DE Case Study & Competency PYQs

Practice Class 12 CBSE Board Previous Year Questions (2008-2026)

Q1 2026 Board (Sample)

00:00

The rate of growth of a population $P$ of a city is proportional to the population at any time $t$. (i) Form the differential equation. (ii) If the population doubles in 50 years, find the time in which it will triple. 4 Marks

Part (i): Form the Equation

$\frac{dP}{dt} \propto P \Rightarrow \frac{dP}{dt} = kP$.

Part (ii): Solve for Time

Step 1: Solve DE

$\frac{dP}{P} = k \, dt \Rightarrow \log P = kt + C \Rightarrow P(t) = P_0 e^{kt}$.

Step 2: Find k

At $t=50, P = 2P_0 \Rightarrow 2P_0 = P_0 e^{50k} \Rightarrow e^{50k} = 2 \Rightarrow k = \frac{\log 2}{50}$.

Step 3: Find time for tripling

$3P_0 = P_0 e^{kt} \Rightarrow 3 = e^{kt} \Rightarrow \log 3 = kt$.

$t = \frac{\log 3}{k} = \frac{\log 3}{(\log 2)/50} = 50 \frac{\log 3}{\log 2}$ years.

50 (log 3 / log 2) years