OCR A Level Further Maths A Pure Core 1 June 2023 question paper Y540

In this question you must show detailed reasoning.

In this question you must show detailed reasoning.

The equation z⁴ + 4z³ + 9z² + 10z + 6 = 0 has roots α, β, γ and δ.

Show that a quartic equation whose roots are α + 1, β + 1, γ + 1 and δ + 1 is

w⁴ + 3w² + 2 = 0.

Hence determine the exact roots of the equation z⁴ + 4z³ + 9z² + 10z + 6 = 0.

Describe geometrically the single transformation consisting of T_A followed by T_B.

By considering the transformation T_A, determine the matrix A⁴²³.

The transformation T_C is represented by the matrix C, where

The region R is defined by the set of points (x, y) satisfying the inequality x² + y² ⩽ 36.

The region R′ is defined as the image of R under T_C.

Find the exact area of the region R ′.

Sketch the region R ′, specifying all the points where the boundary of R ′ intersects the

coordinate axes.

Find the general solution of the differential equation - + 5y = 0.

Hence find the general solution of the differential equation - + 5y = x(4 - 5x).

In this question you must show detailed reasoning.

The power output, p watts, of a machine at time t hours after it is switched on can be modelled by the equation p = 20 - 20 tanh (1.44 t) for t ⩾ 0.

Determine, according to the model, the mean power output of the machine over the first half hour

after it is switched on. Give your answer correct to 2 decimal places.

The engineer applies the equation “ F = ma ” to the model of the motion of P and derives the following differential equation.

When t = 0 the displacement of P is 6 m, and it is travelling towards O with a speed of 5 ms^-1.

Without attempting to solve the differential equation, find the acceleration of P

when t = 0.

The points P, Q and R have coordinates (0, 2, 3), (2, 0, 1) and (1, 3, 0) respectively.

The acute angle between the line segments PQ and PR is θ.

The triangle PQR lies in the plane Π.

Determine an equation for Π, giving your answer in the form ax + by + cz = d, where a, b, c

and d are integers.

In this question you must show detailed reasoning.

Use de Moivre’s theorem to determine constants A, B and C such that

sin⁴θ ≡ A cos 4θ + B cos 2θ + C.

The function f is defined by

The diagram shows the curve with equation y = for 0 ⩽ x < 1 and the asymptote x = 1. The region R is the unbounded region between the curve, the x-axis, the line x = 0 and the line x = 1.

You are given that the area of R is finite.

Determine the exact area of R.