Moderate -0.8 This is a straightforward application of the binomial expansion formula for negative indices with a=2, n=-2. It requires routine substitution into the formula and simplification of coefficients, but involves no problem-solving or conceptual challenges beyond direct recall of the technique.
2 Expand \(\left( 2 + x ^ { 2 } \right) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\), simplifying the coefficients.
Obtain correct unsimplified version of the \(x^2\) or \(x^4\) term of the expansion of \((1 + \frac{1}{2}x^2)^{-2}\) or \((2 + x^2)^{-2}\)
M1
M mark not earned by versions with unexpanded binomial coefficients such as \(\binom{-2}{1}\)
State correct first term \(\frac{1}{4}\)
B1
Obtain next two terms \(-\frac{1}{4}x^2 + \frac{3}{16}x^4\)
A1+A1
OR: Differentiate expression and evaluate \(f(0)\) and \(f'(0)\), where \(f'(x) = kx(2+x^2)^{-3}\)
M1
State correct first term \(\frac{1}{4}\)
B1
Obtain next two terms \(-\frac{1}{4}x^2 + \frac{3}{16}x^4\)
A1+A1
Notes: SR: Answers given as \(\frac{1}{4}(1 - x^2 + \frac{3}{4}x^4)\) earn M1B1A1. SR: Solutions involving \(k(1+\frac{1}{2}x^2)^{-2}\), where \(k = 2, 4\) or \(\frac{1}{2}\) can earn M1 and A1 for a correct simplified term in \(x^2\) or \(x^4\). Allow exact decimal equivalents as coefficients.
Total: [4]
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain correct unsimplified version of the $x^2$ or $x^4$ term of the expansion of $(1 + \frac{1}{2}x^2)^{-2}$ or $(2 + x^2)^{-2}$ | M1 | M mark not earned by versions with unexpanded binomial coefficients such as $\binom{-2}{1}$ |
| State correct first term $\frac{1}{4}$ | B1 | |
| Obtain next two terms $-\frac{1}{4}x^2 + \frac{3}{16}x^4$ | A1+A1 | |
| **OR:** Differentiate expression and evaluate $f(0)$ and $f'(0)$, where $f'(x) = kx(2+x^2)^{-3}$ | M1 | |
| State correct first term $\frac{1}{4}$ | B1 | |
| Obtain next two terms $-\frac{1}{4}x^2 + \frac{3}{16}x^4$ | A1+A1 | |
**Notes:** SR: Answers given as $\frac{1}{4}(1 - x^2 + \frac{3}{4}x^4)$ earn M1B1A1. SR: Solutions involving $k(1+\frac{1}{2}x^2)^{-2}$, where $k = 2, 4$ or $\frac{1}{2}$ can earn M1 and A1 for a correct simplified term in $x^2$ or $x^4$. Allow exact decimal equivalents as coefficients.
**Total: [4]**
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2 Expand $\left( 2 + x ^ { 2 } \right) ^ { - 2 }$ in ascending powers of $x$, up to and including the term in $x ^ { 4 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2003 Q2 [4]}}