| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2022 |
| Session | January |
| Marks | 8 |
| Topic | Generalised Binomial Theorem |
| Type | Two unknowns from two coefficient conditions |
| Difficulty | Standard +0.3 This is a straightforward application of the binomial theorem with two simultaneous equations. Part (i) is routine expansion using the formula. Part (ii) requires multiplying the expansion by (1+bx), equating coefficients to form two linear equations, and solving for a and b. The algebra is simple and the method is standard textbook material, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
5.\\
(i) Expand $( 1 + a x ) ^ { - 4 }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.\\
(ii) The coefficients of $x$ and $x ^ { 2 }$ in the expansion of $( 1 + b x ) ( 1 + a x ) ^ { - 4 }$ are 1 and - 2 respectively. Given that $a > 0$, find the values of $a$ and $b$.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM 2022 Q5 [8]}}