| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Non-zero terms only |
| Difficulty | Standard +0.3 This is a straightforward application of the binomial expansion requiring rewriting the expression in the form (1+u)^n, followed by routine integration of a polynomial and comparison with the exact integral. All steps are standard C4 techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.08d Evaluate definite integrals: between limits |
6 (i) Find the first three non-zero terms of the binomial expansion of $\frac { 1 } { \sqrt { 4 - x ^ { 2 } } }$ for $| x | < 2$.\\
(ii) Use this result to find an approximation for $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$, rounding your answer to\\
4 significant figures.\\
(iii) Given that $\int \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x = \arcsin \left( \frac { 1 } { 2 } x \right) + c$, evaluate $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$, rounding your answer to 4 significant figures.
\hfill \mbox{\textit{OCR MEI C4 2006 Q6 [7]}}