Moderate -0.8 This is a straightforward binomial expansion problem requiring students to equate coefficients of the first three terms. It involves routine application of the binomial theorem formula and solving simple simultaneous equations. The algebra is uncomplicated and the method is standard textbook practice, making it easier than average.
4 The first three terms in the expansion of \(( 2 + a x ) ^ { n }\), in ascending powers of \(x\), are \(32 - 40 x + b x ^ { 2 }\). Find the values of the constants \(n , a\) and \(b\).
\(3^{\text{rd}}\) term \(= n(n-1) \frac{1}{2} .2^{n-2}(ax)^2\)
M1
Allow for one power of 2 and ax
\(a = -\frac{1}{4}\)
A1
co
\(b = 20\)
A1
co
Total: [5]
$(2+ax)^n$ | |
$1^{\text{st}}$ term $= 2^n = 32 \to n = 5$ | B1 | co
$2^{\text{nd}}$ term $= n.2^{n-1}(ax) = -40x$ | M1 | Allow for both binomial coefficients
$3^{\text{rd}}$ term $= n(n-1) \frac{1}{2} .2^{n-2}(ax)^2$ | M1 | Allow for one power of 2 and ax
$a = -\frac{1}{4}$ | A1 | co
$b = 20$ | A1 | co
**Total: [5]**
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4 The first three terms in the expansion of $( 2 + a x ) ^ { n }$, in ascending powers of $x$, are $32 - 40 x + b x ^ { 2 }$. Find the values of the constants $n , a$ and $b$.
\hfill \mbox{\textit{CAIE P1 2006 Q4 [5]}}