Moderate -0.5 This is a straightforward application of the binomial expansion for fractional powers with n=1/3. It requires routine substitution into the formula (1+x)^n = 1 + nx + n(n-1)x²/2! + ..., then stating the standard validity condition |3x| < 1. The calculation is mechanical with no problem-solving required, making it slightly easier than average.
5 Find the first three terms in the binomial expansion of \(\sqrt [ 3 ] { 1 + 3 x }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
Correct binomial coefficients, ie \(1, \frac{1}{3}, \frac{(1/3)(-2/3)}{2}\), not nCr form
\(= 1 + x \ldots\)
A1
\(1+x\ldots\) simplified www in this part
\(\ldots - x^2 + \ldots\)
A1
\(\ldots - x^2\) simplified www in this part, ignore subsequent terms. Using \((3x)^2\) as \(3x^2\) can score M1B1B0. Condone omission of brackets if \(3x^2\) is used as \(9x^2\). Do not allow MR for power 3 or \(-\frac{1}{3}\) or similar
Valid for \(-1 \leq 3x \leq 1 \Rightarrow -\frac{1}{3} \leq x \leq \frac{1}{3}\)
M1
or \(\
A1
or \(\
x\
[5 marks total]
# Question 5:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1+3x)^{\frac{1}{3}} = 1 + \frac{1}{3}(3x) + \frac{\frac{1}{3}\cdot(-\frac{2}{3})}{2!}(3x)^2 + \ldots$ | M1 | Correct binomial coefficients, ie $1, \frac{1}{3}, \frac{(1/3)(-2/3)}{2}$, not nCr form |
| $= 1 + x \ldots$ | A1 | $1+x\ldots$ simplified www in this part |
| $\ldots - x^2 + \ldots$ | A1 | $\ldots - x^2$ simplified www in this part, ignore subsequent terms. Using $(3x)^2$ as $3x^2$ can score M1B1B0. Condone omission of brackets if $3x^2$ is used as $9x^2$. Do not allow MR for power 3 or $-\frac{1}{3}$ or similar |
| Valid for $-1 \leq 3x \leq 1 \Rightarrow -\frac{1}{3} \leq x \leq \frac{1}{3}$ | M1 | or $\|3x\| \leq 1$ oe. Condone inequality signs throughout or say $<$ at one end and $\leq$ at the other |
| | A1 | or $\|x\| \leq \frac{1}{3}$. Correct final answer scores M1A1. Condone $-\frac{1}{3} \leq \|x\| \leq \frac{1}{3}$; $x \leq \frac{1}{3}$ is M0A0. The last two marks are not dependent on the first three |
**[5 marks total]**
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5 Find the first three terms in the binomial expansion of $\sqrt [ 3 ] { 1 + 3 x }$ in ascending powers of $x$. State the set of values of $x$ for which the expansion is valid.
\hfill \mbox{\textit{OCR MEI C4 Q5 [5]}}