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SubjectsMathematics
परिमेय संख्याएँ
In this Class 9 Mathematics topic from the Number Systems chapter, students explore rational numbers as numbers that can be written in the form p/q, where p and q are integers and q is not zero. They learn to represent and compare them on the number line, identify equivalent forms, and perform addition, subtraction, multiplication, and division. The topic also develops understanding of properties such as closure, commutativity, associativity, and distributivity, helping students apply rational numbers accurately in mathematical problems.
Correct answer: B. (\frac{2}{7},-4)
Explanation: \(\frac{2}{7}\) is a ratio of integers, and \(-4\) can be written as \(\frac{-4}{1}\); therefore, both numbers in option B are rational. In option D, \(\sqrt{49}=7\) is rational, but \(\sqrt{2}\) is irrational. Exam tip: Write an integer with denominator 1 to check whether it is rational.
Correct answer: C. \(\sqrt{81}\)
Explanation: \(\sqrt{81}=9\), and 9 can be written as \(\frac{9}{1}\); therefore, it is a rational number. \(\sqrt{21}\) and \(\sqrt{11}\) are not square roots of perfect squares, so they are irrational; \(\pi\) is also irrational. Exam tip: The square root of a perfect square is always an integer and hence rational.
Correct answer: B. \(\sqrt{5}\times\sqrt{5}\)
Explanation: \(\sqrt{5}\times\sqrt{5}=(\sqrt{5})^2=5\), and 5 is a rational number. Option A gives \(\sqrt{6}\), while option C gives \(2\sqrt{7}\); both are irrational. Option D remains \(\pi\), which is irrational. Exam tip: the product of two identical square roots equals the number inside the radical.
Correct answer: B. \( -\frac{1}{3} \)
Explanation: \( -\frac{1}{3} = -0.3333\ldots \). On the number line, \( -0.3333\ldots \) lies to the right of \( -0.34 \), so \( -\frac{1}{3} > -0.34 \), making option B correct. Option A is a common mistake caused by comparing the decimal digits without considering that, for negative numbers, the number closer to zero is greater. Exam tip: convert the fraction to a decimal or write both numbers to the same number of decimal places before comparing them.
Correct answer: B. Rational number
Explanation: \(\sqrt{2}\times\sqrt{8}=\sqrt{2\times8}=\sqrt{16}=4\). Since \(4=\frac{4}{1}\), it is a rational number. Although \(\sqrt{2}\) and \(\sqrt{8}\) are individually irrational, the product of two irrational numbers is not always irrational. Also, \(4\) is neither negative nor non-real. Exam tip: When multiplying square roots, multiply the radicands first and look for a perfect square.
Correct answer: C. \(\frac{9}{4}\)
Explanation: \(\frac{9}{4}\) is the ratio of two integers, 9 and 4, so it is rational. Its value is 2.25, which is not an integer. Both \(-4\) and \(\sqrt{16}=4\) are integers, whereas \(\sqrt{7}\) is irrational. Exam tip: first check whether a number can be written as \(\frac{p}{q}\), where \(p,q\) are integers and \(q\ne0\), then check whether its value is an integer.
Correct answer: C. \(0.\overline{37}\)
Explanation: \(0.\overline{37}=0.373737\ldots\) is a recurring decimal, so it is a rational number. It can be written as \(\frac{37}{99}\), where both numerator and denominator are integers and the denominator is non-zero. In contrast, \(\sqrt{11}\) and \(\sqrt{13}\) are square roots of non-perfect squares and are irrational; \(\pi\) is also irrational. Exam tip: every terminating or recurring decimal is rational.
Correct answer: B. Integers
Explanation: By definition, a rational number is written as \(\frac{m}{n}\), where both \(m\) and \(n\) are integers and \(n\ne0\). Therefore, integers is the correct option. “Natural numbers only” is incorrect because \(m\) can be zero or negative, as in \(-\frac{3}{5}\). Exam tip: in a rational number, the numerator and denominator are integers, but the denominator must not be zero.
