Edexcel A Level Further Maths Paper 3A: Further Pure Mathematics 1 June 2023 question paper 9FM0/3A

Use Simpson’s rule with 4 intervals to find an estimate for

Give your answer to 3 significant figures.

Given that ₀²e^sin2x dx = 3.855 to 4 significant figures,

comment on the accuracy of your answer to part (a).

The vertical height, h m, above horizontal ground, of a passenger on a fairground ride, t seconds after the ride starts, where t 5, is modelled by the differential equation

Given that t = e^x, show that

Hence show that the transformation t = e^x transforms equation (I) into the equation

Hence show that

where A and B are constants.

Given that when t = 1, h = 2.5 and when t = 2, = –1

determine the height of the passenger above the ground 5 seconds after the start of the ride.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use algebra to determine the values of x for which

The ellipse E has equation

Determine the exact value of the eccentricity of E

The points P (4 cosθ, 3 sinθ) and Q (4 cosθ, –3 sinθ) lie on E where 0 < θ < The line l₁ is the normal to E at the point P

Use calculus to show that l₁ has equation

4 x sin θ – 3 y cos θ = 7 sin θ cos θ

The line l₂ passes through the origin and the point Q The lines l₁ and l₂ intersect at the point R

Determine, in simplest form, the coordinates of R

Hence show that, as θ varies, R lies on an ellipse which has the same eccentricity as ellipse E

Show that the substitution t tan transforms the integral

into the integral

Hence determine

Show that

Hence determine the first 3 non-zero terms in ascending powers of x of the Maclaurin series expansion of ln(e^2x cos 3x), giving each coefficient in simplest form.

Use the Maclaurin series expansion for ln(1 + x) to write down the first 4 non-zero terms in ascending powers of x of the Maclaurin series expansion of ln(1 + kx), where k is a constant.

Hence determine the value of k for which

exists.

With respect to a fixed origin O the point A has coordinates (3, 6, 5) and the line l has equation

The points B and C lie on l such that AB = AC = 15

Given that A does not lie on l and that the x coordinate of B is negative,

determine the coordinates of B and the coordinates of C

Hence determine a Cartesian equation of the plane containing the points A, B and C

The point D has coordinates (–2, 1, α), where α is a constant.

Given that the volume of the tetrahedron ABCD is 147

determine the possible values of α

Given that α 0

determine the shortest distance between the line l and the line passing through the points A and D, giving your answer to 2 significant figures.