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Differential Equations 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

Solved CBSE Board Case Study: Population Growth & Rate Models

Case Context: In a biology research lab, the rate of increase in a bacterial culture population P at any time t is proportional to the population present at that time. Let the initial count of bacteria at t = 0 be P_0.

Question (i): Formulate the differential equation and find the general solution for population P(t).

Solution: Since rate of growth is proportional to P, dP/dt = kP where k > 0 is constant. Separating variables: dP/P = k dt. Integrating both sides: ln|P| = kt + C, or P(t) = C * e^(kt). At t = 0, P(0) = P_0, hence C = P_0. Thus, P(t) = P_0 * e^(kt).

Question (ii): If the bacterial population doubles in 4 hours, determine the value of growth constant k and find the population after 12 hours.

Solution: At t = 4 hours, P(4) = 2*P_0. Substituting: 2*P_0 = P_0 * e^(4k) => e^(4k) = 2, so k = (ln 2)/4. After t = 12 hours: P(12) = P_0 * e^(12k) = P_0 * (e^(4k))^3 = P_0 * (2)^3 = 8*P_0. The population increases to 8 times the original count.

CBSE Strategy Note: Case study problems in Section E carry 4 marks (1 + 1 + 2 marks). Always clearly write the initial conditions (t = 0), express constants explicitly in logarithmic form, and check units of time.