#### (i) State whether the given statement is true or false? Justify your answer by an example. $\frac{\sqrt{2}}{3}$ is a rational number.(ii) State whether the given statement is true or false? Justify your answer by an example. There are infinitely many integers between any two integers.(iii) State whether the given statement is true or false? Justify your answer by an example. Number of rational numbers between 15 and 18 is finite.(iv) State whether the given statement is true or false? Justify your answer by an example. There are numbers which cannot be written in the form $\frac{p}{q}$ ,$q \neq 0$, p and q both are integers.(v) State whether the given statement is true or false? Justify your answer by an example. The square of an irrational number is always rational.(vi) State whether the given statement is true or false? Justify your answer by an example.  $\frac{\sqrt{12}}{\sqrt{3}}$ is not a rational number as  $\sqrt{12}$ and $\sqrt{3}$ are not integers.(vii) State whether the given statement is true or false? Justify your answer by an example. $\frac{\sqrt{15}}{\sqrt{3}}$is written in the form $\frac{p}{q}, q \neq 0$and so it is a rational number.

(i)

Solution.

Any number which can be represented in the form of p/q, where q is not equal to zero, is a rational number. So it is basically a fraction with non-zero denominator.
Examples: $4/5, 2/3, -6/7$
Irrational numbers are real numbers which cannot be represented as simple fractions.
Examples: $\sqrt{2},\sqrt{3},\pi$
The given number is $\frac{\sqrt{2}}{3}$.
Here, $\sqrt{2}$ is an irrational number, and 3 is a rational number. We know that when we divide an irrational number by a non-zero rational, it always gives an irrational number.
Hence, the given number is irrational.
Therefore, the given statement is false.

(ii)

Solution.

An integer is a number which can be written without fractional components or decimal representation. They can be positive, negative or zero.
Examples: $\left \{ .....-4,-3,-2,-1,0,1,2,3,4,..... \right \}$
The given statement is “There are infinitely many integers between any two integers.”
This is false, because between two integers (like 1 and 9), there does not exist infinite integers.
Also, if we consider two consecutive integers (like 8 and 9), there does not exist any integer between them.
Therefore, the given statement is false.

(iii)

Solution.

Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. So it is basically a fraction with non-zero denominator.
Examples: $4/5, 2/3, -6/7$
So, it is a type of real number.
The given statement is: Number of rational numbers between 15 and 18 is finite.
If we see the definition of rational numbers as mentioned above, the given statement is false, because between any two rational numbers there exist infinitely many rational numbers.
Here, we have rational numbers between 15 and 18 as:
$16,16.1\left ( =\frac{161}{10} \right ),16.2\left ( =\frac{162}{10} \right ),16.12\left ( =\frac{1612}{100} \right ),.....$ and infinitely more.
Therefore, the given statement is false.

Solution:

There are infinitely many numbers which cannot be written in the form $\frac{p}{q}$ ,$q \neq 0$, p and q both are integers. These numbers are called irrational numbers

(v)

Solution.

Any number which can be represented in the form of p/q where q is not equal to zero is a rational number.
Irrational numbers are real numbers which cannot be represented as simple fractions.
The given statement is: The square of an irrational number is always rational.
This is False, e.g., let us consider irrational numbers $\sqrt{2}$ and $\sqrt[4]{2}$
$(a)(\sqrt{2})^{2}=2,$which is a rational number.
$(b)(\sqrt[4]{2})^{2}=\sqrt{2},$ which is an irrational number.
Hence, square of an irrational number is not always a rational number.
Therefore, the given statement is false.

(vi)

Solution:

$\frac{\sqrt{12}}{\sqrt{3}}=\frac{\sqrt{4 \times 3}}{\sqrt{3}}=\frac{\sqrt{4} \times \sqrt{3}}{\sqrt{3}}=2 \times 1=2$which is a rational number.

(vii)

Solution:

$\frac{\sqrt{15}}{\sqrt{3}}=\frac{\sqrt{5 \times 3}}{\sqrt{3}}=\frac{\sqrt{5} \times \sqrt{3}}{\sqrt{3}}=\sqrt{5}$which is an irrational number.