OCR A Level Further Maths B (MEI) Extra Pure (minor option - choose one with a major, or three alone) June 2022 question paper Y435

In this question you must show detailed reasoning.

Show that the characteristic equation of A is -λ³ + 14λ² - 56λ + 64 = 0.

Use the Cayley-Hamilton theorem to determine A^-1.

A matrix E and a diagonal matrix D are such that A = EDE^-1. The elements in the diagonal of D increase from top left to bottom right.

Determine the matrix D.

A sequence is defined by the recurrence relation 5t_n+1 - 4t_n = 3n² + 28n + 6, for n ⩾ 0, with t₀ = 7.

Find an expression for t_n in terms of n.

Another sequence is defined by v = for n ⩾ 1, where m is a constant.

In each of the following cases determine_nlim_→∞ v_n, if it exists, or show that the sequence is divergent.

m = 3

m = 2

m = 1

where k₁, k₂ and k₃ are constants (which are not necessarily in A) and the operations addition, subtraction and multiplication of numbers have their usual notation and meaning.

• A = ℝ

State the value of e.

Explain whether your answer to part (d) would change in each of the following cases, giving details of any change.

A = Z

A = {2m: m ∈ Z}

A = {n: n ∈ Z, n ⩾ -2}

A surface S is defined by z = f(x, y), where f(x, y) = y e^-(x2+2x+2)y.

Determine the coordinates of any stationary points on S.

Fig. 5.1 shows the graph of z = e^-x2 and Fig. 5.2 shows the contour of S defined by z = 0.25.

Specify a sequence of transformations which transforms the graph of z = e^-x2 onto the graph

of the section defined by z = f (x, 1).

Hence, or otherwise, sketch the section defined by z = f (x, 1).

Using Fig. 5.2 and your answer to part (c), classify any stationary points on S, justifying your

answer.

You are given that P is a point on S where z = 0.

Find, in vector form, the equation of the tangent plane to S at P.

The tangent plane found in part (e) intersects S in a straight line, L.

Write down, in vector form, the equation of L.