2026 HCI P1 Q3

The diagram below shows the graph of \(y=\mathrm{g}'(x)\) and the regions \(A\), \(B\) and \(C\). \(A\) is the region enclosed by the graph, \(x\)-axis and \(y\)-axis. \(B\) is the region enclosed by the graph and \(x\)-axis. \(C\) is the region enclosed by the graph, \(x\)-axis and the line \(x=7\).

The areas of the regions \(A\), \(B\) and \(C\) are 8, 4.5 and 4.5 respectively. The Fundamental Theorem of Calculus states \(\displaystyle \int_a^b\mathrm{f}(x)\,\mathrm{d}x=\mathrm{F}(b)-\mathrm{F}(a)\), where \(\mathrm{F}\) is any antiderivative of \(\mathrm{f}\).

1. It is given that \(\mathrm{g}(0)=10\). By using the Fundamental Theorem of Calculus, find the value of \(\mathrm{g}(7)\).[3]
2. Let \(\mathrm{h}(x)=\mathrm{g}(x^2+1)\). Given \(\mathrm{g}'(2)=-5\), find the gradient of the tangent to the graph of \(\mathrm{h}\) at the point \((1,a)\), where \(a\in\mathbb{R}\). Hence find the equation of this tangent in terms of \(a\).[4]