Question 1

Which of the following is a rational number?

Accepted Answer

A rational number is a number that can be expressed in the form $p/q$, where $p$ and $q$ are integers and $q 
eq 0$. Among the given options, $\sqrt{9} = 3$, which can be written as $3/1$. Therefore, $\sqrt{9}$ is a rational number, while $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ are irrational number

Question 2

The HCF of two numbers 26 and 91 is:

Accepted Answer

To find the HCF of 26 and 91, we find their prime factorizations: $26 = 2 	imes 13$ and $91 = 7 	imes 13$. The common prime factor with the lowest exponent is $13$. Therefore, the HCF of 26 and 91 is 13.

Question 3

What is the LCM of 12 and 15?

Accepted Answer

To find the LCM of 12 and 15, we write their prime factorizations: $12 = 2^2 	imes 3$ and $15 = 3 	imes 5$. The LCM is the product of the highest power of each prime factor involved: $2^2 	imes 3 	imes 5 = 4 	imes 3 	imes 5 = 60$. Thus, the LCM is 60.

Question 4

The decimal expansion of the rational number $13/3125$ will terminate after:

Accepted Answer

A rational number $p/q$ has a terminating decimal expansion if the prime factorization of $q$ is of the form $2^n 5^m$, where $n$ and $m$ are non-negative integers. Here, $3125 = 5^5 = 2^0 5^5$. Since the highest power of 5 is 5, the decimal expansion will terminate after 5 decimal places.

Question 5

Which of the following numbers is irrational?

Accepted Answer

An irrational number cannot be expressed as a simple fraction. Among the options, $\sqrt{4} = 2$ (rational), $\sqrt{9} = 3$ (rational), and $\sqrt{25} = 5$ (rational). However, $\sqrt{7}$ cannot be simplified to a rational number, making it an irrational number.

Real Numbers