Statement I :The rational number between $2. \bar{2} and 2. \bar{3}$ is $\frac{41}{18}$
Statement I : $\frac{2}{3} n + 4$ is a rational number for $n \in z$

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Both A and B are true

Both A and B are false

A is true and B is False

A is false and B is true

answer is A.

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Detailed Solution

(I):

2. \bar{2} = \frac{22 - 2}{9} = \frac{20}{9}

$2. \bar{3} = \frac{23 - 2}{9} = \frac{21}{9}$

$∴$ The rational number between $2. \bar{2} and 2. \bar{3}$

$=$ The rational number between $\frac{20}{9} and \frac{21}{9}$

$= \frac{(\frac{20}{9} + \frac{21}{9})}{2} = \frac{(\frac{20 + 21}{9})}{2} = \frac{41}{18}$

(II): $\frac{2}{3} n + 4, n \in z$

For $n = - 3 \Rightarrow \frac{2}{3} (- 3) + 4$

$n = 0 \Rightarrow \frac{2}{3} \times 0 + 4$

$n = 3 \Rightarrow \frac{2}{3} \times 3 + 4$

$n = 1 \Rightarrow \frac{2}{3} \times 1 + 4 = \frac{2}{3} + 4 = \frac{2 + 12}{3} = \frac{14}{3} \in Q$

So, $\frac{2}{3} n + 4, n \in z$ is a rational number for $n \in z$

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