Complete Summary and Solutions for Mathematical Modelling – NCERT Class XII Mathematics Part I Appendix 2 – Introduction, Types, Construction, Applications

Full Appendix Summary & Detailed Notes - Mathematical Modelling NCERT Appendix 2

A.2.1 Introduction

In class XI, we have learnt about mathematical modelling as an attempt to study some part (or form) of some real-life problems in mathematical terms, i.e., the conversion of a physical situation into mathematics using some suitable conditions. Roughly speaking mathematical modelling is an activity in which we make models to describe the behaviour of various phenomenal activities of our interest in many ways using words, drawings or sketches, computer programs, mathematical formulae etc.

In earlier classes, we have observed that solutions to many problems, involving applications of various mathematical concepts, involve mathematical modelling in one way or the other. Therefore, it is important to study mathematical modelling as a separate topic. In this chapter, we shall further study mathematical modelling of some real-life problems using techniques/results from matrix, calculus and linear programming.

Conceptual Diagram: Mathematical Modelling Cycle (Like Fig A.2.1)

Physical Situation → Mathematical Model (Laws/Symbols) → Solution → Interpretation/Comparison → (If no agreement) Modify → Back to Model.

This iterative process ensures model accuracy, tying to real-world validation.

A.2.2 Why Mathematical Modelling?

Students are aware of the solution of word problems in arithmetic, algebra, trigonometry and linear programming etc. Sometimes we solve the problems without going into the physical insight of the situational problems. Situational problems need physical insight that is introduction of physical laws and some symbols to compare the mathematical results obtained with practical values. To solve many problems faced by us, we need a technique and this is what is known as mathematical modelling.

Examples of problems solved via modelling:

(i) Width of a river (hard to cross).
(ii) Optimal shot-put angle (height, resistance, gravity).
(iii) Height of a tower (inaccessible top).
(iv) Temperature at Sun's surface.
(v) Why heart patients avoid lifts (physiology).
(vi) Mass of Earth.
(vii) Yield of pulses from standing crops (no full harvest).
(viii) Blood volume in body (no full bleed).
(ix) India population 2020 (no wait).

All solved using maths; try non-math first to appreciate power.

Why This Guide Stands Out (For 2025 Exams)

Comprehensive coverage of Appendix 2 (pages 196-207): All principles/steps point-wise with derivations (e.g., tan α model), full examples (Ex1 tower, Ex2/3 matrices), diagrams (Fig A.2.1 cycle). Added 2025 relevance: Modelling in AI simulations. Processes: Situation → Model → Solve → Interpret → Iterate. Proforma: Params → Eqs → Sol → Validate.

A.2.3 Principles of Mathematical Modelling

Mathematical modelling is a principled activity and so it has some principles behind it. These principles are almost philosophical in nature. Some of the basic principles are:

(i) Identify the need for the model.
(ii) List parameters/variables.
(iii) Identify relevant data.
(iv) Identify assumptions.
(v) Identify governing physical principles.
(vi) Identify (a) equations, (b) calculations, (c) solution.
(vii) Identify tests for (a) consistency, (b) utility.
(viii) Identify parameter values to improve.

Leads to steps:

Step 1: Identify physical situation.
Step 2: Convert to math model (params/vars, laws/symbols).
Step 3: Solve math problem.
Step 4: Interpret/compare with observations/experiments.
Step 5: If agreement, accept; else modify and repeat Step 2.

Modelling Steps Derivation (Iterative Process)

Step 1: Situation (real problem).
Step 2: Model (abstract: vars like h, l, α; eq H = h + l tan α).
Step 3: Solve (compute H).
Step 4: Validate (measure actual H, compare).
Step 5: Iterate if mismatch (adjust assumptions, e.g., add wind). Ensures robustness.

Example 1: Height of Tower (Integrated)

Step 1: Find tower height.
Step 2: AB=tower H, PQ=h eye, l=PC=QB, α=elevation: H = h + l tan α.
Step 3: Solve with known h,l,α.
Step 4: If foot inaccessible, β=depression: l = h cot β, substitute.
Step 5: Exact params, accept.

Diagram: Tower Height (Fig A.2.2)

Observer P at h, distance l to B (foot), α to A (top). Tan α = (H-h)/l → H = h + l tan α. If l unknown, use β: tan β = h/l.

Example 2: Raw Materials for Products (Integrated)

Step 1: Firm produces P1,P2,P3 using R1,R2,R3; orders F1,F2.
Step 2: A = orders matrix (2x3), B = resources per unit (3x3).
Step 3: AB = total resources needed (2x3).
Step 4: Compare with available; if exceed, request more/reduce orders.

Example 3: Interpret Model (Integrated)

A = [[3,4,0],[10,15,6]], B = [[7,9,3],[10,20,0],[5,12,7]]. AB = [[165,247,87],[170,220,60]]. Need 335 R1 >330 avail; reduce A to A1=[[9,12,6],[10,20,0]], A1B=[[141,216,78],[170,220,60]] < avail.

Summary: Modelling bridges real to math; iterative for accuracy. Exercises: Model population growth, optimize diets (LPP).

Key Principles, Steps & Terms - Complete Glossary (20+ Terms)

All from NCERT Appendix 2, with examples, relevance. Grouped by principles/steps. Added: Sensitivity analysis (param improvement). Real-life: COVID models. For flashcards.

Mathematical Modelling

Convert real situation to math using conditions/symbols. Ex: Tower height. Relevance: Predict/optimize real phenomena.

Physical Situation

Real-life problem. Ex: Find tower height. Relevance: Starting point for abstraction.

Parameters/Variables

Quantities in model (h, l, α). Relevance: Represent unknowns/measurables.

Assumptions

Simplifications (e.g., flat ground). Relevance: Make solvable; test sensitivity.

Governing Principles

Physical laws (tan α = opp/adj). Relevance: Basis for equations.

Mathematical Model

Equations/symbols from situation. Ex: H = h + l tan α. Relevance: Abstract for solving.

Solution

Compute from model. Ex: AB matrix product. Relevance: Numerical/analytical result.

Interpretation

Compare with real (observations/experiments). Relevance: Validate utility.

Consistency Test

Check model internal logic. Relevance: Avoid contradictions.

Utility Test

Compare predictions with data. Relevance: Practical value.

Iteration

Modify if no agreement. Relevance: Refine model.

Step 1: Identify Situation

Define problem. Ex: Tower height. Relevance: Scope.

Step 2: Convert to Model

Vars/laws. Ex: Tan eq. Relevance: Abstraction.

Step 3: Solve

Math computation. Ex: Matrix mult. Relevance: Output.

Step 4: Interpret/Compare

With real data. Relevance: Agreement.

Step 5: Accept/Modify

Iterate if needed. Relevance: Improvement.

Parameter Improvement

Adjust values for better fit. Relevance: Optimization.

Sensitivity Analysis

How changes affect output. Relevance: Robustness (advanced).

Tip: Flashcards: Step on front, Ex+Relevance back. Depth: Derive tan model from similar triangles. Interlinks: Matrices (Ex2/3) to Ch3. Graphs: Cycle diagram. Flow: Principles → Steps → Apply.

Solved Examples from NCERT - Step-by-Step with MathJax (Ex1-3)

All examples solved in detail, grouped by type. Expanded with steps, diagrams, verification. Mirrors book.

Example 1: Height of Tower (Trigonometry Model)

Step 1: Physical: Find tower height.
Step 2: Vars: H=height, h=eye ht, l=dist, α=elev: \( \tan \alpha = \frac{H - h}{l} \) → \( H = h + l \tan \alpha \).
Step 3: Solve with measured h,l,α.
Step 4: If l unknown, β=depress: \( l = h \cot \beta \), sub.
Step 5: Exact, accept.

Solution: Eq (1): H = h + l tan α. Verification: Measure actual H, error <5% typical.

Example 2: Raw Materials Model (Matrix)

Step 1: Firm P1-3, raw R1-3, orders F1-2.
Step 2: A=orders (2x3), B=per unit (3x3).
Step 3: AB=total needed (2x3).
Step 4: Compare avail.
Step 5: If exceed, modify orders.

Solution: AB gives R1,R2,R3 for each F. Ex: A=[[3,4,0],[10,15,6]], B=[[7,9,3],[10,20,0],[5,12,7]] → AB=[[165,247,87],[170,220,60]].

Example 3: Interpret Ex2 with Data

Avail: R1=330, R2=455, R3=140. AB needs 335>330 R1, etc. Reduce A to A1=[[9,12,6],[10,20,0]] → A1B=[[141,216,78],[170,220,60]] < avail.

Solution: Model suggests reduce orders for feasibility. Verification: Simulate production, matches.

Tip: Always iterate Step 5. For 2025: Model with matrices (4m), interpret (6m). Extend to LPP diet problems.

Interactive Quiz - Master Modelling (10 MCQs)

Test principles/steps/examples; aim 80%+. Instant feedback. 2025 focus.

Quick Revision Notes & Mnemonics (Table)

Concise 1-page; bold keys. Full appendix in 4 rows.

Subtopic	Key Points	Examples	Mnemonics/Tips
Intro & Why	Modelling: Real→Math (symbols/laws). Ex: Tower h, Sun temp. Need: Insight beyond algebra; predict inaccessible (pop2020, blood vol).	Ex: River width, shot angle, Earth mass.	IW: Intro Why. Tip: "Real Mess → Math Clean; Try Non-Math First!".
Principles	8 Principles: Need→Vars→Data→Assum→Laws→Eqs/Calc/Sol→Tests(Consist/Util)→Improve Params.	Vars: h,l,α; Assum: Flat ground.	NVDALECUT: Need Vars Data Assum Laws Eq Calc Util Tests. Tip: Philosophical base.
Steps	1: Situation. 2: Model (vars/laws). 3: Solve. 4: Interpret/Compare. 5: Accept/Modify (iterate).	Ex1: Tan eq solve. Ex2: AB matrix.	SCMS A: Situation Convert Model Solve Accept. Tip: Cycle: No Agree → Loop 2.
Examples	Ex1: Trig model H=h+l tanα. Ex2/3: Matrix AB=resources; interpret/reduce if exceed.	Ex3: 335>330 R1 → Reduce orders.	TM: Trig Matrix. Tip: Validate: Compare predicted vs measured; sensitivity for params.

Overall: IW-NVDALECUT-SCMSA-TM. Flashcards per row. 2025: Steps apply (6m), matrix model (4m).

Complete Summary and Solutions for Mathematical Modelling – NCERT Class XII Mathematics Part I Appendix 2 – Introduction, Types, Construction, Applications

Mathematical Modelling

Full Appendix Summary & Detailed Notes - Mathematical Modelling NCERT Appendix 2

A.2.1 Introduction

Conceptual Diagram: Mathematical Modelling Cycle (Like Fig A.2.1)

A.2.2 Why Mathematical Modelling?

Why This Guide Stands Out (For 2025 Exams)

A.2.3 Principles of Mathematical Modelling

Modelling Steps Derivation (Iterative Process)

Example 1: Height of Tower (Integrated)

Diagram: Tower Height (Fig A.2.2)

Example 2: Raw Materials for Products (Integrated)

Example 3: Interpret Model (Integrated)

Key Principles, Steps & Terms - Complete Glossary (20+ Terms)

Mathematical Modelling

Physical Situation

Parameters/Variables

Assumptions

Governing Principles

Mathematical Model

Solution

Interpretation

Consistency Test

Utility Test

Iteration

Step 1: Identify Situation

Step 2: Convert to Model

Step 3: Solve

Step 4: Interpret/Compare

Step 5: Accept/Modify

Parameter Improvement

Sensitivity Analysis

Solved Examples from NCERT - Step-by-Step with MathJax (Ex1-3)

Example 1: Height of Tower (Trigonometry Model)

Example 2: Raw Materials Model (Matrix)

Example 3: Interpret Ex2 with Data

Interactive Quiz - Master Modelling (10 MCQs)

Quick Revision Notes & Mnemonics (Table)

Group Discussions