Moderate -0.8 This is a straightforward application of the binomial expansion formula for negative/fractional powers. Students need to recall the formula and perform routine algebraic simplification of coefficients, but no problem-solving or conceptual insight is required beyond direct substitution with n=-1/3 and b=-3.
2 Expand \(( 1 - 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
Show correct (unsimplified) version of the \(x\) or the \(x^2\) or the \(x^3\) term
M1
Obtain correct first two terms \(1 + x\)
A1
Obtain correct quadratic term \(2x^2\)
A1
Obtain correct cubic term \(\frac{4}{3}x^3\) (allow \(\frac{2}{3}, 4.67, 4.66\) for the coefficient)
A1
Guidance: The M mark may be implied by correct simplified terms, if no working is shown. It is not earned by unexpanded binomial coefficients involving \(-\frac{1}{2}\), e.g. \(^{-1}C_1\) or \(\begin{pmatrix}-\frac{1}{2}\end{pmatrix}\)
An attempt to divide 1 by the expansion of \((1-3x)^1\) earns M1 if the expansion has a correct (unsimplified) \(x, x^2\), or \(x^3\) term and if the partial quotient contains a term in \(x\). The remaining A marks are awarded as above.
Total: 4 marks
OR route:
Answer
Marks
Guidance
Differentiate and calculate \(f(0), f'(0)\), where \(f(x) = k(1-3x)^{-1}\)
M1
Obtain correct first two terms \(1 + x\)
A1
Obtain correct quadratic term \(2x^2\)
A1
Obtain correct cubic term \(\frac{4}{3}x^3\) (allow \(\frac{2}{3}, 4.67, 4.66\) for the coefficient)
A1
Total: 4 marks
**EITHER route:**
Show correct (unsimplified) version of the $x$ or the $x^2$ or the $x^3$ term | M1 |
Obtain correct first two terms $1 + x$ | A1 |
Obtain correct quadratic term $2x^2$ | A1 |
Obtain correct cubic term $\frac{4}{3}x^3$ (allow $\frac{2}{3}, 4.67, 4.66$ for the coefficient) | A1 |
**Guidance:** The M mark may be implied by correct simplified terms, if no working is shown. It is not earned by unexpanded binomial coefficients involving $-\frac{1}{2}$, e.g. $^{-1}C_1$ or $\begin{pmatrix}-\frac{1}{2}\end{pmatrix}$ | |
An attempt to divide 1 by the expansion of $(1-3x)^1$ earns M1 if the expansion has a correct (unsimplified) $x, x^2$, or $x^3$ term and if the partial quotient contains a term in $x$. The remaining A marks are awarded as above. | | **Total: 4 marks**
**OR route:**
Differentiate and calculate $f(0), f'(0)$, where $f(x) = k(1-3x)^{-1}$ | M1 |
Obtain correct first two terms $1 + x$ | A1 |
Obtain correct quadratic term $2x^2$ | A1 |
Obtain correct cubic term $\frac{4}{3}x^3$ (allow $\frac{2}{3}, 4.67, 4.66$ for the coefficient) | A1 | **Total: 4 marks**
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2 Expand $( 1 - 3 x ) ^ { - \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2002 Q2 [4]}}