| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2015 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Composite substitution expansion |
| Difficulty | Challenging +1.8 This AEA question requires multiple sophisticated techniques: generalised binomial expansion with fractional powers, substitution to connect parts, convergence analysis involving completing the square and applying radius of convergence conditions, and integration of a power series. While each individual step is methodical, the multi-part structure, the need to manipulate the quadratic expression strategically, and the convergence proof requiring inequality manipulation make this substantially harder than standard A-level fare but not at the extreme end of AEA difficulty. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions1.08d Evaluate definite integrals: between limits |
(a) Find the binomial series expansion for $(4 + y)^{\frac{1}{2}}$ in ascending powers of $y$ up to and including the term in $y^3$. Simplify the coefficient of each term.
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(b) Hence show that the binomial series expansion for $(4 + 5x + x^2)^{\frac{1}{2}}$ in ascending powers of $x$ up to and including the term in $x^3$ is
$$2 + \frac{5x}{4} - \frac{9x^2}{64} + \frac{45x^3}{512}$$
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(c) Show that the binomial series expansion of $(4 + 5x + x^2)^{\frac{1}{2}}$ will converge for $-\frac{1}{2} < x \leq \frac{1}{2}$
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(d) Use the result in part (b) to estimate
$$\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{4 + 5x + x^2} \, dx$$
Give your answer as a single fraction.
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\hfill \mbox{\textit{Edexcel AEA 2015 Q4 [15]}}