OCR A Level Further Maths B (MEI) Extra Pure (minor option - choose one with a major, or three alone) June 2023 question paper Y435

A surface is defined in 3-D by z = 3x³ + 6xy + y².

Determine the coordinates of any stationary points on the surface.

A sequence is defined by the recurrence relation 4t_n+1 - t_n = 15n + 17 for n ⩾ 1, with t₁ = 2.

Solve the recurrence relation to find the particular solution for t_n.

Another sequence is defined by the recurrence relation (n + 1)u - u² = 2n - for n ⩾ 1, with u₁ = 2.

Explain why the recurrence relation for u_n cannot be solved using standard techniques

for non-homogeneous first order recurrence relations.

Verify that the particular solution to this recurrence relation is given by u_n = an +

where a and b are constants whose values are to be determined.

A third sequence is defined by v_n = for n ⩾ 1.

Determine_nlim_→∞ v_n.

The set G is given by G = { M: M is a real 2 × 2 matrix and det M = 1 }.

Show that G forms a group under matrix multiplication, ×. You may assume that matrix

multiplication is associative.

Determine whether S is closed under ×.

Determine whether S is a subgroup of (G, ×).

Find a subgroup of (G, ×) of order 2.

By considering the inverse of the non-identity element in any such subgroup, or

otherwise, show that this is the only subgroup of (G, ×) of order 2.

The set of all real 2 × 2 matrices is denoted by H.

With the help of an example, explain why (H, ×) is not a group.

You are given instead that P satisfies the matrix equation I = P² + rP for some rational number r.

Use the Cayley-Hamilton theorem to determine the value of a and the corresponding

value of r.

Hence show that P⁴ = sI + tP where s and t are rational numbers to be determined.

You should not calculate P⁴.