OCR
Further Additional Pure
2022
June
— Question 7
Exam Board
OCR
Module
Further Additional Pure (Further Additional Pure)
Year
2022
Session
June
Topic
Reduction Formulae
7
Differentiate \(\left( 16 + t ^ { 2 } \right) ^ { \frac { 3 } { 2 } }\) with respect to \(t\).
Let \(I _ { n } = \int _ { 0 } ^ { 3 } t ^ { n } \sqrt { 16 + t ^ { 2 } } d t\) for integers \(n \geqslant 1\).
Show that, for \(n \geqslant 3 , \left. ( n + 2 ) \right| _ { n } = 125 \times 3 ^ { n - 1 } - \left. 16 ( n - 1 ) \right| _ { n - 2 }\).
The curve \(C\) is defined parametrically by \(\mathrm { x } = \mathrm { t } ^ { 4 } \cos \mathrm { t }\), \(\mathrm { y } = \mathrm { t } ^ { 4 } \sin \mathrm { t }\), for \(0 \leqslant t \leqslant 3\). The length of \(C\) is denoted by \(L\).
Show that \(\mathrm { L } = \mathrm { I } _ { 3 }\). (You are not required to evaluate this integral.)