OCR A Level Further Maths B (MEI) Core Pure June 2025 question paper Y420

The complex number z satisfies the equation z + 2 i z * + 1 - 4 i = 0.

You are given that z = x + i y, where x and y are real numbers.

Determine the values of x and y.

In this question you must show detailed reasoning.

Find the acute angle between the planes 2 x - y + 2 z = 5 and x + 2 y + z = 8.

Using standard summation formulae, show that, for integers n ⩾ 1,

where a and b are integers to be determined.

You are given that M and N are non-singular 2 × 2 matrices.

Write down the product rule for the inverse matrices of M, N and MN.

The figure below shows the curve with cartesian equation (x² + y²)² = xy.

Show that the polar equation of the curve is r² = a sin b θ, where a and b are positive

constants to be determined.

Determine the exact maximum value of r.

Determine the area enclosed by one of the loops.

In this question you must show detailed reasoning. By first expressing in partial fractions, show that y³ x = ln n, where m and n

are integers to be determined.

The function f (x) is defined as f (x) = ln (1 + x), for x > - 1.

Prove by mathematical induction that the nth derivative of f(x), f⁽ⁿ⁾(x), for all n ⩾ 1, is given

[You are not required to show this series for ln (1 + x) converges for - 1 < x ⩽ 1.]

The figure below shows an Argand diagram with a regular pentagon ABCDE. The point A represents the real number 1. The point B represents the complex number w.

Write down, in terms of w, the complex numbers represented by the points C, D and E.

Write down an equation whose roots are the complex numbers represented by the points

A, B, C, D and E.

Show that the sum of these roots is zero.

Find w. Give your answer in the form r (cos θ + i sin θ), where r > 0 and θ = k π, where

k is a positive constant to be found.

By considering the line segment AB, show that the length of each side of the pentagon is

In this question you must show detailed reasoning.

The lines l₁ and l₂ have equations

where a, b and c are constants.

In the case where c = 0 and l₁ and l₂ intersect at right angles, determine the coordinates of

the point of intersection of the two lines.

Now consider the case where c = 4 and lines l₁ and l₂ are parallel.

Find the values of a and b.

Determine the distance between the two lines.

In this question you must show detailed reasoning.

Determine, in any order, the exact values of the following.

• The four roots of the equation

• The value of c

• The value of d

The gradient of a curve y = f(x) satisfies the differential equation - 2y = 2 + x².

You are given that the curve passes through the point (1, 0).

By solving this differential equation, determine the exact x-coordinate of the stationary point

on the curve y = f (x).

By using the definition of cosh x and sinh x in terms of e^x and e^-x, show that

cosh²x + sinh²x ≡ cosh 2x.

J cosh x sinh x N (b) The transformation T of the plane has associated matrix M, where M = KKK OOO and sinh x cosh x x > 0. L P

Show that T transforms the unit square with coordinates (0, 0), (1, 0), (0, 1) and (1, 1) to a

rhombus of unit area.

You are given that the length of each side of the rhombus is 2 units.

Determine the exact value of x. Give your answer in logarithmic form.

Show that ^3 - e^4iθ^h3 - e^-4iθh = a + b cos 4θ, where a and b are integers to be determined.

The infinite series C and S are defined as follows.

Determine the area of R. Give your answer in the form p + ln q where p and q are integers to

be determined.

The region R is rotated through 2 π radians about the x-axis.

Determine the volume of the solid of revolution formed. Give your answer in the form

A researcher is modelling the height of a particular type of tree over its lifetime.

Data suggests that the maximum possible height of this type of tree over its lifetime is double the height of the tree 5 years after planting.

It is given that, t years after planting a seed for this type of tree, the corresponding height of the tree is h m, and that h = 0 when t = 0.

The researcher first models the height of the tree by assuming that the rate of increase of h proportional to (20 - h), with constant of proportionality 0.2.

Write down the first order differential equation for this model.

Show that this model predicts that the maximum possible height of the tree is 20 m.

Show by integration that h = 20(1 - e^-0.2t).

Determine whether this model’s prediction for the height of the tree 5 years after

planting is consistent with the maximum possible height of the tree being 20 m.

The researcher refines the model for the height of the tree using the following second order differential equation.

Determine the general solution of this second order differential equation.

Show that the refined model also predicts that the maximum possible height of the tree

is 20 m.

Further research determines that the initial rate of growth of this type of tree is 2.9 metres per year.

By applying the initial conditions to find the particular solution of this differential equation, determine whether the refined model’s prediction for the height of the tree 5 years after planting is consistent with the maximum possible height of the tree

being 20 m.