Challenging +1.2 This is a multi-step integration problem requiring finding a normal line equation, determining intersection points, and computing area between curves. While it involves several techniques (differentiation, equation of normal, integration by parts for x ln x), each step follows standard A-level procedures. The 9 marks reflect length rather than exceptional difficulty, and the structured 'show that' format provides clear direction.
\includegraphics{figure_9}
Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x, \quad x > 0\)
The line \(l\) is the normal to \(C\) at the point \(P(e, e)\)
The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis.
Show that the exact area of \(R\) is \(Ae^2 + B\) where \(A\) and \(B\) are rational numbers to be found.
[9]
\includegraphics{figure_9}
Figure 2 shows a sketch of part of the curve $C$ with equation $y = x \ln x, \quad x > 0$
The line $l$ is the normal to $C$ at the point $P(e, e)$
The region $R$, shown shaded in Figure 2, is bounded by the curve $C$, the line $l$ and the $x$-axis.
Show that the exact area of $R$ is $Ae^2 + B$ where $A$ and $B$ are rational numbers to be found.
[9]
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q9 [9]}}