CBSE Class 11 Mathematics Chapter 14: Mathematical Reasoning NCERT Solutions

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CBSE Class 11 Mathematics Chapter 14, Mathematical Reasoning, lays the groundwork for logical thinking in mathematics. This chapter explores the core principles of reasoning, helping students differentiate between statements and non-statements and understand how to determine the truth value of a statement. It examines different sentence types – declarative, interrogative, exclamatory, and imperative – clarifying that only declarative sentences qualify as mathematical statements. Mastering these concepts is vital for developing a robust logical foundation, essential for tackling more complex mathematical ideas and problem-solving techniques. The NCERT Solutions for this chapter offer clear, step-by-step explanations designed to enhance students' comprehension and prepare them effectively for examinations, thereby boosting their overall grasp of mathematical reasoning.

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Quick info

BoardCBSE
ClassClass 11
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 14: Mathematical Reasoning

Chapter summary

Chapter 14, Mathematical Reasoning, focuses on the core concepts of logical statements. The NCERT Solutions cover the definition of a statement, the criteria for a sentence to be a statement (declarative and having a definite truth value), and examples of non-statements like questions, commands, and subjective opinions. It also touches upon the truthfulness of statements based on factual accuracy. These solutions are designed to clarify the basic building blocks of logical arguments in mathematics.

Learning outcomes

  • Understand the definition of a mathematical statement.
  • Differentiate between statements and non-statements.
  • Identify the truth value (true or false) of given sentences.
  • Provide reasons for classifying sentences as statements or non-statements.
  • Recognize subjective and objective sentences in the context of statements.

Topics covered

Paper topics

  • Statements
  • Truth values
  • Declarative sentences
  • Non-statements
  • Subjective sentences
  • Objective sentences
  • Mathematical logic basics
  • Identifying statements
  • Reasons for non-statements
  • Factual accuracy of sentences

Important topics

  • Definition of a statement
  • Distinguishing statements from non-statements
  • Determining truth values
  • Examples of statements and non-statements
  • Reasons for classification

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Questions and Solutions

Question 1

Which of the following sentences are statements? Give reasons for your answer.
  1. There are 35 days in a month.
  2. Mathematics is difficult.
  3. The sum of 5 and 7 is greater than 10.
  4. The square of a number is an even number.
  5. The sides of a quadrilateral have equal length.
  6. Answer this question.
  7. The product of (-1) and 8 is 8.
  8. The sum of all interior angles of a triangle is 180°.
  9. Today is a windy day.
  10. All real numbers are complex numbers.
Solution:

To determine if a sentence is a statement, we need to check if it is a declarative sentence and if it has a definite truth value (either true or false).

  1. Sentence: There are 35 days in a month.

    Reasoning: This is a declarative sentence. We know that the maximum number of days in any month is 31. Therefore, this sentence is factually incorrect. Since it has a definite truth value (False), it is a statement.

  2. Sentence: Mathematics is difficult.

    Reasoning: This sentence expresses a personal opinion. Whether mathematics is difficult or not varies from person to person. It does not have a single, definite truth value. Therefore, it is not a statement.

  3. Sentence: The sum of 5 and 7 is greater than 10.

    Reasoning: This is a declarative sentence. The sum of 5 and 7 is 12. Since 12 is indeed greater than 10, the sentence is factually correct. It has a definite truth value (True). Therefore, it is a statement.

  4. Sentence: The square of a number is an even number.

    Reasoning: This sentence is not always true. For example, the square of 2 is 4 (even), but the square of 3 is 9 (odd). Since the truth value is not definite (it depends on the number chosen), it is not a statement.

  5. Sentence: The sides of a quadrilateral have equal length.

    Reasoning: This sentence is not always true. While a rhombus or a square has equal sides, a general quadrilateral or a rectangle does not necessarily have all sides of equal length. Since its truth value is not definite, it is not a statement.

  6. Sentence: Answer this question.

    Reasoning: This is an imperative sentence, which is a command or an order. It does not assert a fact that can be true or false. Therefore, it is not a statement.

  7. Sentence: The product of (-1) and 8 is 8.

    Reasoning: This is a declarative sentence. The actual product of (-1) and 8 is -8. Since the sentence states that the product is 8, it is factually incorrect. It has a definite truth value (False). Therefore, it is a statement.

  8. Sentence: The sum of all interior angles of a triangle is 180°.

    Reasoning: This is a fundamental geometric fact. The sum of the interior angles of any triangle is always 180°. This sentence is declarative and has a definite truth value (True). Therefore, it is a statement.

  9. Sentence: Today is a windy day.

    Reasoning: This sentence is declarative, but its truth value is not definite. 'Today' refers to a specific day, and 'windy' is a subjective description. Without knowing the specific day and location, we cannot determine if it is true or false. Therefore, it is not a statement.

  10. Sentence: All real numbers are complex numbers.

    Reasoning: This is a declarative sentence. Every real number 'a' can be expressed as a complex number 'a + 0i'. Therefore, the statement is factually correct and has a definite truth value (True). Hence, it is a statement.

Question 2

Give three examples of sentences which are not statements. Give reasons for the answers.
Solution:

A sentence is not a statement if it is not declarative or if it does not have a definite truth value (true or false).

Here are three examples:

  1. Sentence: He is a doctor.

    Reason: This is a declarative sentence, but it is not a statement because the pronoun 'He' is ambiguous. We do not know who 'He' refers to, so we cannot determine if the sentence is true or false. Its truth value is indefinite.

  2. Sentence: Geometry is difficult.

    Reason: This sentence expresses a subjective opinion. The difficulty of geometry varies from person to person. Since it does not have a definite truth value that applies universally, it is not a statement.

  3. Sentence: Where is she going?

    Reason: This is an interrogative sentence (a question). Questions do not assert a fact and therefore cannot be true or false. Additionally, the pronoun 'she' is ambiguous. Hence, it is not a statement.

Common mistakes

  • Confusing opinions or subjective sentences with statements.
  • Failing to recognize that questions, commands, and exclamations are not statements.
  • Incorrectly assigning truth values to sentences that are not statements.
  • Not providing a clear reason for classifying a sentence as a statement or non-statement.

Revision tips

  • Focus on the core definition: a statement must be declarative and have a definite truth value.
  • Practice identifying the truth value of factual sentences.
  • Pay close attention to sentences containing pronouns or vague terms, as they are often not statements.
  • Review the examples of questions, commands, and opinions to reinforce what constitutes a non-statement.

Practice MCQs

Q1. Which of the following is a statement?

Q2. Why is 'Mathematics is difficult' not a statement?

Q3. What is the truth value of the sentence: 'The sum of 5 and 7 is greater than 10'?

Q4. Which of the following is NOT a statement?

Q5. The sentence 'Today is a windy day' is not a statement because:

Frequently asked questions

What is a statement in mathematics?

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

Why are questions and commands not considered statements?

Questions (interrogative sentences) and commands (imperative sentences) do not assert anything that can be true or false. They are requests for information or actions, not declarations of fact.

How do I determine if a sentence is a statement?

Check if the sentence is declarative (makes a statement) and if it can be definitively classified as true or false based on objective criteria.

What does 'truth value' mean?

The truth value of a statement is its truthfulness, which can be either 'True' (T) or 'False' (F).

Are subjective sentences like 'Mathematics is difficult' statements?

No, subjective sentences are not statements because their truth value depends on personal opinion and varies from person to person.

How do these NCERT solutions help with Chapter 14?

These solutions provide rewritten, clear explanations and step-by-step reasoning for each question, helping students understand the concepts of mathematical reasoning and identify statements accurately.

Important links

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