Moderate -0.3 This is a straightforward binomial expansion question requiring standard application of the binomial theorem to find coefficients, then solving for a constant by matching terms. Part (a) is routine expansion, part (b) requires expanding (3+1/x)² and identifying which terms from (2+ax)⁶ contribute to the constant term—a mechanical process with no novel insight needed. Slightly easier than average due to being a standard textbook-style problem with clear steps.
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)
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)\\
\hfill \mbox{\textit{SPS SPS SM Pure 2025 Q2 [7]}}