Moderate -0.8 Part (a) is a straightforward binomial expansion requiring direct application of the formula with small positive integer n=6, calculating only the first 3 terms. Part (b) requires multiplying the result by a linear factor, which is a routine extension involving simple algebraic manipulation. This is a standard C2 textbook exercise with no problem-solving insight required, making it easier than average.
3. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$( 2 - 3 x ) ^ { 6 }$$
giving each term in its simplest form.
(b) Hence, or otherwise, find the first 3 terms, in ascending powers of \(x\), of the expansion of
$$\left( 1 + \frac { x } { 2 } \right) ( 2 - 3 x ) ^ { 6 }$$
\(\left({}^6C_1 \times \ldots \times x\right)\) or \(\left({}^6C_2 \times \ldots \times x^2\right)\) for either the \(x\) term or \(x^2\) term. Requires correct binomial coefficient in any form with correct power of \(x\)
\(= 64 - 576x + 2160x^2 + \ldots\)
A1 A1
1st A1: Either \(-576x\) or \(2160x^2\) (allow \(+-576x\)). 2nd A1: Both \(-576x\) and \(2160x^2\) (do not allow \(+-576x\))
Part (b)
Answer
Marks
Guidance
Answer
Mark
Guidance
\(\left(1 + \frac{x}{2}\right) \times\) (their part (a) answer, at least up to \(x\) term)
M1
Condone missing brackets
\(= 64 - 544x + 1872x^2 + \ldots\)
A1 A1
1st A1: At least 2 terms correct (allow \(+-544x\)). 2nd A1: \(64 - 544x + 1872x^2\). Terms can be listed rather than added
Note: SC: If candidate expands in descending powers of \(x\), only M marks available.
# Question 3:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(2-3x)^6 = 64 + \ldots$ | B1 | 64 seen as the only constant term |
| $\{(2-3x)^6\} = (2)^6 + {}^6C_1(2)^5(-3x) + {}^6C_2(2)^4(-3x)^2 + \ldots$ | M1 | $\left({}^6C_1 \times \ldots \times x\right)$ or $\left({}^6C_2 \times \ldots \times x^2\right)$ for either the $x$ term or $x^2$ term. Requires correct binomial coefficient in any form with correct power of $x$ |
| $= 64 - 576x + 2160x^2 + \ldots$ | A1 A1 | 1st A1: Either $-576x$ or $2160x^2$ (allow $+-576x$). 2nd A1: Both $-576x$ and $2160x^2$ (do not allow $+-576x$) |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\left(1 + \frac{x}{2}\right) \times$ (their part (a) answer, at least up to $x$ term) | M1 | Condone missing brackets |
| $= 64 - 544x + 1872x^2 + \ldots$ | A1 A1 | 1st A1: At least 2 terms correct (allow $+-544x$). 2nd A1: $64 - 544x + 1872x^2$. Terms can be listed rather than added |
**Note:** SC: If candidate expands in descending powers of $x$, only M marks available.
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3. (a) Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of
$$( 2 - 3 x ) ^ { 6 }$$
giving each term in its simplest form.\\
(b) Hence, or otherwise, find the first 3 terms, in ascending powers of $x$, of the expansion of
$$\left( 1 + \frac { x } { 2 } \right) ( 2 - 3 x ) ^ { 6 }$$
\hfill \mbox{\textit{Edexcel C2 2014 Q3 [7]}}