The power output, \(P\) watts, of a certain wind turbine is proportional to the cube of the wind speed \(v\)ms\(^{-1}\).
When \(v = 3.6\), \(P = 50\).
Determine the wind speed that will give a power output of 225 watts. [3]
The first three terms of an arithmetic series are \(9p\), \(8p - 3\), \(5p\) respectively, where \(p\) is a constant.
Given that the sum of the first \(n\) terms of this series is \(-1512\), find the value of \(n\). [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 \(m\).
\(t\)
0
5
10
25
\(m\)
200
160
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. [4]
A student was asked to solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\). The student's attempt is written out below.
\(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\)
\(4\log_3 x - 3 \log_3 x - 2 = 0\)
\(\log_3 x - 2 = 0\)
\(\log_3 x = 2\)
\(x = 8\)
Identify the two mistakes that the student has made. [2]
Solve the equation \(2(\log_3 x)^2 - 3 \log_3 x - 2 = 0\), giving your answers in an exact form. [4]
In this question you must show detailed reasoning.
It is given that the geometric series
$$1 + \frac{5}{3x-4} + \left(\frac{5}{3x-4}\right)^2 + \left(\frac{5}{3x-4}\right)^3 + \ldots$$
is convergent.
Find the set of possible values of \(x\), giving your answer in set notation. [5]
Given that the sum to infinity of the series is \(\frac{2}{3}\), find the value of \(x\). [3]