| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Topic | Generalised Binomial Theorem |
| Type | Product with quadratic or higher term |
| Difficulty | Standard +0.3 This is a straightforward binomial expansion question with standard follow-through parts. Part (i) requires routine application of the binomial series to expand (4-3x)^(-1/2), part (ii) involves simple polynomial multiplication, and part (iii) is basic integration of a quadratic. All techniques are standard A-level fare with no novel insight required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt to find the third term of \(\left(1-\frac{3}{4}x\right)^{-\frac{1}{2}}\) | M1 | |
| \(\frac{1}{2}\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)\left(-\frac{3}{4}x\right)^2\) unsimplified | A1 | |
| \(1+\frac{3}{8}x\) | B1 | |
| Multiply their expansion by \(\frac{1}{2}\left(\frac{1}{2}+\frac{3x}{16}+\frac{27x^2}{256}\right)\) | B1 | |
| \(-\frac{4}{3} < x < \frac{4}{3}\) | B1 | [5] |
| (ii) Attempt to multiply the expansion found in (i) by \(1 + x^2\) or multiply relevant terms. | M1 | |
| \(\frac{1}{2} + \frac{3}{16}x + \frac{155}{256}x^2\) aef | A1 | [2] |
| (iii) Attempt to integrate their expansion and substitute at least 0.1 | M1 | |
| 0.0511 | A1 | [2] |
(i) Attempt to find the third term of $\left(1-\frac{3}{4}x\right)^{-\frac{1}{2}}$ | M1 |
$\frac{1}{2}\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)\left(-\frac{3}{4}x\right)^2$ unsimplified | A1 |
$1+\frac{3}{8}x$ | B1 |
Multiply their expansion by $\frac{1}{2}\left(\frac{1}{2}+\frac{3x}{16}+\frac{27x^2}{256}\right)$ | B1 |
$-\frac{4}{3} < x < \frac{4}{3}$ | B1 | [5]
(ii) Attempt to multiply the expansion found in (i) by $1 + x^2$ or multiply relevant terms. | M1 |
$\frac{1}{2} + \frac{3}{16}x + \frac{155}{256}x^2$ aef | A1 | [2]
(iii) Attempt to integrate their expansion and substitute at least 0.1 | M1 |
0.0511 | A1 | [2]Let $f(x) = \frac{1 + x^2}{\sqrt{4 - 3x}}$
\begin{enumerate}[label=(\roman*)]
\item Obtain in ascending powers of $x$ the first three terms in the expansion of $\frac{1}{\sqrt{4 - 3x}}$ and state the values of $x$ for which this expansion is valid. [5]
\item Hence obtain an approximation to $f(x)$ in the form $a + bx + cx^2$ where $a$, $b$ and $c$ are constants. [2]
\item Use your approximation to estimate $\int_0^{0.1} f(x) dx$. [2]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2010 Q7 [9]}}