Area Ratio or Comparative Area

Question asking for the ratio of two areas, or requiring comparison of areas of sub-regions formed by curves and tangent lines, rather than a single absolute area value.

csat-suneung 2025 Q13 4 marks View
A cubic function $f(x)$ with leading coefficient 1 satisfies $$f(1) = f(2) = 0, \quad f'(0) = -7$$ Let Q be the point where the line segment OP intersects the curve $y = f(x)$ other than P, where O is the origin and $\mathrm{P}(3, f(3))$. Let $A$ be the area enclosed by the curve $y = f(x)$, the $y$-axis, and the line segment OQ, and let $B$ be the area enclosed by the curve $y = f(x)$ and the line segment PQ. What is the value of $B - A$? [4 points]
(1) $\frac{37}{4}$
(2) $\frac{39}{4}$
(3) $\frac{41}{4}$
(4) $\frac{43}{4}$
(5) $\frac{45}{4}$