Moderate -0.8 This is a straightforward application of the binomial expansion formula for negative powers. Students need to factor out the constant (2^{-2}), apply the standard expansion to (1 + 3x/2)^{-2}, and simplify coefficients—a routine procedure requiring only formula recall and careful arithmetic, making it easier than average.
Obtain correct unsimplified version of the \(x\) or \(x^2\) term in the expansion of \((2 + 3x)^{-2}\) or \((1 + \frac{3}{4}x)^{-2}\)
M1
State correct first term \(\frac{1}{4}\)
B1
Obtain the next two terms \(-\frac{3}{4}x + \frac{27}{16}x^2\)
A1 + A1
Guidance notes:
- [The M mark is not earned by versions with symbolic binomial coefficients such as \(\binom{-2}{1}\).]
- [The M mark is earned if division of 1 by the expansion of \((2 + 3x)^2\), with a correct unsimplified \(x\) or \(x^2\) term, reaches a partial quotient of \(a + bx\).]
- [Accept exact decimal equivalents of fractions.]
- [SR: Answer given as \(\frac{1}{4}(1 - 3x + \frac{27}{4}x^2)\) can earn B1M1A1 (if \(\frac{1}{4}\) seen but then omitted, give M1A1).]
- [SR: Solutions involving \(k(1 + \frac{3}{4}x)^{-2}\), where \(k = 2, 4\) or \(\frac{1}{2}\), can earn M1 and A1 for correctly simplifying both the terms in \(x\) and \(x^2\).]
OR:
Answer
Marks
Guidance
Differentiate expression and evaluate \(f(0)\) and \(f'(0)\), where \(f'(x) = k(2 + 3x)^{-3}\)
M1
State correct first term \(\frac{1}{4}\)
B1
Obtain the next two terms \(-\frac{3}{4}x + \frac{27}{16}x^2\)
A1 + A1
Total: 4 marks
Obtain correct unsimplified version of the $x$ or $x^2$ term in the expansion of $(2 + 3x)^{-2}$ or $(1 + \frac{3}{4}x)^{-2}$ | M1 |
State correct first term $\frac{1}{4}$ | B1 |
Obtain the next two terms $-\frac{3}{4}x + \frac{27}{16}x^2$ | A1 + A1 |
**Guidance notes:**
- [The M mark is not earned by versions with symbolic binomial coefficients such as $\binom{-2}{1}$.]
- [The M mark is earned if division of 1 by the expansion of $(2 + 3x)^2$, with a correct unsimplified $x$ or $x^2$ term, reaches a partial quotient of $a + bx$.]
- [Accept exact decimal equivalents of fractions.]
- [SR: Answer given as $\frac{1}{4}(1 - 3x + \frac{27}{4}x^2)$ can earn B1M1A1 (if $\frac{1}{4}$ seen but then omitted, give M1A1).]
- [SR: Solutions involving $k(1 + \frac{3}{4}x)^{-2}$, where $k = 2, 4$ 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 $f'(x) = k(2 + 3x)^{-3}$ | M1 |
State correct first term $\frac{1}{4}$ | B1 |
Obtain the next two terms $-\frac{3}{4}x + \frac{27}{16}x^2$ | A1 + A1 | **Total: 4 marks**
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1 Expand $( 2 + 3 x ) ^ { - 2 }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
\hfill \mbox{\textit{CAIE P3 2007 Q1 [4]}}