OCR A Level Further Maths A Additional Pure Mathematics (optional - choose any two of Y542/Y543/Y544/Y545) June 2023 question paper Y545

The surface S is defined for all real x and y by the equation z = x² + 2xy. The intersection of S with the plane Π gives a section of the surface. On the axes provided in the Printed Answer Booklet, sketch this section when the equation of Π is each of the following.

x = 1

y = 1

A small component for a car is produced in the shape of this surface. The curved surface area of the component must be 8 cm², accurate to within one percent. The engineering process produces such components with a curved surface area accurate to within one half of one percent.

Determine whether all components produced will be suitable for use in the car.

The points A and B have position vectors a = i + p j + q k and b = 2 i + 3 j + 2 k respectively, relative to the origin O.

Determine the value of p and the value of q for which a × b = 2 i + 6 j - 11 k.

The point C has coordinates (d, e, f) and the tetrahedron OABC has volume 7.

Using the values of p and q found in part (a), find the possible relationships between

d, e and f.

Explain the geometrical significance of these relationships.

• Show that {A_n} increases monotonically.

• Show that {A_n} converges to a limit, A, whose exact value should be stated.

The group G consists of the set S = {1, 9,17, 25} under ×₃₂, the operation of multiplication modulo 32.

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

Up to isomorphisms, there are only two groups of order 4.

• C₄, the cyclic group of order 4

• K₄, the non-cyclic (Klein) group of order 4

State, with justification, to which of these two groups G is isomorphic.

List the odd quadratic residues modulo 32.

Given that n is an odd integer, prove that n⁶ + 3n⁴ + 7n² ≡ 11(mod 32).

The surface S has equation z = x sin y + for x > 0 and 0 < y < π.

Determine, as a function of x and y, the determinant of H, the Hessian matrix of S.

Given that S has just one stationary point, P, use the answer to part (a) to deduce the nature

of P.

The coordinates of P are (α, β, γ).

Show that β satisfies the equation β + tan β = 0.

Write down the values of α + β and αβ.

Consider the sequence {S_n}, where S_n = αⁿ + βⁿ for n ⩾ 0.

Determine the values of S₂ and S₃.

Show that S_n+2 = S_n+1 + S_n for n ⩾ 0.

Deduce that S_n is an integer for all n ⩾ 0.

A student models the terms of the sequence {S_n} using the formula T_n = αⁿ.

Explain why this formula is unsuitable for every n ⩾ 1.

Considering the cases n even and n odd separately, state a modification of the formula

T_n = αⁿ, other than T_n = αⁿ + βⁿ, such that T_n = S_n for all n ⩾ 1.

Let f(n) denote the base-n number 2121_n where n ⩾ 3.

For each n ⩾ 3, show that f (n) can be written as the product of two positive integers

greater than 1, a (n) and b (n), each of which is a function of n.

Deduce that f (n) is always composite.

Let h be the highest common factor of a (n) and b (n).

Prove that h is either 1 or 5.

Find a value of n for which h = 5.

For each element x in C show that it has an inverse element in C.