for x ∈ ℝ. [Figure 10.1]

The function f is given by f(x) = 4cos⁴x + cos²x − k for 0 ≤ x ≤ 2π, where k is a constant.

The variables x and y satisfy the equation y = 4^(2x-a), where a is an integer. As shown in the diagram, the graph of ln y against x is a straight ...

Express the equation 7 tan θ + 4 cot θ – 13 sec θ = 0 in terms of sin θ only.

The diagram shows the curve with equation y = 3 sin x − 3 sin 2x for 0 ≤ x ≤ π. The curve meets the x-axis at the origin and at the points with x-c...

A curve has equation x²y + 2y³ = 48. Find the equation of the normal to the curve at the point (4, 2), giving your answer in the form ax + by + c =...

By sketching the graphs of y = |5-2x| and y = 3 ln x on the same diagram, show that the equation |5 – 2x| = 3 ln x has exactly two roots.

A curve has equation y = (9e^(2x) + 16) / (e^x - 1).

The polynomial p(x) is defined by p(x) = 2x³ + 5x² + ax + 2a, where a is an integer.

Given that y = ln x / x², find the exact value of dy/dx when x = e.

Sketch, on the same diagram, the graphs of y = |2x – 9| and y = 5x – 3.

A curve has equation e^(2x) cos 2y + sin y = 1. Find the exact gradient of the curve at the point (0, 1/6 π).

Use the trapezium rule with three intervals to show that the value of ∫_1^4 ln x dx is approximately ln 12.

The polynomial p(x) is defined by p(x) = 2x³ + ax² – 3x – 4, where a is a constant. It is given that (x - 4) is a factor of p(x).

The diagram shows the curve y = 3e^(2x−1). The shaded region is bounded by the curve and the lines x = a, x = a + 1 and y = 0, where a is a constan...

The diagram shows the curves y = √2π – 2x and y = sin²x for 0 ≤ x ≤ π. The shaded region is bounded by the two curves and the line x = 0. Find the ...

Express 3 sin 2θ sec θ + 10 cos(θ – 30°) in the form R sin(θ+ α) where R > 0 and 0° < α < 90°. Give the value of α correct to 2 decimal places.

Given that y = In x x2', find the exact value of dy dx when x = e.

A curve has equation e2x cos 2y + sin y = 1. Find the exact gradient of the curve at the point (0, 1/6π).

Use the trapezium rule with three intervals to show that the value of S 4م In 12. In x dx is approximately [4]

The diagram shows the curve y = 3e2x−1. The shaded region is bounded by the curve and the lines x = a, x = a + 1 and y = 0, where a is a constant. ...

Express 3 sin 2θ sec θ + 10 cos(θ – 30°) in the form R sin(θ + α) where R > 0 and 0° < α < 90°. Give the value of α correct to 2 decimal places.

Solve the equation 2(3^(2x−1)) = 4^(x+1), giving your answer correct to 2 decimal places.

Expand (2-x²)⁻² in ascending powers of x, up to and including the term in x⁴, simplifying the coefficients.

Solve the equation 2 cot 2x + 3 cotx = 5, for 0° < x < 180°.

The variables x and y satisfy the differential equation `dy/dx = xy / (1 + x²)` and y = 2 when x = 0. Solve the differential equation, obtaining a ...

The polynomial ax³ – 10x² + bx + 8, where a and b are constants, is denoted by p(x). It is given that (x – 2) is a factor of both p(x) and p'(x).

Let `I = ∫(from 0 to 3) (27 / (9 + x²)²) dx`.

The complex number u is defined by u = `(√2 – a√2i) / (1 + 2i)`, where a is a positive integer.

The equation of a curve is x³ + y³ + 2xy + 8 = 0.

In the diagram, OABCDEFG is a cuboid in which OA = 2 units, OC = 4 units and OG = 2 units. Unit vectors i, j and k are parallel to OA, OC and OG re...

The curve y = x√sinx has one stationary point in the interval 0 < x < π, where x = a (see diagram). [Figure 10.1]

Solve the equation ln(e2x + 3) = 2x + ln 3, giving your answer correct to 3 decimal places.

Solve the equation 3 cos 2θ = 3 cos θ + 2, for 0° ≤ θ ≤ 360°.

The polynomial ax³ + x² + bx + 3 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 + 2) the re...

The equation of a curve is y = cos³x√sinx. It is given that the curve has one stationary point in the interval 0 < x < ½π. Find the x-coordinate of...

By sketching a suitable pair of graphs, show that the equation ln x = 3x − x² has one real root.

The variables x and y satisfy the differential equation dy/dx = xey-x, and y = 0 when x = 0.

The equation of a curve is x³ + 3x²y – y³ = 3.

Let f(x) = x² + 9x / (3x−1)(x² + 3)