CBSE Class 11 Maths Exemplar Chapter 14: Mathematical Reasoning NCERT Solutions

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This chapter on Mathematical Reasoning for CBSE Class 11 Maths Exemplar provides essential solutions to understand the fundamental concepts of logic. It covers identifying statements, distinguishing between true and false statements, and breaking down compound statements into their component parts. The solutions explain the criteria for a sentence to be a statement and how to analyze complex sentences involving 'and' and 'or' connectives. This resource is designed to help students build a strong foundation in logical thinking, crucial for various mathematical and analytical tasks, and aids in effective exam revision by clarifying key principles.

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BoardCBSE
ClassClass 11
SubjectMaths Exemplar
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 14

Chapter summary

Chapter 14 of the CBSE Class 11 Maths Exemplar focuses on Mathematical Reasoning. This section provides NCERT Solutions that clarify the definition of a statement in logic, differentiating it from questions or commands. It details how to identify whether a given sentence is a statement and its truth value (true or false). The solutions also guide students in decomposing compound statements into simpler component statements and understanding the role of logical connectives like 'and' and 'or'.

Learning outcomes

  • Understand the definition of a statement in mathematical logic.
  • Identify whether a given sentence is a statement or not.
  • Determine the truth value (true or false) of simple statements.
  • Decompose compound statements into their constituent component statements.
  • Analyze the structure of compound statements using logical connectives.

Topics covered

Paper topics

  • Statements in Logic
  • Truth Values
  • Declarative Sentences
  • Compound Statements
  • Component Statements
  • Logical Connectives
  • Identifying Statements
  • Analyzing Mathematical Sentences

Important topics

  • Definition of a Statement
  • Truth Value Determination
  • Component Statements
  • Compound Statements Analysis

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Questions and Solutions

Question 1

Which of the following sentences are statements? Justify your answer.

(i) A triangle has three sides.

(ii) 0 is a complex number.

(iii) Sky is red.

(iv) Every set is an infinite set.

(v) 15 + 8 > 23.

(vi) y + 9 = 7

(vii) Where is your bag?

(viii) Every square is a rectangle.

(ix) Sum of opposite angles of a cyclic quadrilateral is 180°. (x) \sin^2 x + \cos^2 x = 0

Solution: A statement is defined as a declarative sentence that is either true or false, but not both at the same time. Let's analyze each sentence:
  1. (i) A triangle has three sides. This is a declarative sentence and it is true. Therefore, it is a statement.
  2. (ii) 0 is a complex number. This is a declarative sentence and it is true, as 0 can be written as 0 + 0i. Therefore, it is a statement.
  3. (iii) Sky is red. This is a declarative sentence, but it is false (the sky is typically blue). Therefore, it is a statement.
  4. (iv) Every set is an infinite set. This is a declarative sentence, but it is false (e.g., the set {1, 2, 3} is finite). Therefore, it is a statement.
  5. (v) 15 + 8 > 23. This is a declarative sentence. Since 15 + 8 = 23, the inequality 23 > 23 is false. Therefore, it is a statement.
  6. (vi) y + 9 = 7. This is an open sentence because its truth value depends on the value of the variable 'y'. Since the value of 'y' is not specified, we cannot determine if it is true or false. Therefore, it is not considered a statement.
  7. (vii) Where is your bag? This is an interrogative sentence (a question), not a declarative sentence. Therefore, it is not a statement.
  8. (viii) Every square is a rectangle. This is a declarative sentence and it is true, as a square fulfills all the properties of a rectangle. Therefore, it is a statement.
  9. (ix) Sum of opposite angles of a cyclic quadrilateral is 180°. This is a declarative sentence and it is true, according to a property of cyclic quadrilaterals. Therefore, it is a statement.
  10. (x) \sin^2 x + \cos^2 x = 0 . This is a declarative sentence. We know the trigonometric identity \sin^2 x + \cos^2 x = 1. Since 1 \neq 0, the given equation is false for all values of x. Therefore, it is a statement.

Question 2

Find the component statements of the following compound statements.

(i) Number 7 is prime and odd.

(ii) Chennai is in India and is the capital of Tamil Nadu.

(iii) The number 100 is divisible by 3, 11 and 5.

(iv) Chandigarh is the capital of Haryana and UP.

(v) \sqrt{7} is a rational number or an irrational number.

(vi) 0 is less than every positive integer and every negative integer.

(vii) Plants use sunlight, water and carbon dioxide for photosynthesis.

(viii) Two lines in a plane either intersect at one point or they are parallel.

(ix) A rectangle is a quadrilateral or a 5 sided polygon.

Solution: A compound statement is formed by combining two or more simple statements, usually with logical connectives like 'and' (conjunction) or 'or' (disjunction). We need to identify these simple statements (component statements).
  1. (i) Number 7 is prime and odd.
    • p: Number 7 is prime.
    • q: Number 7 is odd.
  2. (ii) Chennai is in India and is the capital of Tamil Nadu.
    • p: Chennai is in India.
    • q: Chennai is the capital of Tamil Nadu.
  3. (iii) The number 100 is divisible by 3, 11 and 5.
    • p: 100 is divisible by 3.
    • q: 100 is divisible by 11.
    • r: 100 is divisible by 5.
  4. (iv) Chandigarh is the capital of Haryana and UP.
    • p: Chandigarh is the capital of Haryana.
    • q: Chandigarh is the capital of UP.
  5. (v) \sqrt{7} is a rational number or an irrational number.
    • p: \sqrt{7} is a rational number.
    • q: \sqrt{7} is an irrational number.
  6. (vi) 0 is less than every positive integer and every negative integer.
    • p: 0 is less than every positive integer.
    • q: 0 is less than every negative integer.
  7. (vii) Plants use sunlight, water and carbon dioxide for photosynthesis.
    • p: Plants use sunlight for photosynthesis.
    • q: Plants use water for photosynthesis.
    • r: Plants use carbon dioxide for photosynthesis.
  8. (viii) Two lines in a plane either intersect at one point or they are parallel.
    • p: Two lines in a plane intersect at one point.
    • q: Two lines in a plane are parallel.
  9. (ix) A rectangle is a quadrilateral or a 5 sided polygon.
    • p: A rectangle is a quadrilateral.
    • q: A rectangle is a 5-sided polygon.

Question 3

Write the component statements of the following compound statements and check whether the compound statement is true or false.

(i) 57 is divisible by 2 or 3.

(ii) 24 is a multiple of 4 and 6.

(iii) All living things have two eyes and two legs.

(iv) 2 is an even number and a prime number.

Solution: We need to identify the component statements and then determine the truth value of the compound statement based on the logical connectives used.
  1. (i) 57 is divisible by 2 or 3.
    • Component statements:
    • p: 57 is divisible by 2.
    • q: 57 is divisible by 3.

    Analysis: 57 is not divisible by 2 (it's an odd number), so p is False (F). 57 is divisible by 3 (since 5 + 7 = 12, which is divisible by 3), so q is True (T). The compound statement uses the connective 'or' (disjunction, p v q). A disjunction is true if at least one of its component statements is true. Since q is true, the compound statement '57 is divisible by 2 or 3' is True.

  2. (ii) 24 is a multiple of 4 and 6.
    • Component statements:
    • p: 24 is a multiple of 4.
    • q: 24 is a multiple of 6.

    Analysis: 24 is indeed a multiple of 4 (24 = 4 × 6), so p is True (T). 24 is also a multiple of 6 (24 = 6 × 4), so q is True (T). The compound statement uses the connective 'and' (conjunction, p ∧ q). A conjunction is true only if both component statements are true. Since both p and q are true, the compound statement '24 is a multiple of 4 and 6' is True.

  3. (iii) All living things have two eyes and two legs.
    • Component statements:
    • p: All living things have two eyes.
    • q: All living things have two legs.

    Analysis: The statement 'All living things have two eyes' is False (e.g., insects have more than two eyes, some animals have no eyes). The statement 'All living things have two legs' is also False (e.g., fish have no legs, snakes have no legs). Since the compound statement uses 'and' (p ∧ q), and both p and q are false, the compound statement 'All living things have two eyes and two legs' is False.

  4. (iv) 2 is an even number and a prime number.
    • Component statements:
    • p: 2 is an even number.
    • q: 2 is a prime number.

    Analysis: The statement '2 is an even number' is True. The statement '2 is a prime number' is also True (it is the smallest and only even prime number). The compound statement uses 'and' (p ∧ q). Since both p and q are true, the compound statement '2 is an even number and a prime number' is True.

Common mistakes

  • Confusing questions or commands with statements.
  • Incorrectly assigning truth values to statements.
  • Failing to identify all component statements in a compound sentence.
  • Misinterpreting the 'and' and 'or' connectives in compound statements.

Revision tips

  • Focus on the definition of a statement: it must be either true or false, but not both.
  • Practice identifying statements by checking if they are declarative sentences.
  • Break down complex sentences into their simplest parts to find component statements.
  • Pay close attention to the logical connectives ('and', 'or') when determining the truth of compound statements.

Practice MCQs

Q1. Which of the following is NOT a statement?

Q2. What is the truth value of the statement 'The Sun is a planet.'?

Q3. Identify a component statement of 'Chennai is in India and is the capital of Tamil Nadu.'

Q4. The statement 'y + 9 = 7' is not considered a statement because:

Q5. Which of the following is a true statement?

Frequently asked questions

What is a statement in mathematical reasoning?

A statement is a declarative sentence that is either true or false, but not both simultaneously. It must have a definite truth value.

How can I identify if a sentence is a statement?

Check if the sentence declares something and has a definite truth value (true or false). Questions, commands, exclamations, and sentences with undefined variables are generally not statements.

What are component statements?

Component statements are the simpler statements that make up a compound statement, often connected by logical words like 'and' or 'or'.

Why is 'y + 9 = 7' not considered a statement?

It is not considered a statement because the truth value depends on the value of 'y', which is not provided. Without a fixed truth value, it cannot be classified as true or false.

How do these solutions help with exam preparation?

These solutions provide clear explanations and step-by-step methods to understand the core concepts of mathematical reasoning, helping students tackle similar problems in exams with confidence.

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