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.
[1]
Explain whether the matrix AB can be formed.
[1]
In this question you must show detailed reasoning.
Write the complex number - 24 + 7i in modulus-argument form.
[3]
[4]
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.
[4]
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.
[2]
Solve the equation 15 sinh x + 16 cosh x - 6 sinh 2 x = 20, giving all your answers in
logarithmic form.
[5]
In this question you must show detailed reasoning.
Show that
where a, b and c are integers whose values are to be determined.
[6]
[2]
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).
[6]
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.
[2]
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.
[3]
By considering 
and 
, make a conjecture for a general formula for 
in
terms of n and a for any integer n ⩾ 1.
[3]
Use induction to prove the formula conjectured in part (a).
[4]
The paper as printed
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