\(\mathbf { 1 }\) & \(\mathbf { 0 }\)
\hline
\end{tabular}
\end{center} Solve the following equations for values of \(x\).
a) \(\quad \ln ( 2 x + 5 ) = 3\)
b) \(\quad 5 ^ { 2 x + 1 } = 14\)
c) \(\quad 3 \log _ { 7 } ( 2 x ) - \log _ { 7 } \left( 8 x ^ { 2 } \right) + \log _ { 7 } x = \log _ { 3 } 81\)
| \(\mathbf { 1 }\) | \(\mathbf { 1 }\) |
The function \(f\) is defined by \(f ( x ) = \frac { 8 } { x ^ { 2 } }\).
a) Sketch the graph of \(y = f ( x )\).
b) On a separate set of axes, sketch the graph of \(y = f ( x - 2 )\). Indicate the vertical asymptote and the point where the curve crosses the \(y\)-axis.
c) Sketch the graphs of \(y = \frac { 8 } { x }\) and \(y = \frac { 8 } { ( x - 2 ) ^ { 2 } }\) on the same set of axes.
Hence state the number of roots of the equation \(\frac { 8 } { ( x - 2 ) ^ { 2 } } = \frac { 8 } { x }\).
| \(\mathbf { 1 }\) | \(\mathbf { 2 }\) |
The position vectors of the points \(A\) and \(B\), relative to a fixed origin \(O\), are given by
$$\mathbf { a } = - 3 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { b } = 5 \mathbf { i } + 8 \mathbf { j }$$
respectively.
a) Find the vector \(\mathbf { A B }\).
b) i) Find a unit vector in the direction of \(\mathbf { a }\).
ii) The point \(C\) is such that the vector \(\mathbf { O C }\) is in the direction of \(\mathbf { a }\). Given that the length of \(\mathbf { O C }\) is 7 units, write down the position vector of \(C\).
c) Calculate the angle \(A O B\).
\section*{
| \(\mathbf { 1 }\) | \(\mathbf { 3 }\) |
a) Find \(\int \left( 4 x ^ { - \frac { 2 } { 3 } } + 5 x ^ { 3 } + 7 \right) \mathrm { d } x\).}
b) The diagram below shows the graph of \(y = x ( x + 6 ) ( x - 3 )\).
\includegraphics[max width=\textwidth, alt={}, center]{631084a7-d827-401a-af0b-bbe1860dc027-7_614_1107_641_470}
Calculate the total area of the regions enclosed by the graph and the \(x\)-axis.
1 4 a) Two variables, \(x\) and \(y\), are such that the rate of change of \(y\) with respect to \(x\) is proportional to \(y\). State a model which may be appropriate for \(y\) in terms of \(x\).
b) The concentration, \(Y\) units, of a certain drug in a patient's body decreases exponentially with respect to time. At time \(t\) hours the concentration can be modelled by \(Y = A \mathrm { e } ^ { - k t }\), where \(A\) and \(k\) are constants.
A patient was given a dose of the drug that resulted in an initial concentration of 5 units.
i) After 4 hours, the concentration had dropped to 1.25 units. Show that \(k = 0 \cdot 3466\), correct to four decimal places.
ii) The minimum effective concentration of the drug is 0.6 units. How much longer would it take for the drug concentration to drop to the minimum effective level?