| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Topic | Generalised Binomial Theorem |
| Type | Finding unknown constant from coefficient |
| Difficulty | Standard +0.3 This question involves standard application of the binomial expansion for fractional powers (a core A-level topic) followed by algebraic manipulation to find an unknown constant. Part (i) is routine recall of the generalized binomial theorem with straightforward coefficient simplification. Part (ii) requires multiplying the expansion by (2+kx) and equating coefficients, which is a standard technique. The question is slightly easier than average as it's methodical with no conceptual surprises, though it does require careful algebraic manipulation across multiple terms. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<1 |
**Question 6(i)**
Attempt $1 + \dfrac{1}{2}x + \dfrac{\left(\frac{1}{2}\right)\left(\frac{-1}{2}\right)}{2}x^2 + \dfrac{\left(\frac{1}{2}\right)\left(\frac{-1}{2}\right)\left(\frac{-3}{2}\right)}{6}x^3$
(allow omission of brackets at this stage) but must reach the $x^3$ term **M1**
Obtain $1 + \dfrac{1}{2}x$ **A1**
Obtain $-\dfrac{1}{8}x^2$ **A1**
Obtain $\dfrac{1}{16}x^3$ **A1**
**Question 6(ii)**
Attempt sum of two relevant terms. Must see the sum of two terms only each giving an $x^3$ result **M1**
Obtain $\dfrac{1}{8} - \dfrac{k}{8} = 1$ **A1**
Obtain $k = -7$ **A1**
**Total: 7 marks**6 (i) Expand $( 1 + x ) ^ { \frac { 1 } { 2 } }$, for $| x | < 1$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying the coefficients.\\
(ii) In the expansion of $( 2 + k x ) ( 1 + x ) ^ { \frac { 1 } { 2 } }$ the coefficient of $x ^ { 3 }$ is 1 . Find the value of $k$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2017 Q6 [7]}}