If z is a non-zero complex number, we define \(L(z)\) by the equation

\[L(z) = \ln \left| z \right| + {\text{i}}\arg (z),{\text{ }}0 \leqslant \arg (z) < 2\pi .\]

(a)     Show that when z is a positive real number, \(L(z) = \ln z\) .

(b)     Use the equation to calculate

(i)     \(L( - 1)\) ;

(ii)     \(L(1 - {\text{i}})\) ;

(iii)     \(L( - 1 + {\text{i}})\) .

(c)     Hence show that the property \(L({z_1}{z_2}) = L({z_1}) + L({z_2})\) does not hold for all values of \({z_1}\) and \({z_2}\) .

Let f be a function with domain \(\mathbb{R}\) that satisfies the conditions,

\(f(x + y) = f(x)f(y)\) , for all x and y and \(f(0) \ne 0\) .

(a)     Show that \(f(0) = 1\).

(b)     Prove that \(f(x) \ne 0\) , for all \(x \in \mathbb{R}\) .

(c)     Assuming that \(f'(x)\) exists for all \(x \in \mathbb{R}\) , use the definition of derivative to show that \(f(x)\) satisfies the differential equation \(f'(x) = k{\text{ }}f(x)\) , where \(k = f'(0)\) .

(d)     Solve the differential equation to find an expression for \(f(x)\) .

A gourmet chef is renowned for her spherical shaped soufflé. Once it is put in the oven, its volume increases at a rate proportional to its radius.

(a)     Show that the radius r cm of the soufflé, at time t minutes after it has been put in the oven, satisfies the differential equation \(\frac{{{\text{d}}r}}{{{\text{d}}t}} = \frac{k}{r}\), where k is a constant.

(b)     Given that the radius of the soufflé is 8 cm when it goes in the oven, and 12 cm when it’s cooked 30 minutes later, find, to the nearest cm, its radius after 15 minutes in the oven.

Find y in terms of x, given that \((1 + {x^3})\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2{x^2}\tan y\) and \(y = \frac{\pi }{2}\) when x = 0.

A certain population can be modelled by the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}t}} = k\,y\cos kt\) , where y is the population at time t hours and k is a positive constant.

(a)     Given that \(y = {y_0}\) when t = 0 , express y in terms of k , t and \({y_0}\) .

(b)     Find the ratio of the minimum size of the population to the maximum size of the population.

Find the equation of the tangent to C at the point (2, e).

Find \(f(x)\).

Determine the largest possible domain of f.

Show that the equation \(f(x) = f'(x)\) has no solution.