| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Two unknowns from two coefficient conditions |
| Difficulty | Standard +0.3 This is a standard binomial expansion question requiring routine application of the formula followed by solving simultaneous equations. Part (i) is straightforward recall, and part (ii) involves algebraic manipulation of two equations in two unknowns—slightly above average due to the multi-step nature but still a textbook exercise with no novel insight required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(1 - 4ax + \ldots\) | B1 | |
| \(\frac{-4,-5}{1.2}(ax)^2\) or \(\frac{-4,-5}{1.2}a^2x^2\) or \(\frac{-4,-5}{1.2}ax^2\) | M1 | Do not accept \(\begin{pmatrix}-4\\2\end{pmatrix}\) unless 10 also appears |
| \(\ldots + 10a^2x^2\) | A1 | 3 |
| (ii) f.t. (their of \(x\)) + \(b\)(their const cf) \(= 1\) | \(\sqrt{B1}\) | Expect \(b - 4a = 1\) |
| f.t. (their of \(x^2\)) + \(b\)(their of \(x\)) \(= -2\) | \(\sqrt{B1}\) | Expect \(10a^2 - 4ab = -2\) Or eliminate '\(a\)' and produce equation in '\(b\)' Or \(6b^2 + 4b = 42\) |
| Attempt to eliminate '\(b\)' and produce equation in '\(a\)' | M1 | |
| Produce \(6a^2 + 4a = 2\) | AEF | |
| \(a = \frac{1}{3}\) and \(b = \frac{7}{3}\) only | A1 | 5 |
(i) $1 - 4ax + \ldots$ | B1 | |
$\frac{-4,-5}{1.2}(ax)^2$ or $\frac{-4,-5}{1.2}a^2x^2$ or $\frac{-4,-5}{1.2}ax^2$ | M1 | Do not accept $\begin{pmatrix}-4\\2\end{pmatrix}$ unless 10 also appears |
$\ldots + 10a^2x^2$ | A1 | 3 |
(ii) f.t. (their of $x$) + $b$(their const cf) $= 1$ | $\sqrt{B1}$ | Expect $b - 4a = 1$ |
f.t. (their of $x^2$) + $b$(their of $x$) $= -2$ | $\sqrt{B1}$ | Expect $10a^2 - 4ab = -2$ Or eliminate '$a$' and produce equation in '$b$' Or $6b^2 + 4b = 42$ |
Attempt to eliminate '$b$' and produce equation in '$a$' | M1 | |
Produce $6a^2 + 4a = 2$ | AEF | |
$a = \frac{1}{3}$ and $b = \frac{7}{3}$ only | A1 | 5 | Made clear to be only (final) answer |6 (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 $b$.\\
\hfill \mbox{\textit{OCR C4 2008 Q6 [8]}}