Moderate -0.8 This is a straightforward application of the binomial expansion formula for fractional powers with n=1/2, requiring only substitution and simplification to find three terms plus stating the standard validity condition |3x|<1. It's slightly easier than average as it's a direct textbook exercise with no problem-solving element, though the fractional power adds minor complexity over integer binomial expansions.
Find the first three terms in the binomial expansion of \(\sqrt{1 + 3x}\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid. [5]
\(\Rightarrow -\frac{1}{3} \leq x \leq \frac{1}{3}\)
Answer
Marks
Guidance
M1
correct binomial coefficients
A1
\(1 + x - ..\)
A1
M1
or \(\
3x\
A1
ie 1, 1/3, (1/3)(-2/3)/2 not nCr form simplified www in this part simplified www in this part, ignore subsequent terms using (3x)³ as 3x² can score M1B1B0 condone omission of brackets if 3x² is used as 9x² do not allow MR for power 3 or -1/3 or similar condone inequality signs throughout or say < at one end and ≤ at the othercondone -1/3 ≤ \
x\
≤ 1/3, x<1/3 is M0A0 the last two marks are not dependent on the first three
Total: [5]
**Answer:** $(1+3x)^{-1} = 1 + \frac{1}{3}(3x) + \frac{1}{2!}\cdot\frac{2}{3^2}(3x)^2 + ...$
$= 1 + x - x^2 + ...$
Valid for $-1 \leq 3x \leq 1$
$\Rightarrow -\frac{1}{3} \leq x \leq \frac{1}{3}$
| M1 | correct binomial coefficients |
| A1 | $1 + x - ..$ |
| A1 | |
| M1 | or $\|3x\| \leq 1$ oe or $\|x\| \leq 1/3$ (correct final answer scores M1A1) |
| A1 | |
| | ie 1, 1/3, (1/3)(-2/3)/2 not nCr form simplified www in this part simplified www in this part, ignore subsequent terms using (3x)³ as 3x² can score M1B1B0 condone omission of brackets if 3x² is used as 9x² do not allow MR for power 3 or -1/3 or similar **condone inequality signs throughout or say < at one end and ≤ at the other** **condone -1/3 ≤ \|x\| ≤ 1/3, x<1/3 is M0A0** the last two marks are not dependent on the first three |
**Total: [5]**
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Find the first three terms in the binomial expansion of $\sqrt{1 + 3x}$ in ascending powers of $x$. State the set of values of $x$ for which the expansion is valid. [5]
\hfill \mbox{\textit{OCR MEI C4 2011 Q2 [5]}}