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Date	May 2017	Marks available	7	Reference code	17M.2.SL.TZ1.S_6
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	S_6	Adapted from	N/A

Question

Let $f(x) = {({x^2} + 3)^7}$ . Find the term in ${x^5}$ in the expansion of the derivative, $f’(x)$ .

Markscheme

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

METHOD 1

derivative of $f(x)$ A2

$7{({x^2} + 3)^6}(x2)$

recognizing need to find ${x^4}$ term in ${({x^2} + 3)^6}$ (seen anywhere) R1

eg $\,\,\,\,\,$ $14x{\text{ (term in }}{x^4})$

valid approach to find the terms in ${({x^2} + 3)^6}$ (M1)

eg $\,\,\,\,\,$ $\left( {\begin{array}{*{20}{c}} 6 \\ r \end{array}} \right){({x^2})^{6 - r}}{(3)^r},{\text{ }}{({x^2})^6}{(3)^0} + {({x^2})^5}{(3)^1} + \ldots$ , Pascal’s triangle to 6th row

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

eg $\,\,\,\,\,$ ${\text{5th term, }}r = 2,{\text{ }}\left( {\begin{array}{*{20}{c}} 6 \\ 4 \end{array}} \right),{\text{ }}{({x^2})^2}{(3)^4}$

correct working (may be seen in expansion) (A1)

eg $\,\,\,\,\,$ $\left( {\begin{array}{*{20}{c}} 6 \\ 4 \end{array}} \right){({x^2})^2}{(3)^4},{\text{ }}15 \times {3^4},{\text{ }}14x \times 15 \times 81{({x^2})^2}$

$17010{x^5}$ A1 N3

METHOD 2

recognition of need to find ${x^6}$ in ${({x^2} + 3)^7}$ (seen anywhere) R1

valid approach to find the terms in ${({x^2} + 3)^7}$ (M1)

eg $\,\,\,\,\,$ $\left( {\begin{array}{*{20}{c}} 7 \\ r \end{array}} \right){({x^2})^{7 - r}}{(3)^r},{\text{ }}{({x^2})^7}{(3)^0} + {({x^2})^6}{(3)^1} + \ldots$ , Pascal’s triangle to 7th row

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

eg $\,\,\,\,\,$ 6th term, $r = 3,{\text{ }}\left( {\begin{array}{*{20}{c}} 7 \\ 3 \end{array}} \right),{\text{ (}}{{\text{x}}^2}{)^3}{(3)^4}$

correct working (may be seen in expansion) (A1)

eg $\,\,\,\,\,$ $\left( {\begin{array}{*{20}{c}} 7 \\ 4 \end{array}} \right){{\text{(}}{{\text{x}}^2})^3}{(3)^4},{\text{ }}35 \times {3^4}$

correct term (A1)

$2835{x^6}$

differentiating their term in ${x^6}$ (M1)

eg $\,\,\,\,\,$ $(2835{x^6})',{\text{ (6)(2835}}{{\text{x}}^5})$

$17010{x^5}$ A1 N3

[7 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5 —Calculus » SL 5.3—Differentiating polynomials, n E Z

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18M.1.AHL.TZ2.H_6b:
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18N.2.SL.TZ0.T_6g:
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19M.1.SL.TZ1.T_15c:
Find the value of $r$ when the area of the curved surface is maximized.
18N.2.SL.TZ0.T_6d:
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19M.1.SL.TZ2.T_15b:
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20N.1.SL.TZ0.T_13c:
Find the value of $k$ .
17M.2.SL.TZ2.T_6a:
Write down the $y$ -intercept of the graph.
17M.2.SL.TZ2.T_6c.i:
Show that $a = 8$ .
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
20N.1.SL.TZ0.T_13a:
Write down $f' (x)$ .
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17N.2.SL.TZ0.T_5e:
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16N.2.SL.TZ0.T_6g:
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Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
18N.2.SL.TZ0.T_6e:
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17M.2.SL.TZ2.T_6e:
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18N.2.SL.TZ0.T_6f:
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