OCR A Level Further Maths B (MEI) Core Pure June 2023 question paper Y420

The complex number a + i b is denoted by z.

Write down z *.

Find Re(i z).

In this question you must show detailed reasoning.

Express w in the form x + i y.

Convert w to modulus-argument form.

In this question you must show detailed reasoning.

Find the angle between the vector 3 i + 2 j + k and the plane - x + 3 y + 2 z = 8.

Using partial fractions and the method of differences, show that

where a and b are integers to be determined.

Deduce the sum to infinity of the series.

In this question you must show detailed reasoning.

Determine the sixth roots of -64, expressed in re^iθ form.

Represent the roots on an Argand diagram.

In this question you must show detailed reasoning.

Determine whether M and N commute under matrix multiplication.

Specify the transformation of the plane associated with each of the following matrices.

M

N

State the significance of the result in part (a) for the transformations associated with M and N.

Use an algebraic method to show that all lines parallel to the x-axis are invariant lines of the

transformation associated with N.

The diagram below shows the curve with polar equation r = a (1 - 2 sin θ) for 0 ⩽ θ ⩽ 2 π, where a is a positive constant.

The curve crosses the initial line at A, and the points B and C are the lowest points on the two loops.

Find the values of r and θ at the points A, B and C.

Find the set of values of θ for the points on the inner loop (shown in the diagram with a

broken line).

Prove by mathematical induction that 8ⁿ - 3ⁿ is divisible by 5 for all positive integers n.

In an electrical circuit, the alternating current I amps is given by I = a sin nt, where t is the time in seconds and a and n are positive constants. The RMS value of the current, in amps, is defined to

Show that the RMS value of the current is amps.

The equation x³ - 4x² + 7x + c = 0, where c is a constant, has roots α, β and α + β.

Determine the roots of the equation.

Find c.

Solve the differential equation cosh - 2y sinh x = cosh x, given that y = 1 when x = 0.

Show that sin⁵θ = a sin 5θ + b sin 3θ + c sin θ, where a, b and c are constants to be determined.

On separate Argand diagrams, show the set of points representing each of the following inequalities.

| z + 2 - 4 i | ⩾ | z - 2 - 6 i |

Three planes have equations

kx - z = 2, - x + ky + 2 z = 1, 2 kx + 2 y + 3 z = 0,

where k is a constant.

By considering a suitable determinant, show that the three planes meet at a point for all

values of k.

Using a matrix method, find, in terms of k, the coordinates of the point of intersection of the

planes.

In this question you must show detailed reasoning.

The point P (4, 1, 0) is equidistant from the plane 2x + y + 2z = 0 and the line = = where b > 0.

Determine the value of b.

Two similar species, X and Y, of a small mammal compete for food and habitat. A model of this competition assumes, in a particular area, the following.

• In the absence of the other species, each species would increase at a rate proportional to the number present with the same constant of proportionality in each case.

• The competition reduces the rate of increase of each species by an amount proportional to the number of the other species present.

So if the numbers of species X and Y present at time t years are x and y respectively, the model gives the differential equations

where k, a and b are positive constants.

Hence find the general solution for y in terms of A, B, k, n, a and t.

Observations suggest that suitable values for the model are k = 0.015, a = 0.04 and b = 0.01. You should use these values in the rest of this question.

When t = 0, the numbers present of species X and Y in this area are x₀ and y₀ respectively.

Use initial values x₀ = 500 and y₀ = 300 with the results in part (b) to determine what the model predicts for each of the following questions.

What numbers of each species will be present after 25 years?

In this question you must show detailed reasoning.

When will the numbers of the two species be equal?

Does either species ever disappear from the area? Justify your answer.

Different initial values will apply in other areas where the two species compete, but previous studies indicate that one species or the other will eventually dominate in any given area.

Identify a relationship between x₀ and y₀ where the model does not predict this

outcome.

Explain what the model predicts in the long term for this exceptional case.