Challenging +1.2 This is a Further Maths question requiring integration to find areas and setting up an equation where two regions have equal area. While it involves a cubic and requires careful setup of the integral equation, the technique is standard: integrate the cubic, apply limits, and solve the resulting equation. The 3-mark allocation suggests a straightforward application of integration skills with some algebraic manipulation, making it moderately above average difficulty but not requiring novel insight.
Four finite regions \(A\), \(B\), \(C\) and \(D\) are enclosed by the curve with equation
$$y = x^3 - 7x^2 + 11x + 6$$
and the lines \(y = k\), \(x = 1\) and \(x = 4\), as shown in the diagram below.
\includegraphics{figure_11}
The areas of \(B\) and \(C\) are equal.
Find the value of \(k\).
[3 marks]
Four finite regions $A$, $B$, $C$ and $D$ are enclosed by the curve with equation
$$y = x^3 - 7x^2 + 11x + 6$$
and the lines $y = k$, $x = 1$ and $x = 4$, as shown in the diagram below.
\includegraphics{figure_11}
The areas of $B$ and $C$ are equal.
Find the value of $k$.
[3 marks]
\hfill \mbox{\textit{AQA Further AS Paper 1 2018 Q11 [3]}}