Answer :
Let's examine the two statements given in the question and determine the correct choice step-by-step.
Statement-1 (Assertion):
- The product of the rational numbers [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex] is [tex]\(\frac{5}{14}\)[/tex], which is a rational number.
Verification of Statement-1:
1. Multiply [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex]:
[tex]\[ \frac{1}{2} \times \frac{5}{7} = \frac{1 \times 5}{2 \times 7} = \frac{5}{14} \][/tex]
2. [tex]\(\frac{5}{14}\)[/tex] is indeed a rational number because it is the ratio of two integers (5 and 14) where the denominator is not zero.
So, Statement-1 is true.
Statement-2 (Reason):
- Rational numbers are closed under multiplication.
- This means that if you multiply any two rational numbers, the result is always a rational number.
Verification of Statement-2:
- By definition, rational numbers are numbers that can be expressed as the quotient or fraction [tex]\(\frac{a}{b}\)[/tex] of two integers [tex]\(a\)[/tex] (the numerator) and [tex]\(b\)[/tex] (the denominator), where [tex]\(b \neq 0\)[/tex].
- When you multiply two rational numbers [tex]\(\frac{p}{q}\)[/tex] and [tex]\(\frac{r}{s}\)[/tex], the result is [tex]\(\frac{p \cdot r}{q \cdot s}\)[/tex].
- Since [tex]\(p\)[/tex], [tex]\(q\)[/tex], [tex]\(r\)[/tex], and [tex]\(s\)[/tex] are integers and [tex]\(q\)[/tex] and [tex]\(s\)[/tex] are not zero, [tex]\(\frac{p \cdot r}{q \cdot s}\)[/tex] is also a rational number.
So, Statement-2 is true.
Relationship between the Statements:
- Statement-2 explains why the product of two specific rational numbers [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex] is also a rational number ([tex]\(\frac{5}{14}\)[/tex]).
- Therefore, Statement-2 is the correct explanation for Statement-1.
Given the above analysis, the correct choice is:
(a) Statement-I and II are true. Statement-II is correct explanation of I.
Statement-1 (Assertion):
- The product of the rational numbers [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex] is [tex]\(\frac{5}{14}\)[/tex], which is a rational number.
Verification of Statement-1:
1. Multiply [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex]:
[tex]\[ \frac{1}{2} \times \frac{5}{7} = \frac{1 \times 5}{2 \times 7} = \frac{5}{14} \][/tex]
2. [tex]\(\frac{5}{14}\)[/tex] is indeed a rational number because it is the ratio of two integers (5 and 14) where the denominator is not zero.
So, Statement-1 is true.
Statement-2 (Reason):
- Rational numbers are closed under multiplication.
- This means that if you multiply any two rational numbers, the result is always a rational number.
Verification of Statement-2:
- By definition, rational numbers are numbers that can be expressed as the quotient or fraction [tex]\(\frac{a}{b}\)[/tex] of two integers [tex]\(a\)[/tex] (the numerator) and [tex]\(b\)[/tex] (the denominator), where [tex]\(b \neq 0\)[/tex].
- When you multiply two rational numbers [tex]\(\frac{p}{q}\)[/tex] and [tex]\(\frac{r}{s}\)[/tex], the result is [tex]\(\frac{p \cdot r}{q \cdot s}\)[/tex].
- Since [tex]\(p\)[/tex], [tex]\(q\)[/tex], [tex]\(r\)[/tex], and [tex]\(s\)[/tex] are integers and [tex]\(q\)[/tex] and [tex]\(s\)[/tex] are not zero, [tex]\(\frac{p \cdot r}{q \cdot s}\)[/tex] is also a rational number.
So, Statement-2 is true.
Relationship between the Statements:
- Statement-2 explains why the product of two specific rational numbers [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex] is also a rational number ([tex]\(\frac{5}{14}\)[/tex]).
- Therefore, Statement-2 is the correct explanation for Statement-1.
Given the above analysis, the correct choice is:
(a) Statement-I and II are true. Statement-II is correct explanation of I.
