An irrational number between 5 and 6 is (a) (1)/(2)(5+6) (b) sqrt(5+6) (c) sqrt(5×6) (d) none of these

Calculate option (a): Calculate option (a): (1)/(2)(5+6). (1/2)(5+6) = (1/2)(11) = 5.5Check rationality of 5.5: Check if 5.5 is irrational. 5.5 can be expressed as a fraction, so it is rational.Calculate option (b): Calculate option (b): sqrt(5+6). sqrt(5+6) = sqrt(11)Check irrationality of sqrt(11): Check if sqrt(11) is irrational. sqrt(11) cannot be expressed as a fraction, so it is irrational.Calculate option (c): Calculate option (c): sqrt(5×6). sqrt(5×6) = sqrt(30)Check irrationality of sqrt(30): Check if sqrt(30) is irrational. sqrt(30) cannot be expressed as a fraction, so it is irrational.Determine irrational numbers: Determine if any of the irrational numbers are between 5 and 6. sqrt(11) ≈ 3.3166 and sqrt(30) ≈ 5.4772; only sqrt(30) is between 5 and 6.

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Q. An irrational number between

5

and

6

\newline

(a)

\frac{1}{2}(5+6)

\newline

(b)

\sqrt{5+6}

\newline

(c)

\sqrt{5 \times 6}

\newline

(d) none of these

Calculate option (a): Calculate option (a): $(\frac{1}{2})(5+6)$ . $(\frac{1}{2})(5+6) = (\frac{1}{2})(11) = 5.5$
Check rationality of $5.5$ : Check if $5.5$ is irrational. $\newline$ $5.5$ can be expressed as a fraction, so it is rational.
Calculate option (b): Calculate option (b): $\sqrt{5+6}$ . $\newline$ $\sqrt{5+6} = \sqrt{11}$
Check irrationality of $\sqrt{11}$ : Check if $\sqrt{11}$ is irrational. $\newline$ $\sqrt{11}$ cannot be expressed as a fraction, so it is irrational.
Calculate option (c): Calculate option (c): $\sqrt{5\times6}$ . $\newline$ $\sqrt{5\times6} = \sqrt{30}$
Check irrationality of $\sqrt{30}$ : Check if $\sqrt{30}$ is irrational. $\newline$ $\sqrt{30}$ cannot be expressed as a fraction, so it is irrational.
Determine irrational numbers: Determine if any of the irrational numbers are between $5$ and $6$ . $\sqrt{11} \approx 3.3166$ and $\sqrt{30} \approx 5.4772$ ; only $\sqrt{30}$ is between $5$ and $6$ .