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Date	November 2020	Marks available	6	Reference code	20N.2.SL.TZ0.S_5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	S_5	Adapted from	N/A

Question

Consider the expansion of ${(3 x^{2} - \frac{k}{x})}^{9}$ , where $k > 0$ .

The coefficient of the term in $x^{6}$ is $6048$ . Find the value of $k$ .

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

valid approach for expansion (must have correct substitution for parameters, but accept an incorrect value for $r$ ). (M1)

eg $(\begin{matrix} 9 \\ r \end{matrix}) {(3 x^{2})}^{9 - r} {(- \frac{k}{x})}^{r}, {(3 x^{2})}^{9} + (\begin{matrix} 9 \\ 1 \end{matrix}) {(3 x^{2})}^{8} {(- \frac{k}{x})}^{1} + (\begin{matrix} 9 \\ 2 \end{matrix}) {(3 x^{2})}^{7} {(- \frac{k}{x})}^{2} + \dots$

valid attempt to identify correct term (M1)

eg $2 (9 - r) - r = 6, {(x^{2})}^{r} {(x^{- 1})}^{9 - r} = x^{6}$

identifying correct term (may be indicated in expansion) (A1)

eg $r = 4, r = 5$

correct term or coefficient in binominal expansion (A1)

eg $(\begin{matrix} 9 \\ 4 \end{matrix}) {(3 x^{2})}^{5} {(- \frac{k}{x})}^{4}, 126 (243 x^{10}) (\frac{k^{4}}{x^{4}}), 30618 k^{4}$

correct equation in $k$ (A1)

eg $(\begin{matrix} 9 \\ 4 \end{matrix}) (243) (k^{4}) x^{6} = 6048 x^{6}, 30618 k^{4} = 6048$

$k = \frac{2}{3}$ (exact) $0.667$ A1 N3

Note: Do not award A1 if additional answers given.

[6 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.9—Binomial theorem where n is an integer

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