GSEB Class 11 Maths Solutions Chapter 14 Mathematical Reasoning Exercise 14.2

Question 1. Write the negation of the following statements:
1. Chennai is the capital of Tamil Nadu.
2. \( \sqrt{2} \) is not a complex number.
3. All triangles are not equilateral.
4. The number 2 is greater than 7.
5. Every natural number is an integer.
Answer:
1. Chennai is not the main city of Tamil Nadu.
2. \( \sqrt{2} \) is a complex number.
3. Every triangle is equilateral.
4. The number 2 is not bigger than 7.
5. Each natural number is not an integer.
In simple words: To find the negation, simply state the opposite of each given statement. If a statement says something 'is', its negation says it 'is not', and vice versa.

Exam Tip: When forming a negation, be careful with universal quantifiers like "all" or "every." The negation of "All A are B" is "Some A are not B," and the negation of "No A are B" is "Some A are B."

Question 2. Are the following pair of statements negation of each other?
(a) The number x is not a rational number. The number x is not an irrational number.
(b) The number x is a rational number. The number x is an irrational number.
Answer:
(a) The opposite of the statement "The number x is not a rational number" is that the number x is a rational number. The second statement says x is not irrational, which is also the same. So, these statements are indeed negations of one another.
(b) The negation of the first statement is that the number x is rational. The number x is not a rational number, or it can also be called an irrational number. The second statement is essentially the same idea. As a result, they are negations of each other.
In simple words: If two statements are negations, one must be true when the other is false. For (a), if x is not rational, it means x is irrational. If x is not irrational, it means x is rational. These are opposites. For (b), a rational number is not an irrational number and vice-versa.

Exam Tip: Remember that a number is either rational or irrational, but not both. This fundamental property helps determine if two statements about a number's rationality/irrationality are negations.

Question 3. Find the component statements of the following compound statements and check whether they are true or false:
1. The number 3 is prime or it is odd.
2. All integers are positive or negative.
3. 100 is divisible by 3, 11 and 5.
Answer:
1. Let p be the statement: The number 3 is prime.
Let q be the statement: The number 3 is odd.
Statements p and q are connected using 'or'. This compound statement is true.

2. Let p represent: All integers are positive.
Let q represent: All integers are negative.
Statements p and q are joined with the word 'or'. Both p and q are false individual statements. Therefore, the overall compound statement is false.

3. Let p be: 100 is divisible by 3.
Let q be: 100 is divisible by 11.
Let r be: 100 is divisible by 5.
Statement p is false, statement q is false, and statement r is true. The combined statement 'p and q and r' is a false statement.
In simple words: Break down each complex sentence into simpler parts. Then, decide if each simple part is true or false. Finally, use the words like 'or' or 'and' to figure out if the whole sentence is true or false.

Exam Tip: For compound statements, an 'or' statement is true if at least one component is true. An 'and' statement is true only if all components are true. Correctly identifying the truth value of each component is crucial.