Mathematical Modelling

Full Appendix Summary & Detailed Notes - Mathematical Modelling Class 11 NCERT

Overview & Key Concepts

Appendix Goal: Translate real-life problems into math models for solutions. Steps: Understand, Formulate (Identify factors, Math description), Solve, Interpret/Validate. Cycle: Refine if needed. Examples: Trig, graphs, LPP, exponential growth. Exam Focus: Processes, assumptions, validation. 2025 Updates: Real-world apps emphasis. Fun Fact: Euler's bridge solved via graphs. Core Idea: Simplify complex realities. Real-World: Blood flow, weather, agriculture. Ties: Trig, graphs, stats.
Wider Scope: Historical prelims, definitions, cycle of modelling. Covers A.2.1 to A.2.3.
Expanded Content: Steps, examples, flowchart.

A.2.1 Introduction

Math models handle real problems via computation. Process: Translate to equations.

A.2.2 Preliminaries

Historical: Ancients used geo for predictions. Ex: Surveyor uses trig for tower height.

A.2.3 What is Mathematical Modelling?

Definition: Study real parts in math terms. Steps: Understand, Identify factors, Math description, Solve, Validate/Interpret. Cycle: Refine if needed. Ex: Pendulum \( T = 2\pi \sqrt{l/g} \).

Summary

Tool for real analysis. Master: Steps, assumptions. Apps: Predictions. Mantra: Validate always.

Why This Guide Stands Out

Steps boxed, examples solved, figures SVG, tables exact, cycle diagram, free 2025 with MathJax.

Key Themes & Tips

Aspects: From geo to growth models.
Tip: Identify essentials; test assumptions.

Exam Case Studies

Model blood flow; validate pendulum; optimize diet.

Project & Group Ideas

Model city traffic via graphs.
Apps: Simulate population in Excel.

Modelling Process Flowchart

Example 1: Tower Height - Solved Steps

Problem: Angle of elevation 40° from 450m away. Find height h.

Step 1: Understand: Parameters: h (height), d=450m (distance), θ=40° (angle).
Step 2: Formulate: Geo: \( \tan \theta = \frac{h}{d} \) → \( h = d \tan \theta \).
Step 3: Solve: \( h = 450 \times \tan 40^\circ \approx 450 \times 0.839 = 377.6 \) m.
Step 4: Interpret: Height ≈ 378 m.

Fig 1: Geometric Representation

Tip: Trig simplifies measurement.

Example 2: Bridge Problem - Solved Steps

Problem: Königsberg 7 bridges; cross each once?

Step 1: Understand: 4 landmasses (A,B,C,D), 7 arcs.
Step 2: Formulate: Graph: Vertices A(3 odd), B(3), C(5), D(3) all odd degree.
Step 3: Solve: Euler: Needs ≤2 odd vertices for Euler path. Here 4 → Impossible.
Step 4: Interpret: Can't; add bridge A-B → 2 odds → Possible (Fig 4).

Fig 2: Königsberg Bridges

Fig 3: Network Graph

Fig 4: Updated with Extra Bridge

Tip: Graph theory for paths.

Simple Pendulum Modelling - Solved Steps

Problem: Find period T.

Understand: Periodic motion; depends on l, g.
Formulate: Identify: Factors: T, m (irrelevant), l (essential), g. Expt: T ∝ √l.
Math Desc: \( T^2 = k l \) → \( k = 4\pi^2 / g \) → \( T = 2\pi \sqrt{l/g} \).
Solve: For l=225cm, T≈3.04s; l=275cm, T≈3.36s (Table 1).
Validate: Expt vs calc error <0.02s (Table 2) → Accept.
Interpret: T ∝ √l, ∝ 1/√g.

l (cm)	T (sec)
225	3.04
275	3.36

Mass (gms)	Length (cms)	Time (secs)
385	275	3.371
385	225	3.056
230	275	3.352
230	225	3.042

Tip: Expts refine factors.

Example 3: Food Mix Optimization - Solved Steps

Problem: Min cost ≥800kg mix, ≥30% protein, ≤5% fibre. Corn: 0.09p/0.02f Rs10; Soy: 0.60p/0.06f Rs20.

Step 1: Understand: Vars: x(corn kg), y(soy kg), z(cost).
Step 2: Formulate: z=10x+20y; x+y≥800; 0.09x+0.6y≥0.3(x+y); 0.02x+0.06y≤0.05(x+y). Simplified: x+y≥800; 0.21x-0.30y≤0; 0.03x-0.01y≥0.
Step 3: Solve: Graph: Min at (470.6, 329.4); z=11294 Rs (Fig 5).
Step 4: Interpret: Use 470.6kg corn, 329.4kg soy for min cost.

Material	Protein	Fibre	Cost per Kg
Corn	0.09	0.02	Rs 10
Soyabean	0.60	0.06	Rs 20

Fig 5: Feasible Region

Tip: LPP for constraints.

Example 4: Population Growth - Solved Steps

Problem: Pop after 10 yrs; P(0)=250M, b=0.02, d=0.01.

Step 1: Understand: P(t+1)=P(t)+B(t)-D(t); rates constant.
Step 2: Formulate: B(t)=b P(t), D(t)=d P(t); P(t+1)=(1+b-d)P(t) → P(t)=P(0) r^t, r=1.01.
Step 3: Solve: P(10)=250M × (1.01)^10 ≈276.16M.
Step 4: Interpret: Approx 276M; assumptions cause fractional error.

Tip: Exponential for growth.

20 Questions & Answers - NCERT Based (Short & Detailed)

Part A: 10 short (1-2 marks), Part B: 10 detailed (4-6 marks). From examples/variations.

Part A: Short Questions (10 Qs)

1. Key steps in modelling?

Answer: Understand, Formulate, Solve, Interpret/Validate

2. Tower height formula?

Answer: \( h = d \tan \theta \)

3. Max odd vertices for Euler path?

Answer: 2

4. Pendulum T formula?

Answer: \( T = 2\pi \sqrt{l/g} \)

5. Food min cost?

Answer: Rs 11294

6. Population growth r?

Answer: r = 1 + b - d

7. Bridge vertices odds original?

Answer: 4

8. Pendulum essential factor?

Answer: l (length)

9. Food constraints number?

Answer: 3

10. P(10) approx?

Answer: 276 million

Part B: Detailed Questions (10 Qs)

1. Steps for pendulum model.

Detailed Answer:

Identify: l essential, m not.
Form: \( T=2\pi\sqrt{l/g} \)
Validate: Error small (Tables 1,2).

2. Explain bridge impossibility.

Detailed Answer:

4 odd vertices; Euler: max 2 for path.
Add A-B: 2 odds, possible.

3. Derive pendulum equation.

Detailed Answer:

Expt: T² ∝ l → T² = k l → k=4π²/g.

4. Food constraints derivation.

Detailed Answer:

Total: x+y≥800
Protein: 0.09x+0.6y ≥0.3(x+y) → 0.21x -0.3y ≤0
Fibre: 0.02x+0.06y ≤0.05(x+y) → -0.03x +0.01y ≥0

5. Population assumptions.

Detailed Answer:

Constant b,d; no migration.
P(t+1)=(1+b-d)P(t)

6. Tower steps detail.

Detailed Answer:

Understand: h,d,θ
Form: tanθ=h/d
Solve: 377.6m
Interpret: 378m

7. Validation importance.

Detailed Answer:

Compare model vs real; refine if error large.
Ex: Pendulum error 0.011s.

8. Modelling cycle.

Detailed Answer:

Assumptions → Form → Solve → Interpret → Validate → If no, back.

9. Food graph point.

Detailed Answer:

Min at intersection (470.6,329.4)
z=10*470.6 +20*329.4=11294

10. Population approx reason.

Detailed Answer:

Assumptions: constant rates, no migration → fractional pop.
Approx to 276M.

Tip: Detail steps/assumptions.

Quick Revision Notes & Mnemonics

Modelling Cycle

Understand → Formulate → Solve → Interpret/Validate → Refine?
Mnemonic: "Ugly Frogs Sing In Velvet" (Understand Formulate Solve Interpret Validate)

Tower

\( h = d \tan\theta \)
Mnemonic: "Height = Distance Tan Angle" (HDTA)

Bridge

≤2 odd vertices for path
Mnemonic: "Odds Even Path" (OEP)

Pendulum

\( T=2\pi\sqrt{l/g} \)
Mnemonic: "Twice Pi Square Root Length Over G" (TPSRLG)

Food

Min z=10x+20y s.t. constraints
Mnemonic: "Corn Soy Protein Fibre" (CSPF)

Population

P(t)=P(0)(1+b-d)^t
Mnemonic: "Birth Death Rate Exponential" (BDRE)

Overall Mnemonic: "Model Real Examples" (MRE). Flashcards for steps.

Key Steps & Processes - All Core

Step	Description	Example	Tip
Understand	Identify parameters	Tower: h,d,θ	List vars
Formulate	Math eq/ineq/graph	Pendulum: \( T=2\pi\sqrt{l/g} \)	Test essentials
Solve	Calc/Graph	Bridge: 4 odds → No	Use tools
Interpret/Validate	Compare real; refine	Pop: Approx 276M	Check errors

Tip: Cycle: If no, back to assumptions.

Mathematical Modelling – NCERT Class 11 Mathematics Appendix 2 – Concepts, Importance, and Applications in Real-world Problems

Full Appendix Summary & Detailed Notes - Mathematical Modelling Class 11 NCERT

Overview & Key Concepts

A.2.1 Introduction

A.2.2 Preliminaries

A.2.3 What is Mathematical Modelling?

Summary

Why This Guide Stands Out

Key Themes & Tips

Exam Case Studies

Project & Group Ideas

Modelling Process Flowchart

Example 1: Tower Height - Solved Steps

Problem: Angle of elevation 40° from 450m away. Find height h.

Fig 1: Geometric Representation

Example 2: Bridge Problem - Solved Steps

Problem: Königsberg 7 bridges; cross each once?

Fig 2: Königsberg Bridges

Fig 3: Network Graph

Fig 4: Updated with Extra Bridge

Simple Pendulum Modelling - Solved Steps

Problem: Find period T.

Example 3: Food Mix Optimization - Solved Steps

Problem: Min cost ≥800kg mix, ≥30% protein, ≤5% fibre. Corn: 0.09p/0.02f Rs10; Soy: 0.60p/0.06f Rs20.

Fig 5: Feasible Region

Example 4: Population Growth - Solved Steps

Problem: Pop after 10 yrs; P(0)=250M, b=0.02, d=0.01.

20 Questions & Answers - NCERT Based (Short & Detailed)

Part A: Short Questions (10 Qs)

1. Key steps in modelling?

2. Tower height formula?

3. Max odd vertices for Euler path?

4. Pendulum T formula?

5. Food min cost?

6. Population growth r?

7. Bridge vertices odds original?

8. Pendulum essential factor?

9. Food constraints number?

10. P(10) approx?

Part B: Detailed Questions (10 Qs)

1. Steps for pendulum model.

2. Explain bridge impossibility.

3. Derive pendulum equation.

4. Food constraints derivation.

5. Population assumptions.

6. Tower steps detail.

7. Validation importance.

8. Modelling cycle.

9. Food graph point.

10. Population approx reason.

Quick Revision Notes & Mnemonics

Modelling Cycle

Tower

Bridge

Pendulum

Food

Population

Key Steps & Processes - All Core

Group Discussions