Edexcel A Level Further Maths Paper 3A: Further Pure Mathematics 1 June 2024 question paper 9FM0/3A

Given that

complete the table with the value of y corresponding to x = 3, giving your answer to 4 significant figures.

In part (b) you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use Simpson’s rule with all the values of y in the completed table to estimate, to 3 significant figures, the value of

Using your answer to part (b) and making your method clear, estimate the value of

Use algebra to determine the values of x for which

Use L’Hospital’s rule to show that

The Taylor series exp ansion of f x about x a is given by

The curve with equationy = f(x) satisfies the differential equation

Given that , 1 is a stationary point of the curve,

determine the nature of this stationary point, giving a reason for your answer.

Show that 2at this stationary point.

Hence determine a series solution for y, in ascending powers of x up to and

including the term in x , giving each coefficient in simplest form.

y e^3x sin x

Use Leibnitz’s theorem to show that

Hence express in the form

where R and α are constants to be determined,R 0 and 0 < α <

The ellipse E has equation

The hyperbola H has equation

where a and b are positive constants.

Given that

• the eccentricity of H is the reciprocal of the eccentricity of E

• the coordinates of the foci of H are the same as the coordinates of the foci of E

determine

the value of a

the value of b

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

Solutions relying on calculator technology are not acceptable.

Use the substitutiont = tan to show that

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

Hence show that

The parabola P has equationy² = 4ax, where a is a positive constant. The point Aat², 2at, wheret 0, lies on P.

Use calculus to show that an equation of the tangent to P at A is

The point B(2k², 4k) and the point C(2k², –4k), where k is a constant, lie on P.

The tangent to P at B and the tangent to P at C intersect at the point D.

Given that the area of the triangle BCD is 432

determine the coordinates of B and the coordinates of C.

The line l₁ has equationr 3 λ 4

where λ and μ are scalar parameters.

The lines l₁ and l₂ intersect at the point P.

Determine the coordinates of P.

Given that the plane Π contains both l₁ and l₂

determine a Cartesian equation for Π.

Determine a Cartesian equation for each of the two lines that

• pass through (0, 0, 0)

• make an angle of 60° with the x-axis

• make an angle of 45° with the y-axis

The motion of a particle P along the x-axis is modelled by the differential equation

where P has displacement x metres from the origin O at time t minutes,t 0

determine the particular solution of the differential equation (I)