OCR A Level Further Maths A Pure Core 1 June 2024 question paper Y540

Given that y = sin^-1(x²), find

Indicate by shading on the Argand diagram in the Printed Answer Booklet the region

representing C₁.

Determine whether the complex number 1.2 + 0.8i is in C₁.

The locus C₂ is the set of complex numbers represented by the interior of the circle with radius 2 and centre 3. The locus C₂ is illustrated on the Argand diagram below.

Use set notation to define C₂.

Determine whether the complex number 1.2 + 0.8i is in C₂.

Find N² in terms of a.

Find det N in terms of a.

In this question you must show detailed reasoning.

The equation 2x³ + 3x² + 6x - 3 = 0 has roots α, β and γ.

Determine a cubic equation with integer coefficients that has roots α²βγ, αβ²γ and αβγ².

In this question you must show detailed reasoning.

By using the definitions of cosh u and sinh u in terms of e^u and e^-u, show that

sinh 2 u ≡ 2 sinh u cosh u.

You are now given that C has exactly one point of inflection.

Use your answer to part (b) to determine the exact coordinates of this point of inflection.

Give your answer in a logarithmic form where appropriate.

Prove by induction that 11 × 7ⁿ - 13ⁿ - 1 is divisible by 3, for all integers n ⩾ 0.

Find the Maclaurin series of (ln(1 + x))² up to and including the term in x⁴.

The diagram below shows parts of the graphs of the curves with equations y = (ln(1 + x))² and y = 2x³.

The curves intersect at the origin, O, and at the point A.

In this question you must show detailed reasoning.

Use your answer to part (a) to determine an approximation for the value of the x-coordinate

of A. Give your answer to 2 decimal places.

Using an integrating factor, determine the first time at which B is stationary according to the

model.

Find, in the same form, a vector equation of the line segment that represents the wire T. The components of the direction vector should be integers whose only positive common factor is 1.

For the construction site to be considered safe, it must pass two tests.

Test 1: The wires S and T need to be at least 5 metres apart at all positions on S and T.

By using an appropriate formula, determine whether the construction site passes Test 1.

A security camera is placed at a point Q on wire S.

Test 2: To ensure sufficient visibility of the construction site, the distance between the security camera and the top of P₃ must be at least 19 m.

Determine whether it is possible to find point Q on S such that the construction site passes

Test 2.