Standard +0.3 This question involves standard exponential curve sketching (routine), solving an exponential equation using logarithms (straightforward application), and algebraic manipulation to express an intersection point in a specific form. While the final part requires careful logarithm manipulation, all techniques are standard A-level Further Maths content with no novel problem-solving required. The multi-part structure and 7 total marks suggest slightly above-average length but not conceptual difficulty.
Sketch the curve \(y = 2^{2x+3}\), stating the coordinates of any points of intersection with the axes. [2]
The point \(P\) on the curve \(y = 3^{3x+2}\) has \(y\)-coordinate equal to 180.
Use logarithms to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. [2]
The curves \(y = 2^{2x+3}\) and \(y = 3^{3x+2}\) intersect at the point \(Q\).
Show that the \(x\)-coordinate of \(Q\) can be written as
$$x = \frac{3\log_3 2 - 2}{3 - 2\log_3 2}.$$
[3]
Sketch the curve $y = 2^{2x+3}$, stating the coordinates of any points of intersection with the axes. [2]
The point $P$ on the curve $y = 3^{3x+2}$ has $y$-coordinate equal to 180.
Use logarithms to find the $x$-coordinate of $P$ correct to 3 significant figures. [2]
The curves $y = 2^{2x+3}$ and $y = 3^{3x+2}$ intersect at the point $Q$.
Show that the $x$-coordinate of $Q$ can be written as
$$x = \frac{3\log_3 2 - 2}{3 - 2\log_3 2}.$$
[3]
\hfill \mbox{\textit{SPS SPS FM 2019 Q8 [7]}}