AQA A Level Maths Paper 1 June 2025 question paper 7357/1

A sequence is defined by

The first four terms of the sequence are

32, – 8, 2, – 0.5

Which one of the following can be used to describe this sequence?

Circle your answer.

Convergent Decreasing Increasing Periodic _

Express

as a single logarithm.

Circle your answer.

The first three terms, in ascending powers of x, of the binomial expansion

where n is a constant.

State the range of values of x for which the expansion is valid.

Circle your answer.

An arithmetic series has first term 3 and common difference – 0.5

Find the sum of the first 250 terms.

It is given that 0 a 1

Sketch the graph with equation

on the axes below.

Show that, for small values of x, the graph with equation

can be approximated by the straight line with equation of the form

y = ax + 1

where a is a constant to be found.

A geometric series, S, has second term 60

The common ratio of S is 0.2

Find the exact value of the sum of the first five terms of S

A different geometric series, T, has second term 60 and positive common ratio r

The sum to infinity of T is T_∞

Show that

Find the maximum value of r – r²

Hence find the range of possible values of T_∞

Fully justify your answer.

Show how

– kt θ = 21 – A e

can be rearranged to obtain

ln (21 – θ) = – kt + ln A

The researcher uses measurements they have recorded to plot the graph of ln(21 – θ) against t as shown in the diagram below.

Use the information on the graph to find the value of A

Give your answer to three significant figures.

Use the information on the graph to find the value of k

Give your answer to three significant figures.

Find the temperature of the dessert when it is initially removed from the freezer.

Give your answer to three significant figures.

The dessert is ready to be eaten when its temperature reaches 4 °C

Use the model to determine the time, after being removed from the freezer, for the dessert to reach this temperature.

Give your answer to the nearest 10 minutes.

The equation of a curve is

The curve has two stationary points.

Use implicit differentiation to show that at the stationary points y =

Hence w that the x-coordinates of the stationary points can be written in the form ±√n where n is an integer to be found.

Functions f and g are defined by

Using set notation, state the range of f

Determine whether f has an inverse.

Fully justify your answer.

The graph of y = g (x), and the line with equation y = x, are shown on the diagram below.

Sketch the graph of y = g^–1(x) on the diagram.

The composite function gf is denoted by h

Write down an expression for h (x)

State the range of h

Find an equation of the tangent to C at the point where t = 0

Find a Cartesian equation for C in the form y = f (x)

Fully justify your answer.

The graph of y = 4xsin 2x for 0 ≤ x ≤ π is shown below.

Show that the shaded area enclosed by the x-axis and the curve is kπ, where k is an integer to be found.

Fully justify your answer.

A curve has equation

The point Q has coordinates (3, 2.5)

The point P on the curve which is closest to the point Q is shown on the diagram below.

Show that the x-coordinate of P satisfies the equation

The Newton–Raphson method is to be used to find an approximate solution to the equation

Show that the Newton–Raphson method generates the iterative formula

Starting with x₀ = 3, use the iterative formula given in part (b) to find the value of x₃

Give your answer to three decimal places.

Hence find the distance PQ

Give your answer to two decimal places.

The triangle ABC is shown in the diagram below. The angle ABC is radians. The angle BCA is x radians.

Explain why angle BAC = – x

Use the sine rule to show that

where p and q are integers.

Hence state the exact value of x when

Use the substitution u = e + 1 to show that

Solve the differential equation

given that y = π when x = 0

Write your answer in the form

where A and B are constants to be found.