10 The constant \(a\) is such that \(\int _ { 1 } ^ { a } x ^ { 2 } \ln x \mathrm {~d} x = 4\).
- Show that \(a = \left( \frac { 35 } { 3 \ln a - 1 } \right) ^ { \frac { 1 } { 3 } }\).
- Verify by calculation that \(a\) lies between 2.4 and 2.8.
- Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
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