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CBSE · Class 12 · Mathematics

Mathematical Modelling (Appendix 2)

2 concepts · 16 practice questions with answers · Free to read

What a model is and the steps of modelling

From daily life

Measuring the height of a tall tower with a tape is impossible. But with an angle and a distance we can build a small mathematical picture of the situation and get the answer. That picture is a model.

Simple definition

Mathematical modelling converts a real situation into mathematics, solves it, and checks the answer. Steps: (1) identify the physical situation, (2) formulate the model with variables and assumptions, (3) solve the mathematics, (4) interpret and compare with observation, (5) if it does not agree, change the assumptions and repeat.

Formula: Situation → Model → Solution → Interpretation → Validation

Rule 1: Build a simple model — State the variables and assumptions.

  • Easy: A car travels at a constant 60 km/h: the distance in 2.5 hours is 60 × 2.5 = 150 km.
  • Medium: A town of 1000 people grows 2% per year: P = 1000 (1.02)ᵗ, which is about 1061 after 3 years.
  • Exam-type: Tower height H = h + l tan α, with eye height h = 1.6 m, distance l = 30 m, α = 30°: H = 1.6 + 30 × 0.577 = about 18.9 m.

Rule 2: Check the model — Compare with data and improve the assumptions.

  • Easy: If the car had to slow down in traffic, the "constant speed" assumption would fail and the model needs changing.
  • Medium: If the town's real population differs from the formula, change the growth rate.
  • Exam-type: If the foot of the tower is not accessible, measure the angle of depression β of its foot and use l = h cot β.
Remember

A model is only as good as its assumptions; always say what you assumed.

Common mistake

Giving the answer without stating the assumptions or checking it.

Memory clue

Situation, symbols, solve, interpret, check, repeat if needed.

Practice questions

Easy
  1. 1What is the first step in mathematical modelling?
  2. 2What do we do if the result of a model does not agree with observation?
  3. 3A car travels at a constant 50 km/h. Find the distance in 3.5 hours.
Medium
  1. 4The eye of an observer is 1.5 m above the ground. The tower is 40 m away and the angle of elevation of its top is 45°. Find the height of the tower.
  2. 5An observer on a balcony 20 m above the ground sees the foot of a tower at an angle of depression 45°. Find the horizontal distance l.
  3. 6With l from the previous question, the angle of elevation of the top of the tower from the balcony is 30°. Find the tower height H = h + l tan α (take tan 30° = 0.577).
Challenging
  1. 7A population of 5000 grows 2% per year. Find it after 2 years.
  2. 8Why is validation an important step?
Show answers (try them yourself first)
  1. 1Identify the physical situation
  2. 2Change the assumptions and repeat from the formulation step
  3. 350 × 3.5 = 175 km
  4. 4H = 1.5 + 40 × tan 45° = 41.5 m
  5. 5l = h cot β = 20 × cot 45° = 20 m
  6. 6H = 20 + 20 × 0.577 = about 31.5 m
  7. 75000 × (1.02)² = 5000 × 1.0404 = 5202
  8. 8It checks whether the model's predictions match real observations, so wrong assumptions can be corrected

Modelling with matrices and calculus

From daily life

A factory must work out how much raw material to buy for the orders of two shops. A table of orders times a table of requirements gives the answer at once.

Simple definition

Matrix models organise data in tables and use multiplication to combine them. Calculus models write a quantity as a function of one variable and use the derivative to find its best value.

Formula: Raw material needed = (orders matrix) × (requirements matrix)

Rule 1: A matrix model — Multiply the order table by the requirement table.

  • Easy: Orders A = [4 2 ; 3 5] (rows: two clients; columns: products P1, P2). Requirements B = [2 1 ; 1 3] (rows: products; columns: materials R1, R2). AB = [10 10 ; 11 18].
  • Medium: So client 1 needs 10 units of R1 and 10 of R2, and client 2 needs 11 and 18.
  • Exam-type: The total need is 21 units of R1 and 28 units of R2.

Rule 2: A calculus model — Express, differentiate, interpret.

  • Easy: 40 m of fence: area A = x (20 − x) is largest at x = 10, so A = 100 m².
  • Medium: A profit model P(x) = −x² + 14x − 24 has P′ = −2x + 14 = 0 at x = 7, so P = 25.
  • Exam-type: The answer must be reasonable: here 0 < x < 20.
Remember

Always interpret the answer in the words of the original problem.

Common mistake

Stopping at the number without saying what it means.

Memory clue

Table times table gives total needs.

Practice questions

Easy
  1. 1With A = [4 2 ; 3 5] and B = [2 1 ; 1 3], find the materials needed by client 1.
  2. 2Find the materials needed by client 2.
  3. 3Find the total amount of R1 needed for both clients.
Medium
  1. 4R1 costs ₹5 and R2 costs ₹3 per unit. Find the material cost for client 2.
  2. 5A firm's cost is C = x² − 8x + 30. Find the least cost.
  3. 6A rectangular plot has perimeter 60 m. Find the largest area.
Challenging
  1. 7Profit is P(x) = −2x² + 24x − 30. Find the maximum profit.
  2. 8Why must a model be validated even if the mathematics is correct?
Show answers (try them yourself first)
  1. 1Row 1 of AB: [10 10]
  2. 2Row 2 of AB: [11 18]
  3. 310 + 11 = 21
  4. 411 × 5 + 18 × 3 = 55 + 54 = ₹109
  5. 5C′ = 2x − 8 = 0 at x = 4, so C = 14
  6. 6Square 15 m × 15 m, area 225 m²
  7. 7P′ = −4x + 24 = 0 at x = 6, so P = −72 + 144 − 30 = 42
  8. 8The assumptions may not match the real world (for example fixed prices), so the result must be checked against real data