| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Topic | Generalised Binomial Theorem |
| Type | Product with linear term |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial expansion for fractional powers with standard follow-up parts: (i) routine calculation of binomial coefficients, (ii) standard validity condition |2x| < 1, (iii) simple multiplication of polynomials. All parts are textbook exercises requiring only methodical application of formulas with no problem-solving insight needed. Slightly easier than average due to the guided structure and routine nature of each step. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| - \( | x | < 0.5\) or equiv.: B1 [1] |
**Part (i)**
- State $1 - (0.5)(2x)$: B1
- State $(0.5)(0.5)(-0.5)(2x)^2$: B1
- Attempt $\dfrac{\left(\frac{1}{2}\right)\left(\frac{-1}{2}\right)\left(\frac{-3}{2}\right)}{3!}(\pm 2x)^3$: M1
- Obtain $-0.5x^3$: A1 **[4]**
**Part (ii)**
- $|x| < 0.5$ or equiv.: B1 **[1]**
**Part (iii)**
- Obtain $2 - x$ correctly by partial expansion of their bracket: B1
- State $a = -2$ correctly by partial expansion of their bracket: B1
- Attempt to multiply $(2 + x)$ and their expansion. Must show at least 7 terms: M1
- State $b = -1.5$: A1 **[4]**
**[Total: 9]**7 (i) Show that the first three terms in the expansion of $( 1 - 2 x ) ^ { \frac { 1 } { 2 } }$ are $1 - x - \frac { 1 } { 2 } x ^ { 2 }$ and find the next term.\\
(ii) State the range of values of $x$ for which this expansion is valid.\\
(iii) Hence show that the first four terms in the expansion of $( 2 + x ) ( 1 - 2 x ) ^ { \frac { 1 } { 2 } }$ are $2 - x + a x ^ { 2 } + b x ^ { 3 }$ and state the values of $a$ and $b$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q7 [9]}}