OCR A Level Further Maths B (MEI) Core Pure November 2021 question paper Y420

In this question you must show detailed reasoning.

Find the gradi of the curve y = 6 arcsin(2x) at the point with x-coordinate . Express the result

Find the modulus and argument of z₁.

Hence express in exact modulus-argument form.

Use a Maclaurin series to find a quadratic approximation for ln (1 + 2 x).

Find the percentage error in using the approximation in part (a) to calculate ln (1.2).

Jane uses the Maclaurin series in part (a) to try to calculate an approximation for ln3.

Explain whether her method is valid.

Determine the exact roots of the equation.

Determine the values of p and q.

On the grid in the Printed Answer Booklet, plot the image OAʹBʹCʹ of the unit square OABC

under the transformation T.

Calculate the value of det M.

Explain the significance of the value of det M in relation to the image OAʹBʹCʹ.

T is equivalent to a sequence of two transformations of the plane.

Specify fully two transformations equivalent to T.

Use matrices to verify your answer.

Show on an Argand diagram the points representing the three cube roots of unity.

The three roots in part (b)(i) are z₁, z₂ and z₃.

By simplifying z₁ + z₂ + z₃, verify that the sum of these roots is zero.

Hence show that sin 20° + sin 140° = sin 100°.

Given that u = λ i + j - 3 k and v = i + 2 j - 2 k, find the following, giving your answers in terms of λ.

u.v

u × v

Hence determine

the acute angle between the planes 2 x + y - 3 z = 10 and x + 2 y - 2 z = 10,

the shortest distance between the lines = = and = = , giving

Fig. 12 shows a rhombus OACB in an Argand diagram. The points A and B represent the complex numbers z and w respectively.

Prove that arg(z + w) = arg z + arg w).

[A copy of Fig. 12 is provided in the Printed Answer Booklet.]

Find the general solution of the differential equation + - 3y = 2e^x.

A curve has polar equation r = a (cos θ + 2 sin θ), where a is a positive constant and 0 ⩽ θ ⩽ π.

Determine the polar coordinates of the point on the curve which is furthest from the pole.

Show that the curve is a circle whose radius should be specified.

Write down the polar coordinates of the centre of the circle.

The equations of three planes are

- 4 x + ky + 7 z = 4, x - 2 y + 5 z = l, 2 x + 3 y + z = 2.

Given that the planes form a sheaf, determine the values of k and l.

Show using exponentials that cosh 2u = 1 + 2 sinh²u.

In a chemical process, a vessel contains 1 litre of pure water. A liquid chemical is then passed into the top of the vessel at a constant rate of a litres per minute and thoroughly mixed with the water. At the same time, the resulting mixture is drawn from the bottom of the vessel at a constant rate of b litres per minute. You may assume that the chemical mixes instantly and uniformly with the water. After t minutes, the mixture in the vessel contains x litres of the chemical.

Show that the proportion of chemical present in the vessel after t minutes is

First, consider the case where b = a.

Solve the differential equation to find x in terms of a and t.

Given that after 1 minute the vessel contains equal amounts of water and chemical, find

the rate of inflow of chemical.

Now consider the case where b = 2 a.

Explain why the differential equation in part (a)(ii) is now invalid for t ⩾

Find the maximum amount of chemical in the vessel.