| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | September |
| Marks | 9 |
| Topic | Exponential Functions |
| Type | Sketch exponential graphs |
| Difficulty | Moderate -0.3 This question tests standard exponential graph sketching (identifying y-intercept and asymptote) and routine trapezium rule application with straightforward follow-up parts involving basic integral properties. While it requires multiple techniques across 9 marks, each component is procedural with no novel insight required—slightly easier than a typical A-level question due to the mechanical nature of the trapezium rule calculation and trivial algebraic manipulations in part (c). |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.06a Exponential function: a^x and e^x graphs and properties1.09f Trapezium rule: numerical integration |
8. (a) Sketch the curve with equation
$$y = a ^ { - x } + 4$$
where $a$ is a constant and $a > 1$\\
On your sketch show
\begin{itemize}
\item the coordinates of the point of intersection of the curve with the $y$-axis
\item the equation of any asymptotes to the curve.\\
(3)\\
(b) Use the trapezium rule with 5 trapeziums to find an approximate value for
\end{itemize}
$$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) d x$$
giving your answer to two significant figures.\\
(3)\\
(c) Using the answer to part (b), find an approximate value for
\begin{enumerate}[label=(\roman*)]
\item $\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x$
\item $\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x + \int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x$
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q8 [9]}}