Moderate -0.3 This is a straightforward application of the binomial theorem requiring factoring out the constant (4^(3/2) = 8), expanding (1 + x/4)^(3/2), and stating the validity condition |x/4| < 1. It's slightly easier than average because it's a direct textbook exercise with clear steps and no problem-solving required, though the fractional power and validity condition add minor complexity beyond basic binomial expansion.
2 Find the first three terms in the binomial expansion of \(( 4 + x ) ^ { \frac { 3 } { 2 } }\). State the set of values of \(x\) for which the expansion is valid.
Dealing with the '4' to obtain \(4^{\frac{3}{2}}(1+\frac{x}{4})^{\frac{3}{2}}\); or expanding as \(4^{\frac{3}{2}} + \frac{3}{2}4^{\frac{1}{2}}x + \binom{3}{2}\binom{1}{2}4^{-\frac{1}{2}}\frac{x^2}{2!}+\ldots\) and having all powers of 4 correct
Correct binomial coefficients for \(n = \frac{3}{2}\), ie \(1, \frac{3}{2}, \frac{3}{2}\cdot\frac{1}{2}\cdot\frac{1}{2!}\). Not nCr form. Independent of coefficient of \(x\). Independent of first M1
# Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(4+x)^{\frac{3}{2}} = 4^{\frac{3}{2}}(1+\frac{1}{4}x)^{\frac{3}{2}}$ | M1 | Dealing with the '4' to obtain $4^{\frac{3}{2}}(1+\frac{x}{4})^{\frac{3}{2}}$; or expanding as $4^{\frac{3}{2}} + \frac{3}{2}4^{\frac{1}{2}}x + \binom{3}{2}\binom{1}{2}4^{-\frac{1}{2}}\frac{x^2}{2!}+\ldots$ and having all powers of 4 correct |
| $= 8(1 + \frac{3}{2}\cdot\frac{1}{4}x) + \frac{3}{2}\cdot\frac{1}{2}\cdot\frac{1}{2!}\cdot(\frac{1}{4}x)^2 + \ldots$ | M1 | Correct binomial coefficients for $n = \frac{3}{2}$, ie $1, \frac{3}{2}, \frac{3}{2}\cdot\frac{1}{2}\cdot\frac{1}{2!}$. Not nCr form. Independent of coefficient of $x$. Independent of first M1 |
| $= 8 + 3x$ | A1 | $8 + 3x$ www |
| $+\frac{3}{16}x^2$ | A1 | $\ldots + \frac{3}{16}x^2$ www. Ignore subsequent terms |
| Valid for $-4 < x < 4$ or $\|x\| < 4$ | B1 | Accept $\leq$ or combination of $<$ and $\leq$, but not $-4 > x > 4$, $\|x\| > 4$, or say $-4 < x$. Condone $-4 < \|x\| < 4$. Independent of all other marks |
**[5 marks total]**
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2 Find the first three terms in the binomial expansion of $( 4 + x ) ^ { \frac { 3 } { 2 } }$. State the set of values of $x$ for which the expansion is valid.
\hfill \mbox{\textit{OCR MEI C4 Q2 [5]}}