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Mathematical Reasoning – CBSE NCERT Study Resources

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11th

11th - Mathematics

Mathematical Reasoning

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Overview of the Chapter: Mathematical Reasoning

This chapter introduces the fundamental concepts of mathematical reasoning, which is a critical skill for solving problems logically and systematically. It covers statements, their truth values, logical connectives, and methods of validating arguments. The chapter also explores quantifiers, implications, and the differences between inductive and deductive reasoning.

Mathematical Reasoning: The process of using logical thinking to analyze and solve mathematical problems.

Key Topics Covered

  • Statements and their truth values
  • Logical connectives (AND, OR, NOT, IMPLIES, IF AND ONLY IF)
  • Quantifiers (Universal and Existential)
  • Validating statements and arguments
  • Implications and contrapositive statements
  • Inductive and deductive reasoning

Detailed Explanation

1. Statements

A statement is a declarative sentence that is either true or false but not both. For example, "The sum of two even numbers is even" is a true statement.

Statement: A sentence that can be classified as true or false without ambiguity.

2. Logical Connectives

Logical connectives are used to combine or modify statements:

  • AND (∧): True only if both statements are true.
  • OR (∨): True if at least one statement is true.
  • NOT (¬): Negates the truth value of a statement.
  • IMPLIES (→): False only when the first statement is true and the second is false.
  • IF AND ONLY IF (↔): True when both statements have the same truth value.

3. Quantifiers

Quantifiers specify the scope of a statement:

  • Universal Quantifier (∀): "For all" or "For every."
  • Existential Quantifier (∃): "There exists" or "For some."

4. Validating Arguments

An argument consists of premises and a conclusion. It is valid if the conclusion logically follows from the premises.

Valid Argument: An argument where the conclusion is true whenever the premises are true.

5. Implications and Contrapositive

An implication "p → q" has a contrapositive "¬q → ¬p," which is logically equivalent to the original implication.

6. Inductive and Deductive Reasoning

  • Inductive Reasoning: Derives general principles from specific observations.
  • Deductive Reasoning: Applies general principles to specific cases to reach a conclusion.

Conclusion

Mathematical reasoning is essential for developing problem-solving skills in mathematics. Understanding statements, logical connectives, and validation methods helps in constructing and analyzing mathematical arguments effectively.

All Question Types with Solutions – CBSE Exam Pattern

Explore a complete set of CBSE-style questions with detailed solutions, categorized by marks and question types. Ideal for exam preparation, revision and practice.

Very Short Answer (1 Mark) – with Solutions (CBSE Pattern)

These are 1-mark questions requiring direct, concise answers. Ideal for quick recall and concept clarity.

Question 1:
Define a mathematical statement.
Answer:

A sentence which is either true or false.

Question 2:
What is the negation of 'All birds can fly'?
Answer:

Some birds cannot fly.

Question 3:
Identify the contrapositive of 'If it rains, then the ground is wet'.
Answer:

If the ground is not wet, then it did not rain.

Question 4:
State the converse of 'If a number is even, then it is divisible by 2'.
Answer:

If a number is divisible by 2, then it is even.

Question 5:
Is 'The square root of 2 is irrational' a mathematical statement?
Answer:

Yes, it is a true statement.

Question 6:
Write the negation of 'For every real number x, x² ≥ 0'.
Answer:

There exists a real number x such that x² < 0.

Question 7:
What is the truth value of '3 + 5 = 8 and 2 + 7 = 10'?
Answer:

False, because the second part is false.

Question 8:
Give an example of a tautology.
Answer:

'It is raining or it is not raining.'

Question 9:
Is 'x + 5 = 10' a mathematical statement? Why?
Answer:

No, because its truth depends on the value of x.

Question 10:
What is the contradiction of 'A triangle has four sides'?
Answer:

'A triangle does not have four sides.'

Question 11:
State the inverse of 'If a number is prime, then it has no divisors other than 1 and itself'.
Answer:

If a number is not prime, then it has divisors other than 1 and itself.

Question 12:
Is 'Every rectangle is a square' true or false?
Answer:

False, because not all rectangles have equal sides.

Question 13:
Define a mathematical statement with an example.
Answer:

A mathematical statement is a declarative sentence that is either true or false but not both.
Example: 'The sum of 3 and 5 is 8.' (True)

Question 14:
What is the negation of the statement: 'All prime numbers are odd'?
Answer:

The negation is: 'There exists at least one prime number that is not odd.'
Note: 2 is a prime number and even, making the original statement false.

Question 15:
Identify the quantifier in the statement: 'There exists a real number x such that x² = 2.'
Answer:

The quantifier is 'There exists', which is an existential quantifier (∃).

Question 16:
Write the contrapositive of: 'If a triangle is equilateral, then it is isosceles.'
Answer:

The contrapositive is: 'If a triangle is not isosceles, then it is not equilateral.'
Note: Contrapositive has the same truth value as the original statement.

Question 17:
State whether the following is a statement: 'Solve the equation x² + 1 = 0.'
Answer:

No, it is not a statement because it is an imperative sentence (command) and cannot be classified as true or false.

Question 18:
What is the converse of: 'If you are over 18, then you can vote.'?
Answer:

The converse is: 'If you can vote, then you are over 18.'
Note: Converse may not always share the truth value of the original statement.

Question 19:
Give an example of a tautology in mathematical reasoning.
Answer:

Example: 'It is raining or it is not raining.'
This is always true regardless of the truth value of individual components.

Question 20:
Identify the connectives used in: 'If it rains, then the ground will be wet.'
Answer:

The connective used is 'If...then...', which represents a conditional statement (→).

Question 21:
Is the statement '√2 is a rational number.' true or false? Justify.
Answer:

The statement is false.
Justification: √2 cannot be expressed as p/q where p and q are integers (q ≠ 0), as proven by contradiction.

Question 22:
What is the inverse of the statement: 'If a number is divisible by 6, then it is divisible by 3.'?
Answer:

The inverse is: 'If a number is not divisible by 6, then it is not divisible by 3.'
Note: Inverse may not always be logically equivalent to the original statement.

Very Short Answer (2 Marks) – with Solutions (CBSE Pattern)

These 2-mark questions test key concepts in a brief format. Answers are expected to be accurate and slightly descriptive.

Question 1:
Define a statement in mathematical reasoning with an example.
Answer:

A statement is a declarative sentence that is either true or false but not both.
Example: 'The sum of 3 and 5 is 8.' This is a true statement.

Question 2:
What is the negation of the statement: 'All prime numbers are odd.'?
Answer:

The negation is: 'Not all prime numbers are odd.'
This means at least one prime number is not odd, such as 2, which is even.

Question 3:
Identify the quantifier in the statement: 'There exists a real number x such that x² = 4.'
Answer:

The quantifier here is 'There exists', which indicates the existence of at least one real number satisfying the condition.

Question 4:
Write the converse of the statement: 'If a number is divisible by 6, then it is divisible by 2 and 3.'
Answer:

The converse is: 'If a number is divisible by 2 and 3, then it is divisible by 6.'
This is also a true statement.

Question 5:
What is a tautology in mathematical reasoning? Give an example.
Answer:

A tautology is a statement that is always true, regardless of the truth values of its components.
Example: 'It is raining or it is not raining.' This is always true.

Question 6:
Check whether the statement 'If x is an integer, then x² ≥ x.' is true or false. Justify.
Answer:

The statement is false.
Counterexample: For x = 0.5 (not an integer), x² = 0.25 < 0.5.
However, if restricted to integers, it holds for all x except x = 0 and x = 1.

Question 7:
Write the contrapositive of the statement: 'If a triangle is equilateral, then it is isosceles.'
Answer:

The contrapositive is: 'If a triangle is not isosceles, then it is not equilateral.'
This is logically equivalent to the original statement.

Question 8:
What is the difference between a necessary and a sufficient condition? Give an example.
Answer:

A necessary condition must be true for a statement to hold, but alone may not guarantee it.
A sufficient condition guarantees the statement's truth.
Example: Being a square (sufficient) implies being a rectangle, but being a rectangle (necessary) does not imply being a square.

Question 9:
Verify if the statement 'For all real numbers x, x² > x.' is true or false.
Answer:

The statement is false.
Counterexample: For x = 0.5, x² = 0.25 < 0.5.
It only holds for x < 0 or x > 1.

Question 10:
Explain the term validity in mathematical reasoning with an example.
Answer:

Validitiy refers to whether an argument's conclusion logically follows from its premises.
Example: Premise 1: All humans are mortal. Premise 2: Socrates is a human. Conclusion: Socrates is mortal.
This is a valid argument.

Short Answer (3 Marks) – with Solutions (CBSE Pattern)

These 3-mark questions require brief explanations and help assess understanding and application of concepts.

Question 1:
Define mathematical statement and give two examples of statements and two examples of non-statements.
Answer:

A mathematical statement is a declarative sentence that is either true or false but not both. It must be unambiguous and verifiable.


Examples of statements:
1. 'The sum of angles in a triangle is 180 degrees.' (True)
2. '7 is an even number.' (False)
Examples of non-statements:
1. 'What is your name?' (Interrogative, not declarative)
2. 'Let’s go to the park.' (Imperative, not verifiable)
Question 2:
Explain the difference between a tautology and a contradiction with an example of each.
Answer:

A tautology is a compound statement that is always true, regardless of the truth values of its components. A contradiction is always false.


Example of tautology:
'p ∨ ¬p' (p OR not p) is always true.
Example of contradiction:
'p ∧ ¬p' (p AND not p) is always false.
Question 3:
Write the converse, inverse, and contrapositive of the statement: 'If a number is divisible by 6, then it is divisible by 2.'
Answer:

Given statement (p → q): 'If a number is divisible by 6, then it is divisible by 2.'


Converse (q → p):
'If a number is divisible by 2, then it is divisible by 6.'
Inverse (¬p → ¬q):
'If a number is not divisible by 6, then it is not divisible by 2.'
Contrapositive (¬q → ¬p):
'If a number is not divisible by 2, then it is not divisible by 6.'
Question 4:
Verify by truth table whether the statement '(p ∧ q) → p' is a tautology, contradiction, or contingency.
Answer:

Constructing the truth table:


| p | q | p ∧ q | (p ∧ q) → p |
|---|---|-------|-------------|
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |

Since '(p ∧ q) → p' is always true, it is a tautology.

Question 5:
Identify the quantifier in the statement 'There exists a real number x such that x² = 2' and rewrite it using the other quantifier.
Answer:

The quantifier in the given statement is 'There exists' (existential quantifier, ∃).


Rewritten using universal quantifier (∀):
'It is not true that for all real numbers x, x² ≠ 2.'

This maintains the original meaning by negating the universal statement.

Question 6:
Define a mathematical statement and give an example of a statement that is true and one that is false.
Answer:

A mathematical statement is a declarative sentence that is either true or false but not both. It must be unambiguous and verifiable.


Example of a true statement: 'The sum of the angles in a triangle is 180 degrees.'
Example of a false statement: 'The square root of 16 is 5.'
Question 7:
Explain the difference between a tautology and a contradiction with suitable examples.
Answer:

A tautology is a statement that is always true, regardless of the truth values of its components. A contradiction is a statement that is always false.


Example of tautology: 'It is raining or it is not raining.' (p ∨ ¬p)
Example of contradiction: 'It is raining and it is not raining.' (p ∧ ¬p)
Question 8:
What is the converse of the statement: 'If a number is divisible by 6, then it is divisible by 2 and 3.'? Is the converse true? Justify.
Answer:

The converse of the given statement is: 'If a number is divisible by 2 and 3, then it is divisible by 6.'


The converse is true. Justification:


If a number is divisible by 2 and 3, it means it is a multiple of both.
Since 2 and 3 are co-prime, their LCM is 6.
Thus, the number must be divisible by 6.
Question 9:
Using truth tables, verify whether the statement '(p → q) ↔ (¬q → ¬p)' is a tautology or not.
Answer:

To verify, construct the truth table for (p → q) ↔ (¬q → ¬p):


  • Column 1: p (T, T, F, F)
  • Column 2: q (T, F, T, F)
  • Column 3: p → q (T, F, T, T)
  • Column 4: ¬q (F, T, F, T)
  • Column 5: ¬p (F, F, T, T)
  • Column 6: ¬q → ¬p (T, F, T, T)
  • Column 7: (p → q) ↔ (¬q → ¬p) (T, T, T, T)

Since the final column is always true, the statement is a tautology.

Question 10:
Identify the quantifiers in the statement: 'For every real number x, there exists a real number y such that x + y = 0.' Rewrite the statement without using quantifiers.
Answer:

The quantifiers in the statement are:


  • Universal quantifier: 'For every' (∀)
  • Existential quantifier: 'There exists' (∃)

Statement without quantifiers:


'Every real number x has a real number y as its additive inverse such that x + y = 0.'

Long Answer (5 Marks) – with Solutions (CBSE Pattern)

These 5-mark questions are descriptive and require detailed, structured answers with proper explanation and examples.

Question 1:
Using mathematical induction, prove that the sum of the first n odd natural numbers is equal to n². Justify each step with logical reasoning.
Answer:
Theoretical Framework

We studied mathematical induction in class, which involves proving a statement for a base case and then assuming it holds for k to prove it for k+1.


Evidence Analysis
  • Base Case: For n=1, sum = 1 = 1². Verified.
  • Inductive Step: Assume true for n=k, i.e., 1+3+...+(2k-1) = k². For n=k+1, sum becomes k² + (2(k+1)-1) = k² + 2k + 1 = (k+1)².

Critical Evaluation

The proof aligns with our textbook’s example of induction, ensuring validity. The assumption and extension are logically sound.

Question 2:
A survey shows 60% of students prefer online learning. If 200 students are selected randomly, find the probability that exactly 120 prefer it. Use binomial distribution and validate assumptions.
Answer:
Theoretical Framework

Our textbook defines binomial distribution as P(X=r) = C(n,r) pʳ (1-p)ⁿ⁻ʳ, where p=0.6, n=200, and r=120.


Evidence Analysis
  • Calculation: P(X=120) = C(200,120) (0.6)¹²⁰ (0.4)⁸⁰ ≈ 0.051 (using approximation).
  • Assumptions: Trials are independent, and p is constant.

Critical Evaluation

The result is consistent with binomial properties. Large n suggests normal approximation, but exact calculation is preferred for precision.

Question 3:
Using mathematical induction, prove that the sum of the first n odd natural numbers is n². Include all steps of the proof.
Answer:
Theoretical Framework

We studied mathematical induction in class, which involves proving a statement for all natural numbers by verifying the base case and inductive step.


Evidence Analysis
  • Base Case (n=1): Sum = 1 = 1², which holds true.
  • Inductive Step: Assume true for n=k, i.e., 1+3+...+(2k-1) = k². For n=k+1, sum becomes k² + (2(k+1)-1) = k² + 2k + 1 = (k+1)².

Critical Evaluation

Our textbook shows induction is rigorous for such proofs. The algebraic manipulation confirms the pattern holds.


Future Implications

This proof reinforces the importance of induction in validating sequences and series.

Question 4:
A survey shows 60% of students like algebra, 40% like geometry, and 20% like both. Using Venn diagrams, find the percentage who like only algebra. Validate with set theory.
Answer:
Theoretical Framework

We studied Venn diagrams and set theory to analyze overlapping data. Let A = algebra lovers (60%), B = geometry lovers (40%), and A∩B = both (20%).


Evidence Analysis
  • Only Algebra (A - A∩B): 60% - 20% = 40%.
  • [Diagram: Two overlapping circles labeled A and B, with 40% in A alone, 20% in intersection, and 20% in B alone.]

Critical Evaluation

Our textbook confirms the formula P(A only) = P(A) - P(A∩B). The result matches the diagram.


Future Implications

This method is vital for real-world surveys, ensuring accurate data segmentation.

Question 5:
A survey shows 60% of students prefer online learning, while 40% favor offline classes. Represent this data using logical connectives and analyze the validity of the conclusion: 'Majority of students prefer online learning.'
Answer:
Theoretical Framework

We studied logical statements where p represents 'students prefer online learning' and q for offline. The survey data implies p is true for 60% and q for 40%.


Evidence Analysis
  • Logical Representation: p ∨ q is true, but p dominates as 60% > 50%.
  • Validity: The conclusion is valid since 60% constitutes a majority.

Critical Evaluation

Our textbook defines a majority as >50%, so the conclusion is mathematically sound. However, real-world factors like sample size could affect generalizability.

Question 6:
Using mathematical induction, prove that the sum of the first n odd natural numbers is n². Explain the significance of this result in combinatorics.
Answer:
Theoretical Framework

We studied mathematical induction in class, which involves proving a base case and an inductive step. Here, the statement is P(n): 1 + 3 + 5 + ... + (2n-1) = n².


Evidence Analysis
  • Base Case (n=1): LHS = 1, RHS = 1² = 1. Hence, P(1) is true.
  • Inductive Step: Assume P(k) is true. For P(k+1), LHS = k² + (2(k+1)-1) = k² + 2k + 1 = (k+1)². Thus, P(k+1) holds.

Critical Evaluation

Our textbook shows this proof aligns with the principle of induction. The result is foundational in combinatorics, as it models perfect square distributions.

Question 7:
A compound statement uses logical connectives 'AND' and 'OR'. Construct truth tables for (p ∧ q) ∨ (~p) and analyze its tautology status. How is this useful in computer science?
Answer:
Theoretical Framework

We learned that truth tables enumerate all possible truth values of propositions. Here, we evaluate (p ∧ q) ∨ (~p).


Evidence Analysis
pqp ∧ q~p(p ∧ q) ∨ (~p)
TTTFT
TFFFF
FTFTT
FFFTT

Critical Evaluation

The statement is not a tautology (not always true). In computer science, such logic gates optimize circuit designs by minimizing redundancies.

Question 8:
Using mathematical induction, prove that the sum of the first n odd natural numbers is n². Justify each step with logical reasoning.
Answer:
Theoretical Framework

We studied mathematical induction in class, which involves proving a statement for all natural numbers. The base case verifies the statement for n=1, and the inductive step assumes it holds for n=k and proves for n=k+1.


Evidence Analysis
  • Base Case: For n=1, sum = 1 = 1² (True).
  • Inductive Hypothesis: Assume sum of first k odd numbers = k².
  • Inductive Step: For n=k+1, sum = k² + (2(k+1)-1) = k² + 2k + 1 = (k+1)².

Critical Evaluation

Our textbook shows induction is rigorous for such proofs. The algebraic manipulation confirms the pattern holds universally.

Question 9:
A survey of 100 students shows 60 like cricket, 50 like football, and 30 like both. Using Venn diagrams and set theory, find the number of students who like neither sport. Validate your answer with a diagram.
Answer:
Theoretical Framework

We studied union and intersection of sets. Let C = cricket lovers (60), F = football lovers (50), and C∩F = both (30). The formula for neither is Total - (C + F - C∩F).


Evidence Analysis
  • Total students = 100.
  • Students liking either sport = 60 + 50 - 30 = 80.
  • Neither = 100 - 80 = 20.

Critical Evaluation

[Diagram: Two overlapping circles labeled C (60) and F (50), intersection shaded as 30.] The Venn diagram visually confirms our calculation aligns with set theory principles.

Question 10:
Using mathematical induction, prove that the sum of the first n odd natural numbers is n². Include all steps of the proof with clear justification.
Answer:
Theoretical Framework

We studied mathematical induction in class, which involves proving a statement for all natural numbers by verifying the base case and inductive step.


Evidence Analysis
  • Base Case (n=1): Sum = 1 = 1². Verified.
  • Inductive Step: Assume true for n=k, i.e., 1+3+...+(2k-1) = k². For n=k+1, sum becomes k² + (2(k+1)-1) = k² + 2k + 1 = (k+1)². Hence, proved.

Critical Evaluation

Our textbook shows this proof as foundational for understanding series summation. The inductive step ensures the statement holds for all n.

Question 11:
A logical equivalence problem: Show that (p → q) ∧ (q → r) is equivalent to (p → r) using truth tables. Justify each step.
Answer:
Theoretical Framework

We learned that logical equivalence can be verified using truth tables by comparing outputs for all input combinations.


Evidence Analysis
pqrp→qq→r(p→q)∧(q→r)p→r
TTTTTTT
TTFTFFF
TFTFTFT
TFFFTFF

Critical Evaluation

The truth table confirms equivalence as outputs match. This aligns with our textbook’s transitive property of implications.

Question 12:
Define mathematical reasoning and explain its importance in problem-solving with an example.
Answer:

Mathematical reasoning refers to the logical process of analyzing, interpreting, and drawing conclusions from mathematical statements or problems. It involves using deductive and inductive reasoning to validate or disprove hypotheses.

Importance:
1. Enhances critical thinking by breaking down complex problems into simpler steps.
2. Ensures accuracy in solutions by validating each step logically.
3. Builds a foundation for advanced topics like calculus and algebra.

Example:
Problem: Prove that the sum of two even numbers is even.
Solution:
Let two even numbers be 2a and 2b, where a and b are integers.
Sum = 2a + 2b = 2(a + b).
Since a + b is an integer, the sum is divisible by 2, hence even.

Question 13:
Differentiate between inductive and deductive reasoning with suitable examples.
Answer:

Inductive Reasoning:
It involves drawing general conclusions from specific observations.
Example:
Observing that 3 + 5 = 8 (even), 7 + 9 = 16 (even), and concluding that the sum of two odd numbers is always even.

Deductive Reasoning:
It involves applying general principles to reach specific conclusions.
Example:
General principle: All even numbers are divisible by 2.
Specific case: 14 is an even number.
Conclusion: 14 is divisible by 2.

Key Difference:
Inductive reasoning may not always be valid (as it relies on patterns), while deductive reasoning guarantees validity if the premises are true.

Question 14:
Explain the concept of contradiction in mathematical reasoning with a proof example.
Answer:

Contradiction is a method of proving a statement by assuming its negation and showing it leads to an impossible or illogical result.

Example: Prove that √2 is irrational.
Proof:
1. Assume √2 is rational, so it can be written as a/b where a and b are co-prime integers.
2. Then, √2 = a/b ⇒ 2 = a²/b² ⇒ 2b² = a².
3. This implies a² is even, so a must be even (let a = 2k).
4. Substituting: 2b² = (2k)² ⇒ 2b² = 4k² ⇒ b² = 2k².
5. Thus, b² is even, so b must be even.
6. But if both a and b are even, they share a common factor 2, contradicting co-primality.
7. Hence, the assumption is false, and √2 is irrational.

Question 15:
Define mathematical reasoning and explain its importance in problem-solving with an example. How does it differ from inductive reasoning?
Answer:

Mathematical reasoning is the process of using logical thinking and structured steps to arrive at a valid conclusion or solution to a problem. It involves analyzing given information, applying mathematical principles, and justifying each step clearly.

Importance in problem-solving:
1. Ensures accuracy by following a systematic approach.
2. Helps in verifying solutions through logical validation.
3. Builds a strong foundation for advanced mathematical concepts.

Example: To prove that the sum of two even numbers is even:
Let a = 2k and b = 2m, where k and m are integers.
Then, a + b = 2(k + m), which is clearly even.

Difference from inductive reasoning:
Mathematical reasoning relies on deductive logic (general to specific), while inductive reasoning draws probable conclusions from specific observations (specific to general).

Question 16:
State and prove the principle of mathematical induction using a real-life example. Explain why this method is essential in mathematics.
Answer:

Principle of Mathematical Induction (PMI):
It is a technique to prove statements for all natural numbers n. It consists of two steps:
1. Base Case: Verify the statement for n = 1.
2. Inductive Step: Assume the statement holds for n = k (inductive hypothesis), then prove it for n = k + 1.

Proof:
Let P(n) be the statement to prove.
1. Base Case: Show P(1) is true.
2. Inductive Step: Assume P(k) is true, then deduce P(k+1).
By PMI, P(n) is true for all n ∈ ℕ.

Real-life example:
Proving that the sum of the first n odd numbers is n²:
1. For n = 1: 1 = 1² (True).
2. Assume true for n = k: 1 + 3 + ... + (2k-1) = k².
For n = k+1: Add (2k+1) to both sides.
LHS becomes k² + (2k+1) = (k+1)² = RHS.

Importance:
PMI is essential for proving universally quantified statements, ensuring generalizability, and avoiding infinite verification.

Question 17:
Define mathematical reasoning and explain its importance in problem-solving with a suitable example. (5 marks)
Answer:

Mathematical reasoning refers to the logical process of analyzing, interpreting, and drawing conclusions from mathematical statements or problems. It involves using deductive and inductive reasoning to validate or disprove hypotheses, ensuring accuracy in solutions.

Importance:
1. Enhances critical thinking by breaking down complex problems into simpler steps.
2. Promotes clarity and precision in formulating arguments.
3. Builds a foundation for advanced topics like calculus and algebra.

Example:
Consider the statement: 'If a number is divisible by 6, it is also divisible by 2 and 3.'
Using mathematical reasoning:
1. Let the number be 6k (where k is an integer).
2. Since 6 = 2 × 3, 6k can be written as 2 × 3k.
3. Thus, the number is divisible by both 2 and 3, proving the statement.

Question 18:
Differentiate between contradiction and tautology in mathematical reasoning with examples. (5 marks)
Answer:

Contradiction and tautology are fundamental concepts in logical reasoning:

1. Tautology:
A statement that is always true, regardless of the truth values of its components.
Example: 'p ∨ ¬p' (p OR NOT p).
Truth table:
- If p is true, p ∨ ¬p = true ∨ false = true.
- If p is false, p ∨ ¬p = false ∨ true = true.

2. Contradiction:
A statement that is always false, irrespective of the truth values of its components.
Example: 'p ∧ ¬p' (p AND NOT p).
Truth table:
- If p is true, p ∧ ¬p = true ∧ false = false.
- If p is false, p ∧ ¬p = false ∧ true = false.

Key Difference:
Tautologies are universally valid, while contradictions are universally invalid, serving as benchmarks in logical proofs.

Question 19:
Define mathematical reasoning and explain its importance in problem-solving with a suitable example. Also, differentiate between inductive and deductive reasoning.
Answer:

Mathematical reasoning is the process of using logical thinking and structured arguments to arrive at conclusions or solve problems. It involves analyzing given information, identifying patterns, and applying mathematical principles to validate results.

Importance in problem-solving:
1. Ensures accuracy by following logical steps.
2. Helps generalize solutions for similar problems.
3. Builds critical thinking and analytical skills.

Example: Proving that the sum of two even numbers is even.
Let two even numbers be 2a and 2b.
Sum = 2a + 2b = 2(a + b), which is clearly divisible by 2. Hence, the sum is even.

Difference between inductive and deductive reasoning:

  • Inductive reasoning: Draws general conclusions from specific observations (e.g., observing patterns in sequences).
  • Deductive reasoning: Applies general principles to reach specific conclusions (e.g., using theorems to solve problems).

Question 20:
State and prove the Principle of Mathematical Induction (PMI). Use it to prove that the sum of the first n odd natural numbers is n².
Answer:

Principle of Mathematical Induction (PMI):
It is a method to prove statements for all natural numbers. It consists of two steps:
1. Base Case: Verify the statement for n = 1.
2. Inductive Step: Assume the statement holds for n = k (inductive hypothesis) and prove it for n = k + 1.

Proof for sum of first n odd numbers:
Statement: 1 + 3 + 5 + ... + (2n - 1) = n².

1. Base Case: For n = 1, LHS = 1, RHS = 1² = 1. True.

2. Inductive Step:
Assume true for n = k, i.e., 1 + 3 + ... + (2k - 1) = k².
For n = k + 1, add the next odd term (2(k + 1) - 1) = (2k + 1) to both sides:
1 + 3 + ... + (2k - 1) + (2k + 1) = k² + (2k + 1)
Simplify RHS: k² + 2k + 1 = (k + 1)².
Thus, the statement holds for n = k + 1.

By PMI, the statement is true for all natural numbers n.

Question 21:
Define mathematical reasoning and explain its importance in problem-solving with a suitable example. Discuss how it differs from inductive reasoning.
Answer:

Mathematical reasoning refers to the logical process of drawing conclusions based on given premises or facts using deductive reasoning. It ensures accuracy and validity in solving problems by following a structured approach.

Importance in problem-solving:
1. Helps in deriving precise solutions by applying logical steps.
2. Eliminates ambiguity by relying on proven mathematical principles.
3. Enhances critical thinking and analytical skills.

Example:
Given: All prime numbers greater than 2 are odd.
Statement: 7 is a prime number greater than 2.
Conclusion: Therefore, 7 is odd.

Difference from inductive reasoning:
1. Deductive reasoning (used in mathematical reasoning) guarantees truth if premises are correct.
2. Inductive reasoning predicts likely outcomes based on patterns but does not ensure certainty.

Question 22:
Prove by contradiction that the square root of 2 is an irrational number. Explain each step clearly and justify the assumptions made.
Answer:

Proof by contradiction:
Assume √2 is a rational number, so it can be written as √2 = a/b, where a and b are co-prime integers (no common factors other than 1).

Step 1: Square both sides:
2 = a²/b²
a² = 2b²

Step 2: This implies a² is even, so a must be even (since the square of an odd number is odd).
Let a = 2k, where k is an integer.

Step 3: Substitute a in the equation:
(2k)² = 2b²
4k² = 2b²
b² = 2k²

Step 4: Now, b² is also even, so b must be even. But this contradicts our assumption that a and b are co-prime (both divisible by 2).

Conclusion: Since the assumption leads to a contradiction, √2 must be irrational.

Case-based Questions (4 Marks) – with Solutions (CBSE Pattern)

These 4-mark case-based questions assess analytical skills through real-life scenarios. Answers must be based on the case study provided.

Question 1:
A survey shows 60% of students prefer online classes, while 30% prefer hybrid. The rest are undecided. If 200 students were surveyed, prove the number of undecided students satisfies mathematical reasoning principles.
Answer:
Problem Interpretation

We must find undecided students using given percentages and total surveyed.

Mathematical Modeling
  • Total preference: 60% (online) + 30% (hybrid) = 90%
  • Undecided: 100% - 90% = 10%
Solution

10% of 200 = 20 students. Our textbook shows percentages must sum to 100%, validating the derivation.

Question 2:
Prove using logical equivalence that the statement 'If a number is divisible by 6, it is even' is equivalent to its contrapositive.
Answer:
Problem Interpretation

We must show P → Q ≡ ¬Q → ¬P for divisibility.

Mathematical Modeling
  • P: divisible by 6, Q: even number
  • Contrapositive: ¬Q → ¬P (If not even, not divisible by 6)
Solution

By logical equivalence laws (NCERT Ch14), both forms share truth tables. Hence, proved.

Question 3:
A compound statement combines two simple statements with 'AND'. Verify if (P ∧ Q) is true when P: '2 is prime', Q: '√2 is irrational' using truth tables.
Answer:
Problem Interpretation

We evaluate truth values of P and Q individually and jointly.

Mathematical Modeling
PQP ∧ Q
TTT
Solution

Both P and Q are true (studied in Number Theory). Thus, (P ∧ Q) is true by conjunction rules.

Question 4:
A school conducted a survey on students' preferred learning methods. Out of 200 students, 80 preferred visual aids, 60 preferred auditory methods, and the rest preferred kinesthetic learning. Using logical reasoning, determine if the statement 'Majority of students prefer non-kinesthetic methods' is valid.
Answer:
Problem Interpretation

We need to verify if the majority (more than 50%) of students prefer non-kinesthetic methods (visual or auditory).


Mathematical Modeling
  • Total students = 200
  • Non-kinesthetic = 80 (visual) + 60 (auditory) = 140
  • Kinesthetic = 200 - 140 = 60

Solution

140 out of 200 students prefer non-kinesthetic methods. Since 140/200 = 70% > 50%, the statement is valid.

Question 5:
Prove using mathematical induction that the sum of the first n odd numbers is n². Assume the statement holds for n = k and show it for n = k + 1.
Answer:
Problem Interpretation

We must prove 1 + 3 + 5 + ... + (2n-1) = n² via induction.


Mathematical Modeling
  • Base case: For n=1, 1=1² (true).
  • Assume true for n=k: Sum = k².

Solution

For n=k+1, sum becomes k² + (2(k+1)-1) = k² + 2k + 1 = (k+1)². Hence, by induction, the statement holds.

Question 6:
A town's population grows exponentially as P(t) = P0ekt, where P0 is the initial population and k is the growth rate. In 2020, the population was 50,000, and by 2023, it reached 55,000.
(i) Derive the value of k.
(ii) Predict the population in 2030.
Answer:
Problem Interpretation

We studied exponential growth models in class. Given P0 = 50,000 and P(3) = 55,000, we need to find k and project future population.


Mathematical Modeling
  • Using P(t) = P0ekt, substitute t = 3: 55,000 = 50,000e3k.
  • Simplify to e3k = 1.1 and take natural log: 3k = ln(1.1).

Solution

Solving gives k ≈ 0.0318. For 2030 (t = 10), P(10) = 50,000e0.318 ≈ 68,000.

Question 7:
A student claims: 'If n is odd, then n2 + 1 is divisible by 4.'
(i) Validate this using n = 5.
(ii) Prove the statement for any odd integer n.
Answer:
Problem Interpretation

Our textbook shows proofs for odd/even properties. We test the claim for n = 5 and generalize.


Mathematical Modeling
  • For n = 5, 52 + 1 = 26, which is not divisible by 4.
  • The claim is false. Instead, we prove n2 − 1 is divisible by 8 for odd n.

Solution

Let n = 2k + 1. Then n2 − 1 = 4k(k + 1). Since k(k + 1) is even, 8 divides the expression.

Question 8:
A farmer wants to fence a rectangular field along a riverbank, using 200 meters of fencing material. The river acts as one side, so no fencing is needed there. Optimize the dimensions to maximize the area. Derive the solution using quadratic functions.
Answer:
Problem Interpretation

We need to maximize the area of a rectangular field with one side along a river, using 200m of fencing for the other three sides.

Mathematical Modeling
  • Let length parallel to river = x, width = y.
  • Constraint: x + 2y = 200 → x = 200 − 2y.
  • Area A = x × y = (200 − 2y)y = −2y2 + 200y.
Solution

This is a quadratic in y with a = −2, b = 200. Maximum occurs at vertex: y = −b/(2a) = 50m. Then x = 100m. Maximum area = 5000m2.

Question 9:
Prove that the sum of the first n odd natural numbers is n2 using mathematical induction. Validate for n = 4.
Answer:
Problem Interpretation

We must prove 1 + 3 + 5 + ... + (2n−1) = n2 via induction.

Mathematical Modeling
  • Base case (n=1): 1 = 12 ✔️
  • Inductive step: Assume true for n=k, then for n=k+1, sum = k2 + (2(k+1)−1) = k2 + 2k + 1 = (k+1)2.
Solution

By induction, the statement holds ∀n∈ℕ. For n=4: 1+3+5+7=16=42 ✔️

Question 10:
In a debate, two students argue about the validity of the statement: 'If a number is divisible by 6, then it is divisible by 2 and 3.' Student A claims it is a tautology, while Student B insists it is a contradiction. Analyze their arguments using mathematical reasoning.
Answer:
Problem Interpretation

We studied that a statement is a tautology if it is always true, and a contradiction if always false. Here, the statement links divisibility rules.


Mathematical Modeling
  • Let p: A number is divisible by 6.
  • q: It is divisible by 2.
  • r: It is divisible by 3.

Solution

The statement is p → (q ∧ r). Since 6 = 2 × 3, p implies both q and r. Thus, the statement is always true, making it a tautology. Student A is correct.

Question 11:
A city's population growth follows P(t) = P₀ert, where P₀ is initial population and r is growth rate. If P₀ = 50,000 and the population doubles in 10 years, derive the value of r using logarithmic principles.
Answer:
Problem Interpretation

Our textbook shows exponential growth models. Here, we need to find r when population doubles (P(t) = 2P₀) at t = 10.


Mathematical Modeling

Given 2P₀ = P₀e10r, we simplify to 2 = e10r.


Solution

Taking natural logs: ln(2) = 10r. Thus, r = ln(2)/10 ≈ 0.0693 (6.93% annual growth). This derivation uses logarithmic properties to solve for the rate.

Question 12:

In a school debate competition, the following statements were made by two participants:

  • Participant A: 'All prime numbers greater than 2 are odd.'
  • Participant B: 'If a number is not divisible by 2, then it is a prime number.'

Using mathematical reasoning, analyze the validity of these statements and justify your answer with examples.

Answer:

Analysis of Participant A's Statement:

The statement 'All prime numbers greater than 2 are odd' is true. Here's why:

  • By definition, a prime number has exactly two distinct divisors: 1 and itself.
  • Any even number greater than 2 is divisible by 2, making it non-prime (e.g., 4, 6, 8).
  • Examples: 3, 5, 7, 11 are all odd and prime.

Analysis of Participant B's Statement:

The statement 'If a number is not divisible by 2, then it is a prime number' is false. Here's why:

  • Not all odd numbers are prime. For example, 9 is not divisible by 2 but is divisible by 3 (9 = 3 × 3).
  • Another example: 15 is odd but not prime (15 = 3 × 5).

Conclusion: Participant A's statement is logically valid, while Participant B's statement is incorrect because it generalizes all odd numbers as primes.

Question 13:

A student claims: 'For any two statements p and q, the compound statement (p ∧ q) ∨ (~p) is always true.' Verify this claim using truth tables and provide a reasoned conclusion.

Answer:

Step 1: Construct the Truth Table

We evaluate the compound statement (p ∧ q) ∨ (~p) for all possible truth values of p and q:


pqp ∧ q~p(p ∧ q) ∨ (~p)
TTTFT
TFFFF
FTFTT
FFFTT

Step 2: Analyze the Results

  • When p is true and q is false, the compound statement evaluates to false.
  • In all other cases, the statement is true.

Conclusion: The student's claim is incorrect because the compound statement is not always true. It fails when p is true and q is false.

Question 14:

In a school debate, three students—Rahul, Priya, and Arun—made the following statements:

  • Rahul: "All rectangles are squares."
  • Priya: "Some squares are not rectangles."
  • Arun: "Every square is a rectangle."

Using the principles of mathematical reasoning, identify which statements are valid and justify your answer with logical reasoning.

Answer:

The validity of the statements can be analyzed using mathematical reasoning based on geometric properties:

  • Rahul's statement: "All rectangles are squares." is invalid.
    Reason: A rectangle has opposite sides equal and all angles 90°, but a square requires all sides to be equal. Not all rectangles meet this criterion.
  • Priya's statement: "Some squares are not rectangles." is invalid.
    Reason: By definition, every square is a rectangle (as it satisfies all properties of a rectangle), but not vice versa.
  • Arun's statement: "Every square is a rectangle." is valid.
    Reason: A square fulfills all conditions of a rectangle (equal opposite sides, 90° angles) with the additional constraint of equal sides.

Thus, only Arun's statement is logically correct.

Question 15:

A mathematics teacher wrote the following compound statement on the board:
"If a number is divisible by 6, then it is divisible by 2 and 3."
Rewrite this statement in its contrapositive form and determine its truth value using logical equivalence.

Answer:

The given statement is a conditional statement of the form "If P, then Q", where:
P: A number is divisible by 6.
Q: It is divisible by 2 and 3.

The contrapositive of this statement is: "If not Q, then not P," which translates to:
"If a number is not divisible by 2 or not divisible by 3, then it is not divisible by 6."

Truth value analysis:
1. Original statement: True (since 6 = 2 × 3, divisibility by 6 implies divisibility by both 2 and 3).
2. Contrapositive: Also true, as it is logically equivalent to the original statement.
Example: The number 5 is not divisible by 2 or 3, and indeed, it is not divisible by 6.

Question 16:
A school is organizing a debate competition. The teacher writes the following statements on the board:
(i) All students who participate in debates are confident.
(ii) Riya is a student who participates in debates.
Based on these statements, the teacher asks if Riya is confident.

Identify the type of mathematical reasoning used here and justify your answer. Also, verify whether the conclusion is valid.

Answer:

The given scenario uses deductive reasoning, a type of mathematical reasoning where conclusions are drawn from general statements or premises.

Justification:
1. The first statement is a general rule: All debate participants are confident (universal premise).
2. The second statement is specific: Riya is a debate participant (particular case).
3. The conclusion (Riya is confident) logically follows from the premises.

Validity: The conclusion is valid because it adheres to the structure of deductive reasoning. If the premises are true, the conclusion must be true.

Note: Deductive reasoning is foundational in mathematics, ensuring conclusions are certain if premises are correct.

Question 17:
In a class, the teacher states:
(i) If a number is divisible by 6, then it is divisible by 2 and 3.
(ii) The number 24 is divisible by 6.
The teacher asks the class to determine if 24 is divisible by 2 and 3 using logical reasoning. Explain the steps and identify the reasoning method.
Answer:

This problem uses logical implication, a key concept in mathematical reasoning.

Steps:
1. Premise 1 establishes a rule: Divisibility by 6 implies divisibility by 2 and 3 (general statement).
2. Premise 2 provides a specific case: 24 is divisible by 6.
3. Applying Premise 1 to Premise 2, we conclude: 24 is divisible by 2 and 3.

Reasoning Method: This is an example of modus ponens, a form of deductive reasoning where:
- If P → Q is true and P is true, then Q must be true.
Here, P = 'number divisible by 6', Q = 'divisible by 2 and 3'.

Verification:
- 24 ÷ 2 = 12 (no remainder) → divisible by 2.
- 24 ÷ 3 = 8 (no remainder) → divisible by 3.
Thus, the conclusion is correct.

Question 18:
A school is organizing a debate competition. The teacher writes the following statement on the board: 'If a student participates in the debate, then they must prepare a speech.'

(a) Identify the hypothesis and conclusion in the given statement.
(b) Write the converse of the statement and check its validity.
Answer:

(a) The given statement is a conditional statement of the form 'If p, then q'. Here:

Hypothesis (p): A student participates in the debate.
Conclusion (q): They must prepare a speech.

(b) The converse of the statement is formed by swapping the hypothesis and conclusion. Thus, the converse is: 'If a student prepares a speech, then they participate in the debate.'

Validity check:
The original statement implies that participation requires preparation, but the converse suggests that preparation guarantees participation, which may not always be true (e.g., a student may prepare but not get selected). Hence, the converse is not necessarily valid.

Question 19:
In a mathematics class, the teacher writes the following statement: 'For all real numbers x, if x² = 4, then x = 2 or x = -2.'

(a) Is this statement true or false? Justify.
(b) Write its contrapositive and verify its truth value.
Answer:

(a) The statement is true. Justification:
The equation x² = 4 has two real solutions: x = 2 and x = -2. No other real numbers satisfy this equation. Thus, the statement holds.

(b) The contrapositive of the statement 'If p, then q' is 'If not q, then not p'. Here:
Original: 'If x² = 4, then x = 2 or x = -2.'
Contrapositive: 'If x ≠ 2 and x ≠ -2, then x² ≠ 4.'

Verification:
The contrapositive is logically equivalent to the original statement. Since the original is true, the contrapositive is also true. For example, if x = 3 (which is neither 2 nor -2), then x² = 9 ≠ 4, validating the contrapositive.

Question 20:
A school is organizing a debate competition. The teacher writes the following statements on the board:

1. If a student participates in the debate, then they must prepare a speech.
2. Riya is participating in the debate.

Based on these statements, answer the following:
a) Identify the hypothesis and conclusion in statement 1.
b) Using logical reasoning, deduce whether Riya must prepare a speech or not.
Answer:

a) In statement 1, the hypothesis is 'a student participates in the debate', and the conclusion is 'they must prepare a speech'.

b) Since Riya is participating in the debate (given in statement 2), the hypothesis of statement 1 is satisfied. Therefore, by logical reasoning, the conclusion must follow: Riya must prepare a speech.

Note: This is an example of Modus Ponens, a valid form of argument in mathematical reasoning.

Question 21:
Consider the following statements:

P: All rectangles are quadrilaterals.
Q: Some quadrilaterals are not rectangles.

a) Write the negation of statement P.
b) Using a Venn diagram, represent the relationship between rectangles and quadrilaterals based on statements P and Q.
Answer:

a) The negation of statement P is: 'Not all rectangles are quadrilaterals' or equivalently, 'There exists at least one rectangle that is not a quadrilateral'.

b) Venn diagram representation:

1. Draw a large circle labeled 'Quadrilaterals'.
2. Inside it, draw a smaller circle labeled 'Rectangles' (since all rectangles are quadrilaterals).
3. Shade the region outside the 'Rectangles' circle but inside the 'Quadrilaterals' circle to represent Q (some quadrilaterals are not rectangles).

Key takeaway: The diagram shows that rectangles are a subset of quadrilaterals, but not all quadrilaterals are rectangles.

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