5 In parts (i), (ii), (iii) of this question you are required to investigate curves with the equation $$x ^ { k } + y ^ { k } = 1$$ for various positive values of \(k\).

1. Firstly consider cases in which \(k\) is a positive even integer.
   (A) State the shape of the curve when \(k = 2\).
   (B) Sketch, on the same axes, the curves for \(k = 2\) and \(k = 4\).
   (C) Describe the shape that the curve tends to as \(k\) becomes very large.
   (D) State the range of possible values of \(x\) and \(y\).
2. Now consider cases in which \(k\) is a positive odd integer.
   (A) Explain why \(x\) and \(y\) may take any value.
   (B) State the shape of the curve when \(k = 1\).
   (C) Sketch the curve for \(k = 3\). State the equation of the asymptote of this curve.
   (D) Sketch the shape that the curve tends to as \(k\) becomes very large.
3. Now let \(k = \frac { 1 } { 2 }\). Sketch the curve, indicating the range of possible values of \(x\) and \(y\).
4. Now consider the modified equation \(| x | ^ { k } + | y | ^ { k } = 1\).
   (A) Sketch the curve for \(k = \frac { 1 } { 2 }\).
   (B) Investigate the shape of the curve for \(k = \frac { 1 } { n }\) as the positive integer \(n\) becomes very large.