OCR A Level Further Maths B (MEI) Core Pure June 2022 question paper Y420

In this question you must show detailed reasoning.

Sketch the polar curve r = a (1 - cos θ), 0 ⩽ θ < 2 π, where a is a positive constant.

Determine the exact area of the region enclosed by the curve.

In this question you must show detailed reasoning.

Sketch the loci on an Argand diagram.

In this question you must show detailed reasoning.

Find the complex numbers represented by the points of intersection.

The function f (x) is defined by f (x) = ln (1 + sinh x).

Given that k lies in the domain of this function, explain why k must be greater than

Find f ′ (x).

Hence find a quadratic approximation to f (x) for small values of x.

Find the percentage error in this approximation when x = 0.1.

The equation

4x⁴ + 16x³ + ax² + bx + 6 = 0, where a and b are real, has roots α β and 3β.

Determine the values of a and b.

On the copy of the Argand diagram in the Printed Answer Booklet, mark the points B

and C representing the complex numbers z₂ and z₃.

Show that z₁ + z₂ + z₃ = 0.

Given now that z₁, z₂ and z₃ are roots of the equation z³ = 8i, find these three roots, giving

your answers in the form a + i b, where a and b are real and exact.

Solve the differential equation (4 - x² - xy = 1, given that y = 1 when x = 0, giving your

answer in the form y = f (x).

Find (i - 2 j) × (2 i + 3 j - k).

Find (3 - e^2iθ)(3 - e^-2iθ) in terms of cos2θ.

Hence show that the sum of the infinite series

In an oscillating system, a particle of mass m kg moves in a horizontal line. Its displacement from its equilibrium position O at time t seconds is x metres, its velocity is vms^-1, and it is acted on by a force 2 mx newtons acting towards O as shown in the diagram.

Initially, the particle is projected away from O with speed 1ms^-1 from a point 2 m from O in the positive direction.

Show that the motion is modelled by the differential equation + 2x = 0.

State the type of motion.

Write down the period of the motion.

Find x in terms of t.

Find the amplitude of the motion.

The motion is now damped by a force 2 mv newtons.

State, giving a reason, whether the system is under-damped, critically damped or

over-damped.

Determine the general solution of this differential equation.

Finally, a variable force 2 m cos 2 t newtons is added, so that the motion is now modelled by the differential equation

Find x in terms of t.

In the long term, the particle is seen to perform simple harmonic motion with a period of just over 3 seconds.

Verify that this behaviour is consistent with the answer to part (c)(i).