Case study questions have become an integral part of the CBSE Class 12 Mathematics examination, particularly in the chapter on Differential Equations. These questions are designed to test students' understanding of real-world applications of differential equations and their ability to interpret and solve problems in context.
Before diving into case study questions, it's essential to review the fundamental concepts of differential equations:
1. Order and degree of a differential equation
2. General and particular solutions
3. Formation of differential equations
4. Methods of solving first-order, first-degree differential equations:
- Variables separable method
- Homogeneous differential equations
- Linear differential equations
Structure
CBSE case study questions typically consist of:
1. A real-world scenario or situation
2. Relevant data or information
3. A set of 4-5 multiple-choice questions (MCQs) based on the given scenario
Sample Case Study
Let's look at a sample case study question on differential equations:
Population Growth Model
A city planner is studying the population growth of a newly developed suburban area. The population at time t (in years) is denoted by P(t). The growth rate is proportional to the current population, with a proportionality constant of 0.05 per year.
Question 1:
The differential equation modeling this population growth is:
a) dP/dt = 0.05P
b) dP/dt = 0.05t
c) dP/dt = 0.05
d) P = 0.05t
Answer: a) dP/dt = 0.05P
Question 2:
If the initial population at t = 0 is 10,000, what is the general solution to this differential equation?
a) P(t) = 10,000e^(0.05t)
b) P(t) = 10,000 0.05t
c) P(t) = 10,000(1 0.05t)
d) P(t) = 10,000 / (1 - 0.05t)
Answer: a) P(t) = 10,000e^(0.05t)
Question 3:
After how many years will the population double?
a) 10 years
b) 13.86 years
c) 20 years
d) 25 years
Answer: b) 13.86 years
(Hint: Solve 2 = e^(0.05t) for t)
Question 4:
What is the population growth rate when the population reaches 15,000?
a) 500 people per year
b) 750 people per year
c) 1000 people per year
d) 1250 people per year
Answer: b) 750 people per year
(Hint: Calculate dP/dt when P = 15,000)
Tips for Solving Case Study Questions
1. Read the scenario carefully and identify the key information.
2. Understand the relationship between variables in the given context.
3. Recognize the type of differential equation involved.
4. Apply the appropriate method to solve the differential equation.
5. Interpret the solution in the context of the given scenario.
6. Double-check your calculations and units.
Drawing Probability in Deck-Building Games
In deck-building games like Dominion or Ascension, the probability of drawing certain cards changes as the game progresses. Differential equations can model this evolution.
Example for Dragon vs Tiger game Card Draw Probability Model
Let P(t) be the probability of drawing a specific card at time t. A simple model for dragon tiger money game:
dP/dt = a(1-P) - bP
Where:
- a is the rate at which the card is added to the deck
- b is the rate at which other cards are added to the deck
This equation models how the probability of drawing a specific card changes as the deck composition evolves.
Case study questions on differential equations provide an excellent opportunity for students to apply their knowledge to real-world situations. By practicing these types of questions, students can improve their problem-solving skills and deepen their understanding of the subject matter.
Remember to review all the concepts in the chapter thoroughly and practice a variety of case study questions to be well-prepared for the CBSE Class 12 Mathematics examination.