OCR A Level Further Maths A Pure Core 2 June 2023 question paper

Write down the transpose of P.

Write down the sum of the two of these matrices which are conformable for addition.

The dimensions of matrix A are 4 by 5. The matrices A and B are conformable for multiplication so that the matrix C = BA can be formed. The matrix C has 6 rows.

Write down the number of columns that C has.

Write down the dimensions of B.

Explain whether the matrix AB can be formed.

In this question you must show detailed reasoning.

Write the complex number - 24 + 7i in modulus-argument form.

form.

In this question you must show detailed reasoning.

A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the shape formed when R is rotated by 2 π radians about the x-axis.

Find the exact value of the volume of precious metal required to make the pendant, according to

the model.

In this question you must show detailed reasoning.

Using the definitions of sinh x and cosh x in terms of exponentials, show that

sinh 2 x ≡ 2 sinh x cosh x.

Solve the equation 15 sinh x + 16 cosh x - 6 sinh 2 x = 20, giving all your answers in

logarithmic form.

In this question you must show detailed reasoning.

Show that

where a, b and c are integers whose values are to be determined.

A surge in the current, I units, through an electrical component at a time, t seconds, is to be modelled. The surge starts when t = 0 and there is initially no current through the component. When the current has surged for 1 second it is measured as being 5 units. While the surge is occurring, I is modelled by the following differential equation.

By using an integrating factor show that, according to the model, while the surge is occurring,

I is given by I = (2t - t²)(sin^-1(t - 1) + 5).

The surge lasts until there is again no current through the component.

Determine the length of time that the surge lasts according to the model.

Determine, according to the model, the rate of increase of the current at the start of the surge.

Give your answer in an exact form.

By considering and , make a conjecture for a general formula for in

terms of n and a for any integer n ⩾ 1.

Use induction to prove the formula conjectured in part (a).