Class 11 Maths Appendix 2 – Mathematical Modelling (NCERT Solutions with Examples) 

Introduction

Mathematical modelling is the process of translating real‑life problems into mathematical form to analyze and solve them. It is widely used in science, engineering, economics, and management. This appendix introduces the concept of modelling, steps involved, and examples such as measuring tower height, bridge problems, pendulum motion, diet optimization, and population growth.

Key Formulas

• Height of tower using trigonometry:

$$\mathrm{<b>h</b>}<b>=</b>\mathrm{<b>d</b>}<b>\cdot </b>\mathrm{<b>tan</b>}<b></b>\mathrm{<b>\theta </b>}$$

• Pendulum period:

$$\mathrm{<b>T</b>}<b>=</b>\mathrm{<b>2</b>}\mathrm{<b>\pi </b>}\sqrt{\frac{\mathrm{<b>l</b>}}{\mathrm{<b>g</b>}}}$$

• Linear programming cost function:

$$\mathrm{<b>z</b>}<b>=</b>\mathrm{<b>10</b>}\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>20</b>}\mathrm{<b>y</b>}$$

• Population growth model:

$$\mathrm{<b>P</b>}<b>(</b>\mathrm{<b>t</b>}<b>)</b><b>=</b>\mathrm{<b>P</b>}<b>(</b>\mathrm{<b>0</b>}<b>)</b><b>(</b>\mathrm{<b>1</b>}<b>+</b>\mathrm{<b>b</b>}<b>-</b>\mathrm{<b>d</b>}{<b>)</b>}^{\mathrm{<b>t</b>}}$$

 Solved NCERT Examples (Step by Step)

Example 1

Find height of tower when distance = 450 m, angle of elevation = 40°. Solution: $h=450\cdot \tan {40}^{\circ }=377.6m\approx 378m$.

Example 2 (Bridge Problem)

Euler proved that crossing all 7 bridges of Königsberg once is impossible because all vertices have odd degree. After adding one bridge, two vertices become even, making traversal possible.

Example 3 (Diet Optimization)

Minimize cost $z=10x+20y$ subject to constraints:

• $$x+y≥800

• $$0.09x+0.6y≥0.3(x+y)

• $$0.02x+0.06y≤0.05(x+y)

Solution: Optimal at $x=470.6,y=329.4$. Minimum cost = Rs 11,294.

Example 4 (Population Growth)

Initial population = 250,000,000, birth rate = 0.02, death rate = 0.01. Growth rate $r=1.01$. Population after 10 years: $P(10)=250,000,000\cdot (1.01{)}^{10}\approx 276,155,531$.

Geometry & Trigonometry Models

1. Find height of building if angle of elevation = 30°, distance = 200 m.

2. A ladder 10 m long makes angle 60° with ground. Find height reached.

3. A kite string of 100 m makes angle 45° with ground. Find height of kite.

4. A man observes top of pole at 60° from 50 m away. Find pole height.

5. A balloon rises vertically. Angle of elevation changes from 30° to 60° as observer walks 100 m. Find balloon height.

Graph Theory & Networks

6. Draw graph model for railway network between 5 cities.

7. Show why a graph with 4 odd vertices cannot have Eulerian path.

8. Add one edge to Königsberg problem to make traversal possible.

9. Model bus routes between 6 stops using vertices and edges.

10. Represent internet connections between 4 servers as graph.

Pendulum & Physics Models

11. Find period of pendulum of length 1 m (g=9.8).

12. Compare periods of pendulums of lengths 1 m and 4 m.

13. Show mass of bob does not affect pendulum period.

14. Find length of pendulum with period 2 sec.

15. If g changes to 9.6, find new period of 1 m pendulum.

Linear Programming & Optimization

16. Minimize cost of diet with wheat and rice subject to protein constraints.

17. A factory produces chairs and tables. Profit function given. Optimize production.

18. A farmer grows wheat and barley. Land and water constraints given. Maximize yield.

19. A transport company runs buses and trucks. Fuel constraints given. Minimize cost.

20. A school buys notebooks and pens. Budget constraint given. Optimize purchase.

Population & Growth Models

21. Population = 1,000,000, birth rate = 0.03, death rate = 0.01. Find after 5 years.

22. Show exponential growth curve for population model.

23. If birth rate = death rate, show population constant.

24. If death rate > birth rate, show population decreases.

25. Current population = 500,000, growth rate = 1.02. Find after 20 years.

Miscellaneous Real‑Life Models

26. Model spread of disease using exponential function.

27. Model cooling of hot object using Newton’s law of cooling.

28. Model traffic flow on highway using linear equations.

29. Model rainfall prediction using probability and statistics.

30. Model electricity consumption in city using regression.

Geometry & Trigonometry Models

1. A tower casts a shadow of 120 m when the sun’s elevation is 45°. Find the tower height.

2. A flagpole stands on a building 20 m high. From a point 30 m away, angle of elevation to top of pole is 60°. Find pole height.

3. A man standing 80 m away observes top of a minaret at 30°. Find minaret height.

4. A rope of length 50 m is tied to a peg on ground and makes 60° with ground. Find vertical height reached.

5. A boy flying a kite holds string 60 m long at angle 30°. Find height of kite above ground.

Graph Theory & Networks

1. Represent metro stations of a city as vertices and tracks as edges. Draw graph model.

2. Show that a graph with all vertices of even degree has Eulerian circuit.

3. Add an edge to a graph with 6 vertices to make Hamiltonian path possible.

4. Model airline routes between 4 cities using directed graph.

5. Represent friendship network of 5 students as graph.

Pendulum & Physics Models

1. Find period of pendulum of length 2 m (g=9.8).

2. Compare periods of pendulums of lengths 2 m and 8 m.

3. Show that pendulum period increases with length.

4. Find length of pendulum with period 3 sec.

5. If g changes to 10, find new period of 1 m pendulum.

Linear Programming & Optimization

1. A company makes pens and pencils. Profit function given. Optimize production.

2. A farmer grows maize and wheat. Land and fertilizer constraints given. Maximize yield.

3. A shopkeeper buys apples and oranges. Budget constraint given. Minimize cost.

4. A transport firm runs cars and vans. Fuel constraint given. Optimize usage.

5. A school buys chalks and markers. Budget constraint given. Optimize purchase.

Population & Growth Models

1. Population = 2,000,000, birth rate = 0.02, death rate = 0.01. Find after 10 years.

2. Show population growth curve when r=1.01.

3. If birth rate < death rate, show population decreases exponentially.

4. Current population = 100,000, growth rate = 1.05. Find after 15 years.

5. Show that if r=1, population remains constant.

15 FAQs with Step‑by‑Step Solutions (Compose Format)

Q1. What is mathematical modelling? Answer: It is the process of representing real‑life problems in mathematical form for analysis and solution.

Q2. What are the steps in modelling? Answer: Formulation, Solution, Interpretation, Validation.

Q3. How is tower height calculated using trigonometry? Answer: $h=d\cdot \tan \theta$.

Q4. Why was the Königsberg bridge problem unsolvable? Answer: All four vertices had odd degree; only two odd vertices allow traversal.

Q5. What formula gives pendulum period? Answer: $T=2\pi \sqrt{\frac{l}{g}}$.

Q6. Does pendulum period depend on mass? Answer: No, it depends only on length and gravity.

Q7. What is objective in diet optimization problem? Answer: Minimize cost while meeting nutrient constraints.

Q8. What is population growth model formula? Answer: $P(t)=P(0)(1+b-d{)}^t$.

Q9. What is validation in modelling? Answer: Comparing model results with real data to check accuracy.

Q10. Why do models have errors? Answer: Due to assumptions and simplifications like ignoring resistance or migration.

Q11. What is linear programming used for? Answer: Optimizing cost or profit under constraints.

Q12. What is Euler’s contribution in modelling? Answer: Introduced graph theory to solve bridge problems.

Q13. What is interpretation step in modelling? Answer: Relating mathematical solution back to real‑life context.

Q14. What is growth rate in population model? Answer: $r=1+b-d$.

Q15. Why is modelling important? Answer: It simplifies complex real‑life problems for better decision making.

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