Question

An irrational number between $5$ and $6$ is:

• A. a) $\sqrt{5+6}$
• B. b) $\frac{1}{2}(5+6)$
• C. c) $\sqrt{5\times 6}$
• D. d) None of these.

Answer 1

The correct option is C

c) $\sqrt{5\times 6}$

Explanation for the correct option:

Option (c): Let $x=\sqrt{5\times 6}$

So, $x^2=5\times 6=30$

Since, 30 is a number which lie between $5^2$ and $6^2$.

Therefore, $\sqrt{30}$ lies between $\sqrt{5^2}$ and $\sqrt{6^2}$.

Thus, $\sqrt{5\times 6}$ is the irrational number which lies between $5$ and $6$.

Explanation for the incorrect options:

Option (a): Let $y=\sqrt{5+6}$

So, $y^2=5+6=11$

Since, 11 does not lie between $5^2$ and $6^2$.

Therefore, $\sqrt{11}$ does not lie between $\sqrt{5^2}$ and $\sqrt{6^2}$.

Thus, $\sqrt{5+6}$ is an irrational number but it does not lie between $5$ and $6$.

Option (b): $\frac{1}{2}(5+6)$

On simplification, we get:

$\frac{1}{2}(5+6)=\frac{11}{2}$ which is a rational number as it is represented in the form of $\frac{p}{q}$where $p$ and $q$ are integers and $q\ne 0$.

Thus, $\frac{1}{2}(5+6)$ is a rational number.

Hence, option $\left(c\right)$ is the correct answer.