5
- 1
3
\end{array} \right] \quad \left[ \begin{array} { l }
2
1
1
\end{array} \right]$$
2 Use the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\)
[0pt]
[2 marks]
3
- Given that
$$\frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } \equiv \frac { A } { ( r + 1 ) ( r + 2 ) } + \frac { B } { ( r + 2 ) ( r + 3 ) }$$
find the values of the integers \(A\) and \(B\)
3 - Use the method of differences to show clearly that
$$\sum _ { r = 9 } ^ { 97 } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { 89 } { 19800 }$$
4 A student states that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \cos x + \sin x } { \cos x - \sin x } \mathrm {~d} x\) is not an improper integral because \(\frac { \cos x + \sin x } { \cos x - \sin x }\) is defined at both \(x = 0\) and \(x = \frac { \pi } { 2 }\)
Assess the validity of the student's argument.
[0pt]
[2 marks]
\(5 \quad \mathrm { p } ( z ) = z ^ { 4 } + 3 z ^ { 2 } + a z + b , a \in \mathbb { R } , b \in \mathbb { R }\)
\(2 - 3 \mathrm { i }\) is a root of the equation \(\mathrm { p } ( \mathrm { z } ) = 0\)
5 - Express \(\mathrm { p } ( z )\) as a product of quadratic factors with real coefficients.
5 - Solve the equation \(\mathrm { p } ( z ) = 0\).