Challenging +1.2 This question requires finding unknown coefficients using calculus conditions (minimum at x=2), then integration to find area, followed by solving simultaneous equations. While it involves multiple techniques (differentiation, integration, algebraic manipulation), each step follows standard A-level procedures without requiring novel insight. The 7-mark allocation and multi-step nature place it above average difficulty, but it remains a structured problem testing routine calculus skills rather than deep problem-solving.
\includegraphics{figure_8}
The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\).
The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\).
Given that the area of the shaded region is \(48\) units\(^2\), determine the \(y\)-coordinate of \(P\). [7]
\includegraphics{figure_8}
The diagram shows the curve with equation $y = 5x^4 + ax^3 + bx$, where $a$ and $b$ are integers. The curve has a minimum at the point $P$ where $x = 2$.
The shaded region is enclosed by the curve, the $x$-axis and the line $x = 2$.
Given that the area of the shaded region is $48$ units$^2$, determine the $y$-coordinate of $P$. [7]
\hfill \mbox{\textit{SPS SPS SM 2025 Q8 [7]}}