Rational Numbers — Statement I-II Type Questions

Statement I: Additive inverse of $\dfrac{-5}{-3} \text{ is } \dfrac{3}{5}$

Statement II: Sum of any rational number and its additive inverse is 0.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

First express the rational number with a positive denominator, then change its sign.

$\Rightarrow \dfrac{-5}{-3} = \dfrac{-5 \times (-1)}{-3 \times (-1)} = \dfrac{5}{3} \\[1em] \Rightarrow \text{Additive inverse} = \dfrac{-5}{3}$

As the additive inverse is $\dfrac{-5}{3}$ and not $\dfrac{3}{5}$, Statement I is false.

For every rational number $\dfrac{p}{q}$, we have $\dfrac{p}{q} + \left(-\dfrac{p}{q}\right) = 0$

So, Statement II is true.

Hence, Option 2 is the correct option.

Statement I: All rational numbers have a multiplicative inverse.

Statement II: Multiplicative inverse is also called reciprocal.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

The rational number 0 has no multiplicative inverse, since there is no rational number which when multiplied by 0 gives 1.

So, Statement I is false.

For a non-zero rational number $\dfrac{p}{q}$, the number $\dfrac{q}{p}$ is called its multiplicative inverse or reciprocal.

So, Statement II is true.

Hence, Option 2 is the correct option.

Statement I: Addition of three rational numbers is associative.

Statement II: Subtraction of three rational numbers is associative.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

For any three rational numbers, we have $\dfrac{p}{q} + \left(\dfrac{r}{s} + \dfrac{t}{u}\right) = \left(\dfrac{p}{q} + \dfrac{r}{s}\right) + \dfrac{t}{u}$

So, Statement I is true.

For any three rational numbers, we have $\dfrac{p}{q} - \left(\dfrac{r}{s} - \dfrac{t}{u}\right) \neq \left(\dfrac{p}{q} - \dfrac{r}{s}\right) - \dfrac{t}{u}$

So, Statement II is false.

Hence, Option 1 is the correct option.

Statement I: Multiplication of any rational number by 0 is 0.

Statement II: Division of any rational number by 0 is not defined.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

When a rational number is multiplied by zero, the product is always zero, i.e. $\dfrac{p}{q} \times 0 = 0 = 0 \times \dfrac{p}{q}$

So, Statement I is true.

Divisibility of a rational number by zero is not defined, i.e. $\dfrac{p}{q} \div 0$ does not exist.

So, Statement II is true.

Hence, Option 3 is the correct option.