OCR C4 2016 June — Question 7 6 marks

Exam BoardOCR
ModuleC4 (Core Mathematics 4)
Year2016
SessionJune
Marks6
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TopicGeneralised Binomial Theorem
TypeFinding unknown power and constant
DifficultyStandard +0.3 This is a straightforward application of the binomial expansion formula requiring students to equate coefficients of x and x² to form two simultaneous equations, then solve for n and k. The validity condition |kx| < 1 is standard bookwork. While it requires careful algebraic manipulation, it follows a well-practiced procedure with no novel insight needed, making it slightly easier than average.
Spec1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions
7 Given that the binomial expansion of \(( 1 + k x ) ^ { n }\) is \(1 - 6 x + 30 x ^ { 2 } + \ldots\), find the values of \(n\) and \(k\). State the set of values of \(x\) for which this expansion is valid.
Question 7:
AnswerMarks Guidance
AnswerMarks Guidance
\(nk = -6\) soiB1 allow \(nkx = -6x\) and/or \(\frac{n(n-1)k^2}{2!}x^2 = 30x^2\) for first two marks
\(\frac{n(n-1)k^2}{2!} = 30\) soiB1 NB \(\frac{n(n-1)\times 36}{2\times n^2} = 30\) oe; \((-\frac{6}{k})(\frac{-6}{k}-1)k^2 = 60\) oe
substitution of \(n = \pm\frac{6}{k}\) or \(k = \pm\frac{6}{n}\) or \(k = \pm\sqrt{\frac{60}{n(n-1)}}\) to eliminate one variableM1 allow omission of brackets
\(n = -1.5\)A1 e.g. allow \(-\frac{6}{4}\)
\(k = 4\)A1
expansion valid for \(x < \frac{1}{4}\) or \(-\frac{1}{4} < x < \frac{1}{4}\) 
## Question 7:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $nk = -6$ soi | B1 | allow $nkx = -6x$ and/or $\frac{n(n-1)k^2}{2!}x^2 = 30x^2$ for first two marks |
| $\frac{n(n-1)k^2}{2!} = 30$ soi | B1 | NB $\frac{n(n-1)\times 36}{2\times n^2} = 30$ oe; $(-\frac{6}{k})(\frac{-6}{k}-1)k^2 = 60$ oe |
| substitution of $n = \pm\frac{6}{k}$ or $k = \pm\frac{6}{n}$ or $k = \pm\sqrt{\frac{60}{n(n-1)}}$ to eliminate one variable | M1 | allow omission of brackets |
| $n = -1.5$ | A1 | e.g. allow $-\frac{6}{4}$ |
| $k = 4$ | A1 | |
| expansion valid for $|x| < \frac{1}{4}$ or $-\frac{1}{4} < x < \frac{1}{4}$ | B1FT | FT their $k$ |

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7 Given that the binomial expansion of $( 1 + k x ) ^ { n }$ is $1 - 6 x + 30 x ^ { 2 } + \ldots$, find the values of $n$ and $k$. State the set of values of $x$ for which this expansion is valid.

\hfill \mbox{\textit{OCR C4 2016 Q7 [6]}}
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