The complex numbers \(u\) and \(w\) are such that
$$u - w = 2 \mathrm { i } \quad \text { and } \quad u w = 6$$
Find \(u\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities
$$| z - 2 - 2 \mathbf { i } | \leqslant 2 , \quad 0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi \quad \text { and } \quad \operatorname { Re } z \leqslant 3$$
A tank containing water is in the form of a hemisphere. The axis is vertical, the lowest point is \(A\) and the radius is \(r\), as shown in the diagram. The depth of water at time \(t\) is \(h\). At time \(t = 0\) the tank is full and the depth of the water is \(r\). At this instant a tap at \(A\) is opened and water begins to flow out at a rate proportional to \(\sqrt { h }\). The tank becomes empty at time \(t = 14\).
The volume of water in the tank is \(V\) when the depth is \(h\). It is given that \(V = \frac { 1 } { 3 } \pi \left( 3 r h ^ { 2 } - h ^ { 3 } \right)\).
Show that \(h\) and \(t\) satisfy a differential equation of the form
$$\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { B } { 2 r h ^ { \frac { 1 } { 2 } } - h ^ { \frac { 3 } { 2 } } } ,$$
where \(B\) is a positive constant.
Solve the differential equation and obtain an expression for \(t\) in terms of \(h\) and \(r\).
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.