Establish bounds or inequalities

7 It is given that, for non-negative integers \(n , I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } x \mathrm {~d} x\).

1. Show that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geqslant 2\).
2. Explain why \(I _ { 2 n + 1 } < I _ { 2 n - 1 }\).
3. It is given that \(I _ { 2 n + 1 } < I _ { 2 n } < I _ { 2 n - 1 }\). Take \(n = 5\) to find an interval within which the value of \(\pi\) lies.

7 Let \(I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x\). Show that, for all positive integers \(n\), $$I _ { n } = n I _ { n - 1 } - 1$$ Find the exact value of \(I _ { 4 }\). By considering the area of the region enclosed by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = ( 1 - x ) ^ { 4 } \mathrm { e } ^ { x }\) in the interval \(0 \leqslant x \leqslant 1\), show that $$\frac { 65 } { 24 } < \mathrm { e } < \frac { 11 } { 4 }$$

1. A complex number \(z\) is represented by the point \(P\) in the complex plane.

Given that \(z\) satisfies $$| z - 6 | = 2 | z + 3 i |$$

1. show that the locus of \(P\) passes through the origin and the points - 4 and - 8 i
2. Sketch on an Argand diagram the locus of \(P\) as \(z\) varies.
3. On your sketch, shade the region which satisfies both $$| z - 6 | \geqslant 2 | z + 3 i | \text { and } | z | \leqslant 4$$