Moderate -0.3 This is a straightforward application of the binomial expansion requiring students to factor out the constant (rewrite as 2^(-1)(1+3x/4)^(-1/2)) and apply the standard formula. It's slightly easier than average because it only asks for three terms with no further manipulation or application required.
1 Expand \(\frac { 1 } { \sqrt { } ( 4 + 3 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
EITHER: Obtain a correct unsimplified version of the \(x\) or \(x^2\) term of the expansion of \((4+3x)^{-1}\) or \((1+\frac{1}{3}x)^{-1}\)
M1
State correct first term \(\frac{1}{2}\)
B1
Obtain the next two terms \(-\frac{3}{16}x + \frac{27}{256}x^2\)
A1 + A1
OR: Differentiate and evaluate \(f(0)\) and \(f'(0)\), where \(f'(x) = k(4+3x)^{-2}\)
M1
State correct first term \(\frac{1}{2}\)
B1
Obtain the next two terms \(-\frac{3}{16}x + \frac{27}{256}x^2\)
A1 + A1
[4]
[Symbolic coefficients, e.g. \(\left(-\frac{1}{2}\right)\) are not sufficient for the M or B mark.]
EITHER: Obtain a correct unsimplified version of the $x$ or $x^2$ term of the expansion of $(4+3x)^{-1}$ or $(1+\frac{1}{3}x)^{-1}$ | M1 |
State correct first term $\frac{1}{2}$ | B1 |
Obtain the next two terms $-\frac{3}{16}x + \frac{27}{256}x^2$ | A1 + A1 |
OR: Differentiate and evaluate $f(0)$ and $f'(0)$, where $f'(x) = k(4+3x)^{-2}$ | M1 |
State correct first term $\frac{1}{2}$ | B1 |
Obtain the next two terms $-\frac{3}{16}x + \frac{27}{256}x^2$ | A1 + A1 | [4]
[Symbolic coefficients, e.g. $\left(-\frac{1}{2}\right)$ are not sufficient for the M or B mark.]
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1 Expand $\frac { 1 } { \sqrt { } ( 4 + 3 x ) }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2012 Q1 [4]}}