| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Form (1+bx)^n expansion |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial expansion formula for (1+bx)^n with n=1/2 and b=-8, requiring routine differentiation of the general term and substitution. Part (b) is simple arithmetic verification, and part (c) involves basic calculator work. The question is slightly easier than average because it follows a standard template with clear scaffolding and no problem-solving insight required. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \((1-8x)^{-1} = 1 + \left(\frac{1}{2}\right)(-8x) + \frac{\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)}{2}(-8x)^2 + \frac{\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{3!}(-8x)^3 + \ldots = 1 - 4x - 8x^2 - 32x^3 - \ldots\) | M1 A1; A1; A1 |
| (b) | \(\sqrt{(1-8x)} = \sqrt{1 - \frac{8}{100}} = \sqrt{\frac{92}{100}} = \sqrt{\frac{23}{25}} = \frac{\sqrt{23}}{5}\) | M1; cso A1 |
| (c) | \(1 - 4x - 8x^2 - 32x^3 = 1 - 4(0.01) - 8(0.01)^2 - 32(0.01)^3 = 1 - 0.04 - 0.0008 - 0.000032 = 0.959168\) then \(\sqrt{23} = 5 \times 0.959168 = 4.79584\) | M1; M1; cao A1 |
(a) | $(1-8x)^{-1} = 1 + \left(\frac{1}{2}\right)(-8x) + \frac{\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)}{2}(-8x)^2 + \frac{\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{3!}(-8x)^3 + \ldots = 1 - 4x - 8x^2 - 32x^3 - \ldots$ | M1 A1; A1; A1 | (4 marks)
(b) | $\sqrt{(1-8x)} = \sqrt{1 - \frac{8}{100}} = \sqrt{\frac{92}{100}} = \sqrt{\frac{23}{25}} = \frac{\sqrt{23}}{5}$ | M1; cso A1 | (2 marks)
(c) | $1 - 4x - 8x^2 - 32x^3 = 1 - 4(0.01) - 8(0.01)^2 - 32(0.01)^3 = 1 - 0.04 - 0.0008 - 0.000032 = 0.959168$ then $\sqrt{23} = 5 \times 0.959168 = 4.79584$ | M1; M1; cao A1 | (3 marks)
**Total: [9]**
---\begin{enumerate}
\item (a) Find the binomial expansion of
\end{enumerate}
$$\sqrt { } ( 1 - 8 x ) , \quad | x | < \frac { 1 } { 8 }$$
in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each term.\\
(b) Show that, when $x = \frac { 1 } { 100 }$, the exact value of $\sqrt { } ( 1 - 8 x )$ is $\frac { \sqrt { } 23 } { 5 }$.\\
(c) Substitute $x = \frac { 1 } { 100 }$ into the binomial expansion in part (a) and hence obtain an approximation to $\sqrt { } 23$. Give your answer to 5 decimal places.\\
\hfill \mbox{\textit{Edexcel C4 2010 Q1 [9]}}