AQA A Level Maths Paper 1 June 2023 question paper 7357/1

Find the coefficient of x⁷ in the expansion of (2x 3)⁷

Circle your answer.

Circle your answer.

The curve with equation y ln x is transformed by a stretch parallel to the x-axis with scale factor 2

Find the equation of the transformed curve.

Circle your answer.

Given that is a small angle, find an approximation for cos 2

Circle your answer.

The trapezium rule with 6 ordinates (5 strips) is to be used to find an approximation for the shaded area.

The values required to obtain this approximation are shown in the table below.

Use the trapezium rule with 6 ordinates (5 strips) to find an approximate value for the shaded area.

Give your answer to four decimal places.

Using your answer to part (a) deduce an estimate for dx

Show that the equation

2 log₁₀ x log₁₀ 4 log₁₀ (x 8)

has exactly one solution.

Fully justify your answer.

Given that n is a positive integer, express

as a single fraction not involving surds.

Hence, deduce that

is a rational number for all positive integer values of n

Show that

The points P and Q have coordinates (6, 15) and (12, 19) respectively.

Find the coordinates of the midpoint of PQ

Find the equation of the perpendicular bisector of PQ

Give your answer in the form ax by c where a, b and c are integers.

A circle passes through the points P and Q

The centre of the circle lies on the line with equation 2x 5y 30

Find the equation of the circle.

The circle intersects the coordinate axes at n points.

State the value of n

The curve with equation

for 360 x 360 is shown below.

Point A on the curve has coordinates (a, 0.5)

Find the value of a

State the value of sin (180 a)

Find the exact value of sin (b 180)

Find the exact value of cos b

The nth term of a sequence is u_n

The sequence is defined by

Find an expression, in terms of p, for u₂

It is given that u₃ 382

Show that p satisfies the equation

It is given that the sequence is a decreasing sequence.

Find the value of u₄ and the value of u₅

The limit of u_n as n tends to infinity is L

Write down an equation for L

Find the value of L

One of the rides at a theme park is a room where the floor and ceiling both move up and down for 10 seconds.

At time t seconds after the ride begins, the distance f metres of the floor above the ground is

At time t seconds after the ride begins, the distance c metres of the ceiling above the ground is

The ride is shown in the diagram below.

Show that the initial distance between the floor and ceiling is 8 metres.

Show that the distance d metres between the floor and ceiling at time t is given by

where R and are positive constants to be found.

Hence, find the minimum distance between the ceiling and the floor.

Give your answer to the nearest centimetre.

The function f is defined by

The curve with equation y f (x) is shown below.

State the value of a

On the diagram above, sketch the curve with equation

and

sketch the line with equation

Explain why the solution to the equation

must also be a solution to the equation

Use the Newton-Raphson method with x₀ 0 to find an approximate solution, x₃, to the equation

Give your answer to four decimal places.

Given that

Hence find

The area, A, bounded by the curve with equation y 2^x, the x-axis, the y-axis and the line x 4 is approximated using eight rectangles of equal width as shown in the diagram below.

Show that the exact area of the largest rectangle is

The areas of these rectangles form a geometric sequence with common ratio

Find the exact value of the total area of the eight rectangles. Give your answer in the form k(1 2) where k is a rational number.

More accurate approximations for A can be found by increasing the number, n, of rectangles used.

Find the exact value of the limit of the approximations for A as n

The curve with equation

has a single stationary point at P as shown in the diagram below.

Show that the y-coordinate of P satisfies the equation

Hence, find the coordinates of P

Given that

find the values of A and B

An empty container, in the shape of a cuboid, has length 1.6 metres, width 1.25 metres and depth 0.5 metres, as shown in the diagram below.

The container has a small hole in the bottom.

Water is poured into the container at a rate of 0.16 cubic metres per minute.

At time t minutes after the container starts to be filled, the depth of water is d metres and water leaks out at a rate of 036d² cubic metres per minute.

At time t minutes after the container starts to be filled, the volume of water in the container is V cubic metres.

Show that

Hence, find t in terms of V

Determine how long it takes to fill the container with water.

Give your answer to the nearest minute.