Moderate -0.8 This is a straightforward application of the binomial expansion requiring students to rewrite (2+x)^{-3} as 2^{-3}(1+x/2)^{-3} and apply the standard formula. It involves routine algebraic manipulation with negative indices but no problem-solving or conceptual insight beyond direct formula application.
1 Expand \(\frac { 1 } { ( 2 + x ) ^ { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
EITHER: Obtain correct unsimplified version of the \(x\) or \(x^2\) term in the expansion of \((2+x)^3\) or \(\left(1+\frac{1}{2}x\right)^{-3}\)
M1
State correct first term \(\frac{1}{8}\)
B1
Obtain next two terms \(-\frac{3}{16}x + \frac{3}{16}x^2\)
A1 + A1
Accept exact decimal equivalents of fractions
[The M mark is not earned by versions with unexpanded binomial coefficients such as \(\binom{-3}{1}\)]
[SR: Answers given as \(\frac{1}{8}\left(1-\frac{3}{2}x+\frac{3}{2}x^2\right)\) can earn M1B1A1.]
[SR: Solutions involving \(k\left(1+\frac{1}{2}x\right)^3\), where \(k = 2, 8\) or \(\frac{1}{2}\), can earn M1 and A1√ for correctly simplifying both the terms in \(x\) and \(x^2\).]
OR: Differentiate expression and evaluate \(f(0)\) and \(f'(0)\), where \(f'(x) = k(2+x)^a\)
M1
State correct first term \(\frac{1}{8}\)
B1
Obtain next two terms \(-\frac{3}{16}x + \frac{3}{16}x^2\)
A1 + A1
Accept exact decimal equivalents of fractions
Total: 4 marks
**EITHER:** Obtain correct unsimplified version of the $x$ or $x^2$ term in the expansion of $(2+x)^3$ or $\left(1+\frac{1}{2}x\right)^{-3}$ | M1 | |
State correct first term $\frac{1}{8}$ | B1 | |
Obtain next two terms $-\frac{3}{16}x + \frac{3}{16}x^2$ | A1 + A1 | Accept exact decimal equivalents of fractions |
| | | [The M mark is not earned by versions with unexpanded binomial coefficients such as $\binom{-3}{1}$] |
| | | [SR: Answers given as $\frac{1}{8}\left(1-\frac{3}{2}x+\frac{3}{2}x^2\right)$ can earn M1B1A1.] |
| | | [SR: Solutions involving $k\left(1+\frac{1}{2}x\right)^3$, where $k = 2, 8$ or $\frac{1}{2}$, can earn M1 and A1√ for correctly simplifying both the terms in $x$ and $x^2$.] |
**OR:** Differentiate expression and evaluate $f(0)$ and $f'(0)$, where $f'(x) = k(2+x)^a$ | M1 | |
State correct first term $\frac{1}{8}$ | B1 | |
Obtain next two terms $-\frac{3}{16}x + \frac{3}{16}x^2$ | A1 + A1 | Accept exact decimal equivalents of fractions |
| | | **Total: 4 marks** |
1 Expand $\frac { 1 } { ( 2 + x ) ^ { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2004 Q1 [4]}}