Differential Equations Case Study Questions Class 12 Maths Chapter 9

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Last Updated on July 23, 2025 by XAM CONTENT

Hello students, we are providing case study questions for class Class 12 Maths. Case study questions are the new question format that is introduced in CBSE board. The resources for case study questions are very less. So, to help students we have created chapterwise case study questions for class Class 12 Maths. In this article, you will find case study questions for cbse class Class 12 Maths chapter 9 Differential Equations.

Chapter	Differential Equations
Type of Questions	Case Study Questions
Nature of Questions	Competency Based Questions
Board	CBSE
Class	Class 12
Subject	Maths
Useful for	Class 12 Studying Students
Answers provided	Yes
Difficulty level	Mentioned
Important Link	Class 12 Maths Chapterwise Case Study

Table of Contents

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Case Study Questions on Differential Equations

Case Study:
A scientist is studying the cooling of a hot liquid in a lab. She observes that the temperature T (in °C) of the liquid at time t (in minutes) cools according to Newton’s Law of Cooling, which states:

\[ \frac{dT}{dt} = -k(T – T_a) \]

where \( T_a \) is the ambient room temperature and \( k \) is a positive constant.

The liquid is initially at 80°C and the room temperature \( T_a = 30^\circ\text{C} \).
After 10 minutes, the liquid temperature is found to be 60°C.

Questions:

Formulate the differential equation and solve it to get the expression for \( T \) in terms of \( t \).
Find the value of the constant \( k \) (correct to two decimal places).
How long will it take for the liquid to cool down to 40°C? (Round your answer to the nearest minute.)
Briefly interpret how such equations are helpful in real life.

Solutions:

1. Formulation and Solution of the Differential Equation:

\[ \frac{dT}{dt} = -k(T – 30) \] This is a separable equation. Rewriting: \[ \frac{dT}{T – 30} = -k\,dt \] Integrate both sides: \[ \int \frac{dT}{T – 30} = -k \int dt \] \[ \ln|T – 30| = -kt + C \] Now, exponentiate both sides: \[ |T – 30| = e^{-kt + C} = Ae^{-kt} \] Let \( A = e^C \) (a positive constant). So, \[ T – 30 = Ae^{-kt} \] \[ T(t) = 30 + A e^{-kt} \] Using the initial condition \( T(0) = 80 \): \[ 80 = 30 + A \implies A = 50 \] So, \[ \boxed{T(t) = 30 + 50e^{-kt}} \]

2. Finding \( k \) using the data after 10 minutes:

Given \( T(10) = 60 \): \[ 60 = 30 + 50 e^{-10k} \] \[ 30 = 50 e^{-10k} \implies \frac{3}{5} = e^{-10k} \] Take the natural logarithm on both sides: \[ \ln\left(\frac{3}{5}\right) = -10k \] \[ k = -\frac{1}{10} \ln\left(\frac{3}{5}\right) \approx -\frac{1}{10} \times (-0.5108) \approx 0.0511 \] To two decimal places, \[ \boxed{k = 0.05} \]

3. Time required for temperature to reach 40°C:

We want \( T(t) = 40 \): \[ 40 = 30 + 50 e^{-kt} \implies 10 = 50 e^{-kt} \implies \frac{1}{5} = e^{-kt} \] Take the log: \[ \ln\left(\frac{1}{5}\right) = -kt \implies t = -\frac{1}{k} \ln\left(\frac{1}{5}\right) \] Insert \( k \approx 0.05 \): \[ \ln\left(\frac{1}{5}\right) \approx \ln(0.2) \approx -1.6094 \] \[ t = -\frac{1}{0.05} \times (-1.6094) = 20 \times 1.6094 \approx 32.19 \] Therefore, it will take approximately \[ \boxed{32 \text{ minutes}} \] for the liquid to cool to 40°C.

4. Interpretation:
Such differential equations help predict how systems like temperature, medicine dosage, or chemical concentrations change over time. They allow scientists and engineers to model real-world processes and make informed decisions using mathematical analysis.

We hope the given case study questions for Differential Equations Class Class 12 helps you in your learning.

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Topics from which case study questions may be asked

Introduction to Differential Equations
Basic Concepts and Definitions
Order and Degree of a Differential Equation
General and Particular Solutions of a Differential Equation
Formation of Differential Equations
Methods of Solving First Order, First Degree Differential Equations
Variables Separable Method
Homogeneous Differential Equations
Linear Differential Equations
Applications of Differential Equations in Real-Life Problems

Differential Equations is a fundamental concept in Class 12 Maths, helping students understand its role in advanced mathematical reasoning and problem-solving.

Frequently Asked Questions (FAQs) on Differential Equations Case Study Questions

Q1: What is a case study question in mathematics?

A1: A case study question in mathematics is a problem or set of problems based on a real-life scenario or application. It requires students to apply their understanding of mathematical concepts to analyze, interpret, and solve the given situation.

Q2: How should students tackle case study questions in exams?

A2: To tackle case study questions effectively, students should:
Read the problem carefully: Understand the scenario and identify the mathematical concepts involved.
Break down the problem: Divide the case study into smaller parts to manage the information better.
Apply relevant formulas and theorems: Use the appropriate mathematical tools to solve each part of the problem.

Q3: Why are case study questions included in the Class 12 Maths curriculum?

A3: Case study questions are included to bridge the gap between theoretical knowledge and practical application. They help students see the relevance of what they are learning and prepare them for real-life situations where they may need to use these mathematical concepts.

Q4: Are there any online resources or tools available for practicing Differential Equations case study questions?

A12: We provide case study questions for CBSE Class 12 Maths on our website. Students can visit the website and practice sufficient case study questions and prepare for their exams. If you need more case study questions, then you can visit Physics Gurukul website. they are having a large collection of case study questions for all classes.