3 \includegraphics[max width=\textwidth, alt={}, center]{b4a4082c-f3cd-47c5-8673-680dae9a22bd-04_684_455_260_845} The diagram shows the curve \(y = 2 + \mathrm { e } ^ { - 2 x }\). The curve crosses the \(y\)-axis at the point \(A\), and the point \(B\) on the curve has \(x\)-coordinate 1 . The shaded region is bounded by the curve and the line segment \(A B\). Find the exact area of the shaded region.

## Question 3:

| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain form $ax + be^{-2x}$ | M1 | |
| Obtain correct $2x - \frac{1}{2}e^{-2x}$ | A1 | |
| Apply limits to obtain $\frac{5}{2} - \frac{1}{2}e^{-2}$ | A1 | |
| Attempt to find area of relevant trapezium | M1 | |
| Obtain $\frac{5}{2} + \frac{1}{2}e^{-2}$ and subtract to obtain $e^{-2}$ or exact equivalent | A1 | |

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3\\
\includegraphics[max width=\textwidth, alt={}, center]{b4a4082c-f3cd-47c5-8673-680dae9a22bd-04_684_455_260_845}

The diagram shows the curve $y = 2 + \mathrm { e } ^ { - 2 x }$. The curve crosses the $y$-axis at the point $A$, and the point $B$ on the curve has $x$-coordinate 1 . The shaded region is bounded by the curve and the line segment $A B$.

Find the exact area of the shaded region.\\

\hfill \mbox{\textit{CAIE P2 2020 Q3 [5]}}