OCR A Level Further Maths A Additional Pure Mathematics (optional - choose any two of Y542/Y543/Y544/Y545) June 2025 question paper Y545

A student claims that the order of 5 modulo 37 is 5.

Without calculation, explain why this student’s claim must be incorrect.

Show that the order of 5 modulo 37 is 36.

The points P, Q, R and S have coordinates (1, 2, 4), (3, 1, 6), (4, 8, - 2) and (5, 5, 5) respectively.

Find, in exact form, the area of triangle PQR.

Find, in exact form, the volume of tetrahedron PQRS.

Use your answers to part (a) and part (b) to deduce the shortest distance from S to the plane containing P, Q and R. Give your answer in exact form.

[You may use the formula V = Ah, where V is the volume of a pyramid, A is the area of one

of its bases and h is the height relative to the base.]

Let N = 10 a + b and M = a - 3 b, where a and b are positive integers and 0 ⩽ b ⩽ 9.

By considering 3 N + M, prove that 31 | N if and only if 31 | M.

Use a procedure based on the result of part (a) to show that 31 | 7 167 758.

In this question you must show detailed reasoning.

The surface S is given by the equation z = 4x²y - 3xy² - 5x for all real values of x and y.

The point P (1, 1, - 4) lies on S.

Find the equation of the tangent plane to S at P.

The point Q is a stationary point of S with positive x - and y-coordinates.

Find, in exact form, the coordinates of Q.

Find the value of the determinant of H, the Hessian matrix of S, at Q.

Deduce the nature of the stationary point Q.

Determine the solution of the linear congruence 3 x ≡ 7 (mod 13). Give your answer in the

form x ≡ a (mod 13), where 0 ⩽ a < 13.

Hence or otherwise, determine the solution of the simultaneous linear congruences

3 x ≡ 7 (mod 13), 7 x ≡ 3 (mod 31).

In this question, the set S is {2, 4, 6, 8}, and ×₁₀ is the operation of multiplication modulo 10.

The group G consists of the set S, under the operation ×₁₀.

Complete the Cayley table for G given in the Printed Answer Booklet.

State the identity element of G.

State the order of each non-identity element of G.

State, with justification, the order of H.

State also the identity element of H.

State all the elements of H of order 2.

Using the result of part (a)(iii), or otherwise, explain why all the (non-identity) elements

which do not have order 2 must have order 4.

State whether or not H is cyclic. You must give a reason for your answer.

In this question you must show detailed reasoning.

A curve is defined parametrically by x = cos³t, y = sin³t, for 0 ⩽ t ⩽ . When the curve is rotated through 2π about the x-axis, a surface of revolution is formed with surface area A.

Determine the exact value of A.

Determine, in terms of q, the general solution of the first-order recurrence relation.

Hence explain why, when q is close to zero, G_n is approximately constant and equal to

1.25.

In the case where G₁ = 1.22 and G₂ = 1.2378, find the values of q and G₀.

The population of a species of fish F_n is being modelled by the second-order recurrence relation

F_n+2 = 1.25F_n+1 - rF_n+1F_n,

where r is a positive constant and n is the number of weeks since the start of the breeding season.

You are given that the value of r is such that the model predicts that the population approaches a limiting value, L.

Determine L in terms of r.

Determine what happens if a term of the sequence is greater than

Explain why the modified recurrence relation F_n+2 = INT(1.25F_n+1 - rF_n+1F_n) may be

preferable when modelling this fish population.

When r = 0.0001, both F_n+2 = 1.25F_n+1 - rF_n+1F_n and F_n+2 = INT(1.25F_n+1 - rF_n+1F_n) predict that the fish population approaches a limiting value.

By finding the limiting value for each model, when F₀ = 2450 and F₁ = 2475, compare

the two models for the fish population.