Edexcel A Level Maths Paper 1: Pure Mathematics 1 June 2025 question paper 9MA0/01

The point P(6, –4) lies on the curve with equationy = f(x), x Find the point to which P is mapped when the curve with equationy = f(x) is transformed to the curve with equation

y = f(x + 2)

y = f^–1(x)

y = 2f(x) – 3

The circle C has equation

State the coordinates of the centre of C.

Find the radius of C writing your answer as a fully simplified surd.

Determine, giving a reason, whether or not the origin lies inside the circle C.

The first three terms of an arithmetic sequence are 6 k, 10 and 2 k where k is a constant.

Find the value of k.

Hence find the value of the sum of the first 50 terms of this sequence.

Given that

• f(x) = 2x³ + 3x² – 16x + 16

• f(–4) = 0

write f(x) in the form (x + a) Q (x) where a is a constant and Q(x) is a quadratic expression.

Hence prove that –4 is the only real root of the equation f(x) = 0 (Solutions relying on calculator technology are not acceptable.)

Hence show that

where k is an integer to be found. (Solutions relying on calculator technology are not acceptable.)

A scientist is monitoring the flight of sea birds after they leave their nests on a cliff. The height above the sea, h metres, of one of the birds is modelled by the equation

where t seconds is the time after the bird leaves its nest and A, B and T are positive constants. Given that

• the bird was 17.6 m above the sea exactly 4 seconds after leaving its nest

• the bird was 11.9 m above the sea exactly 9 seconds after leaving its nest

find a complete equation for the model. Find, according to the model,

the height of the bird’s nest above the sea,

the limitation on the value of T.

Figure 1 shows a sketch of a curve C with equationy = f(x), where f(x) is a quartic expression in x. The curve

• has maximum turning points at (–1, 0) and (5, 0)

• crosses the y ‑axis at (0, –75)

• has a minimum turning point at x = 2

Find the set of values of x for which

writing your answer in set notation.

Find the equation of C. You may leave your answer in factorised form. The curve C₁ has equation y = f(x) + k, where k is a constant. Given that the graph of C₁ intersects the x‑axis at exactly four places,

find the range of possible values for k.

A student was asked to prove that the square of any number can be expressed in the form 5n or 5n ± 1 where n The start of the student’s proof is shown in the box below.

Identify and correct an algebraic error in the box above.

Show the calculations and statements that are required to complete the proof.

A racing car is driven along a straight road. Figure 2 shows a graph of the speed of the car as it travels along the road. The car starts from rest and is driven for T seconds before stopping. The speed of the car is modelled by the equation

where t seconds is the time after the car starts to move. According to the model,

find the value of T, giving your answer to one decimal place,

show that the maximum speed of the car occurs when

Using the iteration formula

find the value of t₃ to 3 decimal places,

find, by repeated iteration, the time taken for the car to reach maximum speed.

Given that

• PQ → = 2i + 8j – 2k

• QR → = 6i + 6k

find → PR.

Hence show that triangle PQR is both right-angled and isosceles.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The value of a car, £V, is modelled by the equation

where A and k are positive constants and t is the age of the car in years. Given that

• the initial value of the car was £20 000

• the value of the car was £12 000 when it was 2.5 years old

find a complete equation for the model, giving the exact value of A and the value of k to 3 significant figures.

Show that the rate of change in the value of the car can be expressed in the form – k (V – 1500)

State a limitation of this model.

Figure 3 shows a sketch of the curve C with equation

and the straight line l with equation y = 2 x – 10

Verify that C and l intersect where x = 6 The curve and line also intersect at the point Q shown in Figure 3.

Show that the x coordinate of Q is a solution of

Using algebra and showing all stages of working, find the exact x coordinate of Q.

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Show that

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Show that

Hence solve, for 0 θ 180

giving your answers to one decimal place, where appropriate.

Figure 4 shows the plan view for the design of a stage. The shape of this design consists of a sector of a circle AOB joined to a rectangle OBCD. Given that

• the radius of the sector is r metres and angle AOB is θ radians

• the length and width of the rectangle are r metres and metres respectively

• the total area of the stage is 240 m²

show that the perimeter of the stage, P metres, is given by

You must make your method clear. Using algebraic differentiation,

find the value of r for which P has a stationary value.

Prove, by further differentiation, that this value of r gives the minimum perimeter of the stage.

Figure 5 shows a sketch of the curve C with equation

The region R, shown shaded in Figure 5, ounded by C, the line with equation x = 1,

Use the substitution x = 2 sinu to show that the area of R is given by

where a, b and k are constants to be found.

Hence, using algebraic integration, find the exact area of R. Give your answer in simplest form.