OCR A Level Further Maths A Additional Pure Mathematics (optional - choose any two of Y542/Y543/Y544/Y545) November 2021 question paper Y545

In this question you must show detailed reasoning.

Express the number 41723₁₀ in hexadecimal (base 16).

List all the proper subgroups of G.

State another group of order 6 to which G is isomorphic.

The points P, Q and R have position vectors p = 2 i + j + 5 k, q = i - j + k and r = 2 i + j + t k respectively, relative to the origin O.

Determine the value(s) of t in each of the following cases.

The line OR is parallel to p × q.

The volume of tetrahedron OPQR is 13.

Solve the simultaneous linear congruences x ≡ 1 (mod 3), x ≡ 5 (mod 11), 2 x ≡ 5 (mod 17).

The surface S has equation x² + y² + z² = xyz - 1.

Deduce that S has no stationary point.

Show that, for n ⩾ 2, n I_n = (n - 1)I_n-2.

Use this reduction formula to deduce the exact value of I₈.

Solve the second-order recurrence system H_n+2 = 5H_n+1 - 4H_n with H₀ = 3, H₁ = 7 for

n ⩾ 0.

Write down the quadratic residues modulo 10.

By considering the sequence {H_n} modulo 10, prove that H_n is never a perfect square.

{u_n} is constant.

{u_n} is periodic, with period 2.

{u_n} is periodic, with period 4.

A student wishes to model the saddle of a horse. They use a surface described by a function of the form z = f (x, y) with a saddle point at the origin O. The z-axis is vertically upwards. The x - and y-axes lie in a horizontal plane, with the x-axis across the horse and the y-axis along the length of the horse (see diagram).

The arc AOB is part of a parabola which lies in the yz-plane. The arc COD is part of a parabola which lies in the xz-plane. The saddle is symmetric in both the xz-plane and yz-plane.

The length of the saddle, the distance AB, is to be 0.6 m with both A and B at a height of 0.27 m above O. The width of the saddle, the distance CD, is to be 0.5 m with both C and D at a depth of 0.4 m below O.

On separate diagrams, sketch the sections x = 0 and y = 0.

Determine a function f that describes the saddle. [You do not need to state the domain of

function f.]