OCR A Level Further Maths A Pure Core 2 June 2025 question paper

In this question you must show detailed reasoning.

Determine, in either order

• a.b

• a × b.

You are given that -7 - 5i is one root of the equation x³ + 10x² + 18x - 296 = 0.

Write down another complex root of the equation x³ + 10x² + 18x - 296 = 0.

Using your answer to part (a), express x³ + 10x² + 18x - 296 as a product of a real linear

factor and a real quadratic factor.

Verify that det (AB) = det (A) × det (B).

Use matrices A and B to demonstrate that matrix multiplication is not commutative.

The transformation represented by matrix A is denoted by T.

Show that the point (2, - 5) is not an invariant point under T.

Find the matrix which represents the inverse transformation of T.

In this question you must show detailed reasoning.

Determine the sum of all cube numbers from 216 to 512 000 inclusive.

In this question you must show detailed reasoning.

Give your answers in the form a + b i where a and b are exact.

State the relationship between the two arguments of the two square roots found in part (a).

One of the regions bounded by two polar curves, C₁ and C₂, is used to model the face of a flat earring.

The polar equations of C₁ and C₂ are

C₁:r = 2θ

C₂:r = θ²

where 0 ⩽ θ ⩽ π.

The curves C₁ and C₂ are shown in the diagram below with the region used to model the earring

You are given that C₁ and C₂ intersect at the pole O.

Find the other point of intersection of C₁ and C₂. Give your answer in polar coordinates.

In this question you must show detailed reasoning.

Determine the area of the face of the earring.

In this question you must show detailed reasoning.

A function f (x) is defined by f (x) = x sinh 2 x.

Using the formula given in part (a), determine the exact value of the coefficient of x⁸ in the

Maclaurin series for x sinh 2 x.

Use the Maclaurin series for e^x to verify your answer to part (b).

When light hits a certain photo-sensitive cell, at time t = 0, there is an electrical response in the cell which is denoted by y (t) where both y and t are measured in suitable units. A student wishes to model this response, y.

In an attempt to model y, the student sets up the differential equation

which is subject to the following conditions.

• y = 0 when t = 5

• y ⩾ 0 for all t ⩾ 0

By substituting into a suitable differential equation, verify that y = (A + Bt)e^-3t is a

complementary function of the differential equation (*).

Determine the particular solution for y in terms of t.

Using your answer to part (b), find the value of y immediately after the light hits the cell.

The cell can be considered to be operating properly if y < 4 when t = 1.

Discuss whether the cell can be inferred to be operating properly.

Find, as a percentage of sinh u, the difference between sinh u and cosh u for the following values of u.

• u = 2

• u = 5

A function f is defined for all integers n by f (n) = sinh (0.01 n) - 5cosh (0.005 n) - 9tanh n.

In this question you must show detailed reasoning.

With the help of suitable approximations, use an algebraic method to determine the smallest value of n for which f (n) > 100. You should verify your answer, once found, by direct

calculation. You may assume that the required value of n is large.