OCR A Level Further Maths A Pure Core 2 November 2021 question paper Y541

In this question you must show detailed reasoning.

The complex numbers z₁ and z₂ are given by z₁ = 3 - 7i and z₂ = 2 + 4i.

Express each of the following as exact numbers in the form a + b i.

3z₁ + 4z₂

z₁z₂

Write z₁ in modulus-argument form giving the modulus in exact form and the argument

correct to 3 significant figures.

Find the position vector of the point of intersection of l₁ and Π.

Find the acute angle between l₁ and Π.

A is the point on l₁ where λ = 1.

l₂ is the line with the following properties.

• l₂ passes through A

• l₂ is perpendicular to l₁

• l₂ is parallel to Π

Find, in vector form, the equation of l₂.

In this question you must show detailed reasoning.

In this question you must show detailed reasoning.

Using the definition of cosh x in terms of exponentials, show that cosh 2x ≡ 2cosh² x - 1.

Solve the equation cosh 2 x = 3 cosh x + 1, giving all your answers in exact logarithmic form.

Define the transformation represented by A.

Show that the area of any object shape is invariant under the transformation represented by A.

In this question you must show detailed reasoning.

form a + ln b + c π where a is an integer, and b and c are both rational.

Find an expression for v in terms of t for t ⩾ 0.

By considering the behaviour of v as t becomes large explain why, according to the model, P ’s

speed must reach a maximum value for some t > 0.

Determine the maximum speed considered in part (c).

Determine the greatest value of t for which the model is valid.

By considering A, A², A³ and A⁴ make a conjecture about the form of the matrix Aⁿ in terms

of n for n ⩾ 1.

Use induction to prove the conjecture made in part (a).

In this question you must show detailed reasoning.

By using an appropriate Maclaurin series prove that if x > 0 then e^x > 1 + x.

Using the inequality in part (b), and by making a suitable choice for t, determine which is

greater, e^π or π^e.