OCR A Level Further Maths B (MEI) Extra Pure (minor option - choose one with a major, or three alone) June 2025 question paper Y435

A binary operation, ○, is defined on real numbers a and b by a ○ b = 3 ab.

Prove that ○ is associative over the real numbers ℝ.

Determine the value of e, the identity element for ○.

Explain why ℝ is closed under ○.

Explain why ℝ under ○ does not form a group. Justify your answer.

Determine the general solution of the recurrence relation 25u_n+2 - 35u_n+1 + 12u_n = c, n ⩾ 0

where c is a constant.

The manager of a company is using the recurrence relation in part (a) to model the number of containers held by the company using the sequence u₀, u₁, u₂, … which satisfies this recurrence relation.

The quantity u₀ represents the number of containers held by the company at the start of January 2000. The quantity u_n, for n ⩾ 1, represents the number of containers held by the company at the end of month n, where the first month is January 2000. If necessary, the quantity u_n is rounded to the nearest integer.

The company holds 20 containers at the start of January 2000 and 70 containers at the end of January 2000.

In an initial model the manager uses c = – 10.

Determine the particular solution for u_n when c = –10.

The model predicts a single peak value for u_n.

Find, by direct calculation, the largest number of containers held by the company

according to the model.

Show that the model in part (b)(i) breaks down in the 19th month.

In a second model the manager uses c = 10 instead of c = – 10.

Show that the second model predicts that eventually the number of containers held by the

company will be a constant k, where k is to be determined.

M is the group ({0, 1, 2, 3, 4, 5, 6, 7, 8}, +₉) where +₉ denotes the binary operation of addition modulo 9.

Write down the identity element of M. Justify your answer.

By considering the order of M, explain why there can be no subgroup of M of order 5.

You are given that M is a cyclic group.

Find all possible generators of M.

You are given that there is only one proper, non-trivial subgroup of M, denoted by (H, +₉).

Write down H.

Show that there can be no other proper, non-trivial subgroups of M.

A surface, S, is defined in 3-D by z = f(x, y) where f(x, y) = x³ - 12xy² + 96y² + 30.

You are given that the point (0, 0, 30) on S is a stationary point.

Determine the coordinates of any other stationary points on S.

By sketching the section z = f (x, 0) on the coordinate axes provided in the Printed Answer

Booklet, determine the nature of the stationary point (0, 0, 30) on S.

The normal to S at the point where x = 3 and y = 2 passes through the point (a, 482, 295) where a is a constant.

Determine the value of a.

The contour given by z = 542 comprises a straight line and a curve. The curved portion of the contour is denoted by C. You are given that there are two points on C where the x coordinate is 8.

Determine the values of the y coordinates at these two points on C.

In this question you must show detailed reasoning.

By finding the characteristic equation of A in terms of a, show that, for any value of a

1 is always one of the eigenvalues of A.

eigenvalues of A are not 1.

Hence show that either some of the eigenvalues of A must be non-real or some of the

elements of A must be non-real.

You are now given the following information.

• a and b are both real.

• A represents a rotation by an acute angle in 3-D space.

• e is a real eigenvector of A associated with the eigenvalue 1.

Explain the significance of e, and the fact that its associated eigenvalue is 1, in relation

to the transformation that A represents.

Explain geometrically why the other eigenvectors of A can not be real.

Find a vector equation for the axis of rotation of the transformation that A represents.