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Differential Equations · Paper 2 — Free Maths AA HL Practice Questions

What this topic covers

Calculus is where AA HL stops rewarding memorised algebra and starts rewarding process. Every mark scheme in this topic is built around specific words — show that, hence, determine, justify — and examiners are trained to withhold marks the instant your working skips a step, even when your final answer is correct. The four subtopics here build on each other in a strict chain: limits define continuity, continuity is the gatekeeper for differentiability, derivatives feed every application (tangents, optimisation, kinematics), and integration — the reverse process — feeds differential equations. Learn it in that order and the HL-only material (implicit differentiation, integration by parts, volumes of revolution, and the whole of differential equations) stops looking like extra content and starts looking like the natural next question.

Read the full Calculus lesson notes →

1MasterySAQ-SSolving first-order differential equationss7 marksPaper 2~11 min

A chemical reaction is modelled by the differential equation

\dfrac{dC}{dt} = -kC^2

, where

C(t)

is the concentration of a reactant in

\text{mol\,L}^{-1}

at time

t

seconds, and

k

is a positive constant.

(a)

Show that

C(t) = \dfrac{1}{kt + \dfrac{1}{C_0}}

, where

C_0 = C(0)

. [3 marks]

(b)

Given that

C(0) = 2.0\,\text{mol\,L}^{-1}

and

C(10) = 0.50\,\text{mol\,L}^{-1}

, calculate the value of

k

. [2 marks]

(c)

Hence calculate the time, in seconds, at which the concentration reaches

0.10\,\text{mol\,L}^{-1}

. Give your answer correct to 3 significant figures. [2 marks]

Solutions

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