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Question
In the expansion of \({(3x – 2)^{12}}\) , the term in \({x^5}\) can be expressed as \(\left( {\begin{array}{*{20}{c}}
{12}\\
r
\end{array}} \right) \times {(3x)^p} \times {( – 2)^q}\) .
(a) Write down the value of \(p\) , of \(q\) and of \(r\) .
(b) Find the coefficient of the term in \({x^5}\) .
Write down the value of \(p\) , of \(q\) and of \(r\) .
Find the coefficient of the term in \({x^5}\) .
Answer/Explanation
Markscheme
(a) \(p = 5\) , \(q = 7\) , \(r = 7\) (accept \(r = 5\)) A1A1A1 N3
[3 marks]
(b) correct working (A1)
eg \(\left( {\begin{array}{*{20}{c}}
{12}\\
7
\end{array}} \right) \times {(3x)^5} \times {( – 2)^7}\) , \(792\) , \(243\) , \( – {2^7}\) , \(24634368\)
coefficient of term in \({x^5}\) is \( – 24634368\) A1 N2
Note: Do not award the final A1 for an answer that contains \(x\).
[2 marks]
Total [5 marks].
\(p = 5\) , \(q = 7\) , \(r = 7\) (accept \(r = 5\)) A1A1A1 N3
[3 marks]
correct working (A1)
eg \(\left( {\begin{array}{*{20}{c}}
{12}\\
7
\end{array}} \right) \times {(3x)^5} \times {( – 2)^7}\) , \(792\) , \(243\) , \( – {2^7}\) , \(24634368\)
coefficient of term in \({x^5}\) is \( – 24634368\) A1 N2
Note: Do not award the final A1 for an answer that contains \(x\).
[2 marks]
Total [5 marks]
Question
Consider the expansion of \({(2x + 3)^8}\).
Write down the number of terms in this expansion.
Find the term in \({x^3}\).
Answer/Explanation
Markscheme
9 terms A1 N1
[1 mark]
valid approach to find the required term (M1)
eg\(\;\;\;\left( {\begin{array}{*{20}{c}} 8 \\ r \end{array}} \right){(2x)^{8 – r}}{(3)^r},{\text{ }}{(2x)^8}{(3)^0} + {(2x)^7}{(3)^1} + \ldots \), Pascal’s triangle to \({{\text{8}}^{{\text{th}}}}\) row
identifying correct term (may be indicated in expansion) (A1)
eg\(\;\;\;{{\text{6}}^{{\text{th}}}}{\text{ term, }}r = 5,{\text{ }}\left( {\begin{array}{*{20}{c}} 8 \\ 5 \end{array}} \right),{\text{ (2x}}{{\text{)}}^3}{(3)^5}\)
correct working (may be seen in expansion) (A1)
eg\(\;\;\;\left( {\begin{array}{*{20}{c}} 8 \\ 5 \end{array}} \right){(2x)^3}{(3)^5},{\text{ }}56 \times {2^3} \times {3^5}\)
\(108864{x^3}\;\;\;\)(accept \(109000{x^3}\)) A1 N3
[4 marks]
Notes: Do not award any marks if there is clear evidence of adding instead of multiplying.
Do not award final A1 for a final answer of \(108864\), even if \(108864{x^3}\) is seen previously.
If no working shown award N2 for \(108864\).
Question
The third term in the expansion of \({(x + k)^8}\) is \(63{x^6}\). Find the possible values of \(k\).
Answer/Explanation
Markscheme
valid approach to find the required term (M1)
eg\(\;\;\;\left( {\begin{array}{*{20}{c}} 8 \\ r \end{array}} \right){x^{8 – r}}{k^r}\), Pascal’s triangle to \({{\text{8}}^{{\text{th}}}}\) row, \({x^8} + 8{x^7}k + 28{x^6}{k^2} + \ldots \)
identifying correct term (may be indicated in expansion) (A1)
eg\(\;\;\;\left( {\begin{array}{*{20}{c}} 8 \\ 2 \end{array}} \right){x^6}{k^2},{\text{ }}\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right){x^6}{k^2},{\text{ }}r = 2\)
setting up equation in \(k\) with their coefficient/term (M1)
eg\(\;\;\;28{k^2}{x^6} = 63{x^6},{\text{ }}\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right){k^2} = 63\)
\(k = \pm 1.5{\text{ (exact)}}\) A1A1 N3
[5 marks]