(a) Find the consumer's surplus for the demand function $p = 25 - x - x^2$, where the prevailing market price $p_0 = 19$.

OR

(b) Solve the following initial value differential equation : $(x - 1) \frac{dy}{dx} = 2xy$, when $y(2) = 1$.

This question tests two distinct calculus applications through internal choice, and a student should attempt whichever they are more confident in.

Concept — Consumer's Surplus (a)

• Consumer's surplus is the monetary gain to buyers who were willing to pay more than the actual market price. Geometrically it is the area between the demand curve and the horizontal price line $p = p_0$, from $x = 0$ to the equilibrium quantity x₀.
• The single formula $CS = \int_0^{x_0} f(x) dx - p_0 \cdot x_0$ captures this area; the definite integral is the total willingness-to-pay and p₀·x₀ is the amount actually paid.

Concept — Separable ODE (b)

• An equation of the form $g(y) dy = h(x) dx$ is solved by integrating both sides. The algebraic trick of writing $2x/(x-1) = 2 + 2/(x-1)$ (splitting an improper rational function) is the key step and the most common place students get stuck.

Exam trap

• In (a), students forget to reject the negative root $x = -3$, or subtract p₀·x₀ before integrating.
• In (b), forgetting the +C or applying the initial condition incorrectly loses the accuracy mark.

Real-world

• Consumer's surplus is a core idea in welfare economics used to measure how much a market price benefits consumers; governments use it when analysing the impact of taxes and subsidies.