Mathematical Modelling
Appendix 2: Mathematics - Ultimate Study Guide | NCERT Class 12 Notes, Solved Examples, Quiz 2025
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).
