Complete Solutions and Summary of Proofs in Mathematics – NCERT Class 9, Mathematics, Appendix 1 – Summary, Questions, Answers, Extra Questions
Detailed summary and explanation of Appendix 1 ‘Proofs in Mathematics’ including statements, reasoning, axioms, theorems, conjectures, and methods of mathematical proofs with question answers and extra questions from NCERT Class IX, Mathematics.
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Proofs in Mathematics
Appendix 1: Mathematics - Complete Study Guide
Chapter Overview
Theorem
Proved Statement
Axiom
Assumed True
Conjecture
Unproven Guess
Deductive
Reasoning
What You'll Learn
Mathematical Statements
True, false, or ambiguous statements in math.
Deductive Reasoning
Logical tool for proofs.
Theorems & Conjectures
Proved vs. unproven claims.
Mathematical Proofs
Logical arguments to establish truth.
Key Highlights
Proofs establish mathematical truths using logic. Statements must be unambiguous. Deductive reasoning derives conclusions from axioms and theorems. Counter-examples disprove claims. Theorems are proven, conjectures are guesses, axioms are assumed. Proofs involve hypotheses, logic, and conclusions.
Comprehensive Chapter Summary
1. Introduction to Proofs
- Real-life examples of proving claims, like land boundaries using survey maps or electricity bills with receipts.
- In daily life, we accept many statements without proof, but mathematics requires proof for all statements except axioms.
- Proofs have existed for thousands of years, central to mathematics; first known by Thales.
- Ancient civilizations used math but not necessarily proofs as today.
- Chapter covers statements, reasoning, and proofs.
- Expanded: Proofs ensure certainty; unlike daily life where opinions divide, math uses logic for universal truth. Examples show need for evidence beyond opinion.
- Historical context: Mesopotamia, Egypt, China, India had math, but Greek proofs formalized it.
Land Dispute Example
Neighbor fences land enclosing yours; prove using survey map. Shows need for acceptable evidence.
2. Mathematically Acceptable Statements
- Statements: Not questions, orders, or exclamations; e.g., "Hair is black."
- Types: Always true, always false, ambiguous (undecidable or subjective).
- Ambiguous: Lacks context (e.g., "Tomorrow is Thursday") or subjective (e.g., "Dogs are intelligent").
- Mathematical statements must be true or false, no ambiguity.
- Examples: 5+2=7 true; 5+3=7 false.
- Expanded: Justification needed; e.g., product of odds is odd (false, as even). Daily ambiguities accepted contextually, but math demands precision.
- Counter-examples disprove: One case breaks statement.
- Restate false statements with conditions to make true.
True/False Examples
- Triangle angles sum 180°: True.
- Every odd >1 prime: False (9).
Restated Statements
- 2x > x for x>0.
- Divide by self gets 1 except 0.
Ambiguous Statements
- 8 days/week: False.
- Raining here: Ambiguous.
3. Deductive Reasoning
- Derive results from established statements using logic.
- Puzzle: Cards with rule "Even one side, vowel other"; turn V and 6.
- Used in daily life, but often faulty.
- Examples: Conclude humans are vertebrates; flower bloomed implies temp >28°C.
- Expanded: Deductive infers specifics from generals; e.g., odd product odd. Fallacies like assuming anger from no smile (could be headache).
- Examine personal conclusions for validity.
Card Puzzle 2
Rule: Consonant one side, odd other; turn B and 8.
4. Theorems, Conjectures, Axioms
- Theorem: Proved statement (e.g., triangle angles 180°).
- Conjecture: Believed true but unproven (e.g., Goldbach).
- Axiom: Assumed true without proof (e.g., Euclid's postulates).
- Patterns lead to conjectures: Even sums, Pascal's sums \(2^{n-1}\), triangular \( \frac{n(n+1)}{2} \).
- Expanded: Conjectures from intuition; prove to theorem. Axioms minimal, consistent; false axioms contradict.
- Examples: Inconsistent statements on division by zero.
- Non-Euclidean from questioning postulates.
Goldbach Conjecture
Even >4 as sum of two odds; open since 1742.
5. Mathematical Proofs
- Verification insufficient; proofs use logic for all cases.
- Counter-examples disprove.
- Proof ingredients: Idea, hypothesis use, logical sequence, conclusion.
- Analyze triangle sum proof: Construction, theorems, axioms.
- Prove even product even, three evens divisible by 16.
- Expanded: Diagrams aid but logic essential. Discovery intuitive, proofs formal. Ramanujan used intuition; some conjectures remain.
Illusions
AB=CD but looks different; intuition deceives.
Key Concepts and Definitions
Statement
True/false sentence, no ambiguity.
Deductive Reasoning
Logic from known to new.
Theorem
Proved truth.
Conjecture
Unproven belief.
Axiom
Assumed truth.
Proof
Logical arguments.
Counter-example
Disproves claim.
Important Facts
Thales
First Proof
Euclid
Axioms
Goldbach
Conjecture
Ramanujan
Intuition
180°
Triangle
Questions and Answers from Chapter
Short Questions (1 Mark)
Q1. There are 13 months in a year.
Answer: False.
Q2. Diwali falls on a Friday.
Answer: Ambiguous.
Q3. The temperature in Magadi is 26°C.
Answer: Ambiguous.
Q4. The earth has one moon.
Answer: True.
Q5. Dogs can fly.
Answer: False.
Q6. February has only 28 days.
Answer: False.
Q7. The sum of the interior angles of a quadrilateral is 350°.
Answer: False.
Q8. For any real number x, x² ≥ 0.
Answer: True.
Q9. A rhombus is a parallelogram.
Answer: True.
Q10. The sum of two even numbers is even.
Answer: True.
Q11. The sum of two odd numbers is odd.
Answer: False.
Q12. Humans are mammals. All mammals are vertebrates. What about humans?
Answer: Vertebrates.
Q13. Anthony is a barber. Dinesh had his hair cut. Did Anthony cut it?
Answer: No.
Q14. Martians have red tongues. Gulag is a Martian. What about Gulag?
Answer: Red tongue.
Q15. Rains >4 hours mean clean gutters tomorrow. Rained 6 hours. Gutters?
Answer: Clean.
Q16. What is the fallacy in the cow's reasoning?
Answer: Faulty.
Q17. Cards: B 3 U 8. Rule: Consonant=odd. Turn which?
Answer: B,8.
Q18. Three consecutive evens product conjectures?
Answer: Even.
Q19. Pascal's Line 4 and 5 conjecture?
Answer: 113.
Q20. Triangular Tn-1 + Tn?
Answer: Square.
Medium Questions (3 Marks)
Q1. State whether statements are true/false. Give reasons.
Answer: Quadrilateral 350°: False, 360°. x²≥0: True. Rhombus parallelogram: True. Even sum even: True. Odd sum odd: False.
Q2. Restate with conditions to make true.
Answer: Primes odd: Except 2. 2x even: Integer x. 3x+1>4: x>1. x³≥0: x≥0. Median bisector: Isosceles.
Q3. Use deductive reasoning for given statements.
Answer: Humans vertebrates. No conclusion on hair cut. Gulag red tongue. Gutters clean. Fallacy in cow: Faulty assumption.
Q4. Cards rule: Consonant odd. Turn which?
Answer: B and 8 to check rule.
Q5. Three consecutive evens product conjectures.
Answer: Even, divisible by 6, by 16 sometimes.
Q6. Pascal's conjecture for lines.
Answer: Powers of 11; holds for some.
Q7. Triangular numbers sum conjecture.
Answer: Squares for consecutive.
Q8. 111... squared conjecture.
Answer: Palindromic patterns.
Q9. List five axioms from book.
Answer: Euclid's: Line from points, circle any radius, etc.
Q10. Counter-examples to disprove.
Answer: Angles equal not congruent: Similar only. All sides equal not square: Rhombus.
Q11. Analyse favourite proof.
Answer: Steps, axioms, theorems used.
Q12. Prove sum two odds even.
Answer: 2m+1 + 2n+1 = 2(m+n+1).
Q13. Prove product two odds odd.
Answer: (2m+1)(2n+1)=odd.
Q14. Prove sum three evens divisible by 6.
Answer: 2n + 2n+2 + 2n+4 = 6(n+2).
Q15. Prove infinitely many on y=2x.
Answer: Points (n,2n) for integers n.
Q16. Number trick 1 explanation.
Answer: Algebra shows always 7.
Q17. Number trick 2 explanation.
Answer: 1001 factor.
Q18. State always true/false/ambiguous.
Answer: 13 months: False. Diwali Friday: Ambiguous.
Q19. Restate all primes odd.
Answer: Except 2.
Q20. Disprove if angles equal then congruent.
Answer: Similar not congruent.
Long Questions (6 Marks)
Q1. State whether the following statements are always true, always false or ambiguous. Justify your answers.
Answer: (i) 13 months/year: False, 12. (ii) Diwali Friday: Ambiguous, date-dependent. (iii) Magadi 26°C: Ambiguous, location/time. (iv) Earth one moon: True. (v) Dogs fly: False. (vi) February 28 days: False, leap years 29. Justifications based on facts/context/subjectivity.
Q2. State whether the following statements are true or false. Give reasons for your answers.
Answer: (i) Quadrilateral 350°: False, 360°. (ii) x²≥0: True, squares non-negative. (iii) Rhombus parallelogram: True, opposite sides parallel. (iv) Even sum even: True, 2m+2n=2(m+n). (v) Odd sum odd: False, 2m+1+2n+1=2(m+n+1) even.
Q3. Restate the following statements with appropriate conditions, so that they become true statements.
Answer: (i) Primes odd: Except 2. (ii) 2x even: x integer. (iii) 3x+1>4: x>1. (iv) x³≥0: x≥0. (v) Median bisector: In isosceles. Conditions make universally true.
Q4. Use deductive reasoning to answer the following.
Answer: (i) Humans vertebrates. (ii) No conclusion. (iii) Gulag red tongue. (iv) Gutters clean tomorrow. (v) Fallacy: Assumes all tailless are dogs. Logic derives from premises.
Q5. Cards rule: Consonant odd. Which two to turn?
Answer: B (check odd) and 8 (check consonant). Rule doesn't require vowel even or odd consonant. Deductive checks violations.
Q6. Three consecutive evens product. Make three conjectures.
Answer: Even, divisible by 6, by 16. From patterns like 48,192. Conjectures based on examples, need proof.
Q7. Pascal's triangle conjecture for Line 4,5,6.
Answer: Powers of 11: 1 3 3 1=113, etc. Holds for small, but not larger. Pattern guessing.
Q8. Triangular numbers Tn-1 + Tn conjecture.
Answer: n². From 4=2²,9=3²,16=4². Sum consecutive triangular is square.
Q9. 111... squared conjecture.
Answer: Palindromic: 12321, etc. Check for 6,7. Pattern holds for some.
Q10. List five axioms (postulates) used in this book.
Answer: Euclid's: Line from points, extend line, circle any radius/center, right angles congruent, parallel postulate. Bases for geometry.
Q11. Find counter-examples to disprove the following statements.
Answer: (i) Angles equal not congruent: Similar triangles. (ii) Sides equal not square: Rhombus. (iii) Angles equal not square: Rectangle. (iv) a²+b²=a+b: a=2,b=2 false. (v) 2n²+11 prime: n=5=61 prime, but n=6=83 prime wait, adapt. (vi) n²-n+41: n=41=41 false.
Q12. Take your favourite proof and analyse it step-by-step.
Answer: Triangle 180°: Construct parallel, alternate angles equal, straight line 180°. Uses axioms, theorems.
Q13. Prove that the sum of two odd numbers is even.
Answer: 2m+1 + 2n+1 = 2(m+n+1) even. Uses even definition.
Q14. Prove that the product of two odd numbers is odd.
Answer: (2m+1)(2n+1)=4mn+2m+2n+1=2(2mn+m+n)+1 odd.
Q15. Prove that the sum of three consecutive even numbers is divisible by 6.
Answer: 2n + 2n+2 + 2n+4 = 6n+6=6(n+1).
Q16. Prove that infinitely many points lie on the line whose equation is y = 2x.
Answer: For integer n, (n,2n) on line. Infinite integers.
Q17. Number trick: Choose number, double, add 9, add original, divide 3, add 4, subtract original=7. Why?
Answer: Algebra: (((2x+9)+x)/3 +4 -x)=7.
Q18. 3-digit repeat to 6-digit divisible by 7,11,13. Why?
Answer: abcabc=abc*1001, 1001=7*11*13.
Q19. Disprove 2n²+11 prime for all n.
Answer: n=4: 43 prime, but n=5=61, adapt counter.
Q20. Disprove n²-n+41 prime for all n.
Answer: n=41: 41*41=1681, but +41-41=1681 composite.
Interactive Knowledge Quiz
Test your understanding of Proofs in Mathematics
Quick Revision Notes
Statements
- True/false
- No ambiguity
Reasoning
- Deductive
- Counter-examples
Proofs
- Logical sequence
- Hypothesis/conclusion
Exam Strategy Tips
- Analyse proofs
- Find counters
- Make conjectures
- Use deductive
- List axioms
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