Edexcel A Level Further Maths Paper 3A: Further Pure Mathematics 1 November 2021 question paper 9FM0/3A

The ellipse E has equation

Find

the coordinates of the foci of E,

the equations of the directrices of E.

Use the substitution t = tan to prove the identity

Use the substitution t = tan to determine the exact value of

giving your answer in simplest form.

Figure 1 shows a sketch of the curve with equation y = f(x) where

Use algebra to determine the values of x for which

A small aircraft is landing in a field.

In a model for the landing the aircraft travels in different straight lines before and after it lands, as shown in Figure 2.

The vector v_A is in the direction of travel of the aircraft as it approaches the field. The vector v_L is in the direction of travel of the aircraft after it lands. With respect to a fixed origin, the field is modelled as the plane with equation

x – 2 y + 25 z = 0

and

Write down a vector n that is a normal vector to the field.

When the aircraft lands it remains in contact with the field and travels in the direction v_L The vector v_L is in the same plane as both v_A and n as shown in Figure 2.

Determine a vector which has the same direction as v_L

State a limitation of the model.

The parabola C has equation

and the hyperbola H has equation

Write down the equations of the asymptotes of H.

The line l₁ is normal to C and parallel to the asymptote of H with positive gradient. The line l₂ is normal to C and parallel to the asymptote of H with negative gradient.

Determine

an equation for l₁

an equation for l₂

The lines l₁ and l₂ meet H at the points P and Q respectively.

Find the area of the triangle OPQ, where O is the origin.

The Taylor series expansion of x about x a is given by

Given that

Determine the Taylor series expansion about x = 1 of

in ascending powers of (x – 1), up to and including the term in (x – 1)³

Give each coefficient in simplest form.

Use this series expansion to evaluate

explaining your reasoning clearly.

With respect to a fixed origin O, the line l has equation

The point A lies on l such that the direction cosines of OA → with respect to the i, j and k axes

Determine the coordinates of the point A.

A community is concerned about the rising level of pollutant in its local pond and applies a chemical treatment to stop the increase of pollutant.

The concentration, x parts per million (ppm), of the pollutant in the pond water t days after the chemical treatment was applied, is modelled by the differential equation

When the chemical treatment was applied the concentration of pollutant was 3 ppm.

Use the iteration formula

once to estimate the concentration of the pollutant in the pond water 6 hours after the chemical treatment was applied.

Show that the transformation u = x³ transforms the differential equation (I) into the differential equation

Determine the general solution of equation (II)

Hence find an equation for the concentration of pollutant in the pond water t days after the chemical treatment was applied.

Find the percentage error of the estimate found in part (a) compared to the value predicted by the model, stating if it is an overestimate or an underestimate.