AQA A Level Maths Paper 1 June 2024 question paper 7357/1

The function f is defined by f(x) = ex + 1 for x ℝ

Find an expression for f–1(x)

Tick () one box.

f–1(x) = ln (x – 1)

f–1(x) = ln (x) – 1

3 $qhe=expression

can be written as

State the value of A

Circle your answer.

3 4 7 9

One of the diagrams below shows the graph of y = arccos x

Identify the graph of y = arccos x

Tick () one box.

Show that

can be written as

where n is a positive integer.

Show that, for small values of θ measured in radians

where A, B and C are constants to be found.

Use your answer to part (a) to find an approximation for

cos 0.28 + 2 sin 0.21 – tan 0.14

Give your answer to three decimal places.

An arithmetic sequence has 300 terms.

The first term of the sequence is –7 and the last term is 32

Find the sum of the 300 terms.

A school holds a raffle at its summer fair.

There are nine prizes.

The total value of the prizes is £1260

The values of the prizes form an arithmetic sequence.

The top prize has the highest value, and the bottom prize has the least value.

The value of the top prize is six times the value of the bottom prize.

Find the value of the top prize.

It is given that

f (x) = x (x – a)(x – 6)

where 0 < a < 6

Sketch the graph of y = f(x) on the axes below.

Sketch the graph of y = f(–2x) on the axes below.

The terms, u_n, of a periodic sequence are defined by

State the period of the sequence.

It is given that

P(x) = 4x³ + 8x² + 11x + 4

Use the factor theorem to show that (2x + 1) is a factor of P(x)

Express P(x) in the form

P (x) = (2 x + 1)(ax 2 + bx + c)

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

The equation x 3 = e 6 – 2 x

has a single solution, x = α

By considering a suitable change of sign, show that α lies between 0 and 4

Show that the equation x3 = e6–2x can be rearranged to give

Explain why the iterative formula

fails to converge to α when the starting value is x₁ = 0

Show that the expression

sin 2 θ cosec θ + cos 2 θ sec θ

can be written as

4 cos θ – sec θ

where sinθ ≠ 0 and cosθ ≠ 0

Explain why the student should reject one of their values for cosθ in Step 4.

State the correct solutions to the equation

Figure 2 below shows a 1.5 metre length of pipe.

The symmetrical cross-section of the pipe is shown below, in Figure 3, where x and y are measured in centimetres.

Use the trapezium rule, with the values shown in the table below, to find the best estimate for the volume of the pipe.

The function f is defined by

The function g is defined by

g (x) = ln x

where g has its greatest possible domain.

Using set notation, state the range of f

State the domain of g

The composite function h is given by

Write down an expression for h(x) in terms of x

Determine if h has an inverse.

Fully justify your answer.

Use a suitable substitution to show that

can be written as

where a is a constant to be found.

Hence, or otherwise, show that

A graph has the equation

A student uses four rectangles to approximate the area under the graph between the lines x = 0 and x = 4

The rectangles are all the same width.

All the rectangles are drawn under the curve as shown in the diagram below.

The total area of the four rectangles is A

The student decides to improve their approximation by increasing the number of rectangles used.

Explain why the value of the student’s improved approximation will be greater than A, but less than

A curve has equation

y 3 e 2 x + 2 y – 16 x = k

where k is a constant.

The curve has a stationary point on the y-axis.

Determine the value of k

A gardener stores rainwater in a cylindrical container.

The container has a height of 130 centimetres.

The gardener empties the water from the container through a hose.

The hose is attached 5 centimetres from the bottom of the container.

At time t minutes after the hose is switched on, the depth of water, h centimetres, in the container decreases at a rate which is proportional to h – 5

Initially the container of water is full, and the depth of water is decreasing at a rate of 1.5 centimetres per minute.

Show that

Solve the differential equation

to find an expression for h in terms of t

Find the time taken for the container to be half empty.

Give your answer to the nearest minute.