Product with unknown constant to determine

9. (a) Write down the binomial expansion of \(( 2 - x ) ^ { 6 }\) up to and including the term in \(x ^ { 2 }\).
(b) Given that $$( 1 + a x ) ( 2 - x ) ^ { 6 } \equiv 64 + b x + 336 x ^ { 2 } + \ldots$$ find the values of the constants \(a , b\).

3. (a) Find and simplify the first three terms in the expansion of \(( 2 - 5 x ) ^ { 5 }\) in ascending powers of \(x\).
(b) In the expansion of \(( 1 + a x ) ^ { 2 } ( 2 - 5 x ) ^ { 5 }\), the coefficient of \(x\) is 48 . Find the value of \(a\).
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3. (i) Find and simplify the first three terms in the binomial expansion of \(( 2 + a x ) ^ { 6 }\) in ascending powers of \(x\).
(ii) In the expansion of \(( 3 - 5 x ) ( 2 + a x ) ^ { 6 }\), the coefficient of \(x\) is 64 . Find the value of \(a\).
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2. (a) Find, in terms of \(a\), the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 + a x ) ^ { 6 }$$ where \(a\) is a non-zero constant. Give each term in simplest form. $$f ( x ) = \left( 3 + \frac { 1 } { x } \right) ^ { 2 } ( 2 + a x ) ^ { 6 }$$ Given that the constant term in the expansion of \(\mathrm { f } ( x )\) is 576
(b) find the value of \(a\). \section*{3. In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has equation $$y = 4 x ^ { \frac { 1 } { 2 } } + 9 x ^ { - \frac { 1 } { 2 } } + 3 \quad x > 0$$ (a) Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each term in simplest form.
(b) Hence find the \(x\) coordinate of the stationary point of \(C\).
(c) Determine the nature of the stationary point of \(C\), giving a reason for your answer.
(d) State the range of values of \(x\) for which \(y\) is decreasing.
(Total for Question 3 is 7 marks)