SPS
SPS FM Pure
2025
February
Q3
3
\end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r }
2 & 0 & 3
1 & - 1 & 3
\end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { l l }
1 & 3
\end{array} \right)$$
Calculate all possible products formed from two of these three matrices.
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2. The complex number \(z\) satisfies the equation \(z ^ { 2 } - 4 \mathrm { i } z ^ { * } + 11 = 0\).
Given that \(\operatorname { Re } ( z ) > 0\), find \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers.
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3.
Prove by mathematical induction that \(\sum _ { r = 1 } ^ { n } ( r \times r ! ) = ( n + 1 ) ! - 1\) for all positive integers \(n\).
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SPS
SPS FM Pure
2025
February
Q4
4. The cubic equation
$$2 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 12 = 0$$
has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are ( \(\alpha + 3\) ), ( \(\beta + 3\) ) and \(( \gamma + 3 )\), giving your answer in the form \(p w ^ { 3 } + q w ^ { 2 } + r w + s = 0\), where \(p , q , r\) and \(s\) are integers to be found.
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SPS
SPS FM Pure
2025
February
Q12
12. The population density \(P\), in suitable units, of a certain bacterium at time \(t\) hours is to be modelled by a differential equation. Initially, the population density is zero, and its long-term value is 5 .
The model uses the differential equation
$$\frac { d P } { d t } - \frac { P } { t \left( 1 + t ^ { 2 } \right) } = \frac { t e ^ { - t } } { \sqrt { 1 + t ^ { 2 } } }$$
Find \(P\) as a function of \(t\). [You may assume that \(\lim _ { t \rightarrow \infty } t e ^ { - t } = 0\) ].
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SPS
SPS FM Pure
2025
February
Q13
13. (a) Write down the Maclaurin series of \(\mathrm { e } ^ { x }\), in ascending power of \(x\), up to and including the term in \(x ^ { 3 }\)
(b) Hence, without differentiating, determine the Maclaurin series of
$$\mathrm { e } ^ { \left( \mathrm { e } ^ { x } - 1 \right) }$$
in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient in simplest form.
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SPS
SPS SM Pure
2025
February
Q2
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)