Moderate -0.8 This is a straightforward C2 binomial expansion question requiring routine application of the formula for the first 4 terms, followed by a standard substitution (x=0.1) for numerical approximation. The mechanics are direct with no problem-solving insight needed, making it easier than average but not trivial due to the arithmetic involved.
3. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of
$$\left( 1 + \frac { x } { 4 } \right) ^ { 8 }$$
giving each term in its simplest form.
(b) Use your expansion to estimate the value of \(( 1.025 ) ^ { 8 }\), giving your answer to 4 decimal places.
M1: correct binomial coefficient combined with correct power of \(x\). A1: two completely correct unsimplified terms
\(= \ldots + \frac{7}{4}x^2 + \frac{7}{8}x^3\) or \(+1.75x^2 + 0.875x^3\)
A1 (4)
A1: needs fully simplified \(\frac{7}{4}x^2\) and \(\frac{7}{8}x^3\)
Part (b)
Answer
Marks
Guidance
Working
Marks
Guidance
States or implies \(x = 0.1\)
B1
B1: states or uses \(x=0.1\) or \(\frac{x}{4} = \frac{1}{40}\)
Substitutes their value of \(x\) (provided it is \(<1\)) into series from (a)
M1
M1: substituting value of \(x\) where \(0 < x < 1\) into expansion (e.g. 0.1 correct, or 0.01, 0.00625, or even 0.025, but not 1 nor 1.025 — earns M0)
\(1 + 0.2 + 0.0175 + 0.000875 = 1.2184\)
A1 cao (3)
A1: printed cao (not answers which round to); answer with no working at all is B0, M0, A0
# Question 3:
## Part (a)
| Working | Marks | Guidance |
|---------|-------|----------|
| $(1+\frac{x}{4})^8 = 1 + 2x + \ldots$ | B1 | B1 must be simplified |
| $+ \frac{8\times7}{2}\left(\frac{x}{4}\right)^2 + \frac{8\times7\times6}{2\times3}\left(\frac{x}{4}\right)^3$ | M1, A1 | M1: correct binomial coefficient combined with correct power of $x$. A1: two completely correct unsimplified terms |
| $= \ldots + \frac{7}{4}x^2 + \frac{7}{8}x^3$ or $+1.75x^2 + 0.875x^3$ | A1 (4) | A1: needs fully simplified $\frac{7}{4}x^2$ and $\frac{7}{8}x^3$ |
## Part (b)
| Working | Marks | Guidance |
|---------|-------|----------|
| States or implies $x = 0.1$ | B1 | B1: states or uses $x=0.1$ or $\frac{x}{4} = \frac{1}{40}$ |
| Substitutes their value of $x$ (provided it is $<1$) into series from (a) | M1 | M1: substituting value of $x$ where $0 < x < 1$ into expansion (e.g. 0.1 correct, or 0.01, 0.00625, or even 0.025, but **not** 1 nor 1.025 — earns **M0**) |
| $1 + 0.2 + 0.0175 + 0.000875 = 1.2184$ | A1 cao (3) | A1: printed cao (not answers which round to); answer with no working at all is B0, M0, A0 |
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3. (a) Find the first 4 terms of the binomial expansion, in ascending powers of $x$, of
$$\left( 1 + \frac { x } { 4 } \right) ^ { 8 }$$
giving each term in its simplest form.\\
(b) Use your expansion to estimate the value of $( 1.025 ) ^ { 8 }$, giving your answer to 4 decimal places.\\
\hfill \mbox{\textit{Edexcel C2 2012 Q3 [7]}}