| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2022 |
| Session | February |
| Marks | 11 |
| Topic | Exponential Functions |
| Type | Sketch exponential graphs |
| Difficulty | Moderate -0.8 This question tests standard A-level techniques: sketching an exponential function, applying the trapezium rule with given strip widths, and manipulating logarithms to prove a given result. All parts are routine applications of core methods with no problem-solving insight required, making it easier than average but not trivial due to the multi-step logarithm manipulation in part (iii). |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules1.09f Trapezium rule: numerical integration |
\begin{enumerate}[label=(\roman*)]
\item Sketch the curve $y = \left(\frac{1}{2}\right)^x$, and state the coordinates of any point where the curve crosses an axis. [3]
\item Use the trapezium rule, with 4 strips of width 0.5, to estimate the area of the region bounded by the curve $y = \left(\frac{1}{2}\right)^x$, the axes, and the line $x = 2$. [4]
\item The point $P$ on the curve $y = \left(\frac{1}{2}\right)^x$ has $y$-coordinate equal to $\frac{1}{6}$. Prove that the $x$-coordinate of $P$ may be written as
$$1 + \frac{\log_{10} 3}{\log_{10} 2}.$$ [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2022 Q5 [11]}}