SPS SPS SM (SPS SM) 2024 October

1. The power output, \(P\) watts, of a certain wind turbine is proportional to the cube of the wind speed \(\mathrm { vms } ^ { - 1 }\).

When \(v = 3.6 , P = 50\). Determine the wind speed that will give a power output of 225 watts.

2. Solve the inequalities

1. \(3 - 8 x > 4\),
2. \(( 2 x - 4 ) ( x - 3 ) \leqslant 12\).

3. The first three terms of an arithmetic series are \(9 p , 8 p - 3,5 p\) respectively, where p is a constant. Given that the sum of the first \(n\) terms of this series is - 1512 , find the value of \(n\).
[0pt] [BLANK PAGE]

4. The quadratic equation \(k x ^ { 2 } + ( 3 k - 1 ) x - 4 = 0\) has no real roots. Find the set of possible values of \(k\).
[0pt] [BLANK PAGE]

6. The mass of a substance is decreasing exponentially. Its mass is \(m\) grams at time \(t\) years. The following table shows certain values of \(t\) and \(m\).

1. Find the values missing from the table.
2. Determine the value of \(t\), correct to the nearest integer, for which the mass is 50 grams.
   [0pt] [BLANK PAGE]

7. A student was asked to solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\). The student's attempt is written out below. $$\begin{aligned} & 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0
& 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0
& \log _ { 3 } x - 2 = 0
& \log _ { 3 } x = 2
& x = 8 \end{aligned}$$

1. Identify the two mistakes that the student has made.
2. Solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\), giving your answers in an exact form.
   [0pt] [BLANK PAGE]

8. In this question you must show detailed reasoning. It is given that the geometric series $$1 + \frac { 5 } { 3 x - 4 } + \left( \frac { 5 } { 3 x - 4 } \right) ^ { 2 } + \left( \frac { 5 } { 3 x - 4 } \right) ^ { 3 } + \ldots$$ is convergent.

1. Find the set of possible values of \(x\), giving your answer in set notation.
2. Given that the sum to infinity of the series is \(\frac { 2 } { 3 }\), find the value of \(x\).
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]