Rational Numbers Test

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Rational Numbers Test - 25

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Weekly Quiz Competition

Question 1
1 / -0

The additive identity of rational number is ______.

A
$$0$$

B
$$1$$

C
$$-1$$

D
$$2$$

Solution

The set of rational numbers forms a group with respect to the addition of rational number in which the additive identity is $$0$$.
As $$a+0=0+a=a$$ for all rational $$a$$.
So the additive identity is $$0$$.
Hence option (A) is correct
Question 2
1 / -0

In the number line, the rational numbers represented by $$'A'$$ is

A
$$\dfrac {2}{8}$$

B
$$\dfrac {3}{8}$$

C
$$\dfrac {1}{8}$$

D
$$\dfrac {2}{3}$$

Solution

Consider the given figure,
Distance between $$0$$ to $$3$$ is divided in equal $$8$$ part

So, each part is equal to $$=\dfrac{3}{8}$$

Hence, point $$A$$ is at $$=\dfrac{3}{8}$$unit

Hence, this is the answer.
Question 3
1 / -0

Between two rational numbers, there exists-

A
No rational number

B
Only one rational number

C
Infinite numbers of rational numbers

D
No irrational number

Solution

Between two rational numbers there are infinitely many rational number for example between $$4$$ and $$5$$ there are $$4.1, 4.2, .4.22, 4.223 .....$$
Hence, $$(C)$$ is the correct answer.
Question 4
1 / -0

How many rational numbers are there between two rational numbers?

A
$$1$$

B
$$0$$

C
unlimited

D
$$100$$

Solution

Consider $$a\,\&\,b$$ as rational numbers
Then a rational number $$n_1$$ can be find in between $$a\,\&\,b$$ such that $$a<n_1<b$$
using the formula $$n_1=\dfrac{a+b}{2}$$

similarly, again we can find a rational number between $$a\,\&\,n_1$$ and $$n_1\,\&\,b$$
and so on...

Hence, between any two distinct rational numbers there are infinitely many rational numbers.
Question 5
1 / -0

Which of the following is an example of distribute property of multiplication over addition for rational numbers ?

A
$$\dfrac{-1}{4} \times \left \{ \dfrac{2}{3} + \left ( \dfrac{-4}{7} \right ) \right \} = \left [ -\dfrac{1}{4} \times \dfrac{2}{3} \right ] + \left [ -\dfrac{1}{4} \times \left ( \dfrac{-4}{7} \right ) \right ]$$

B
$$\dfrac{-1}{4} \times \left \{ \dfrac{2}{3} + \left ( \dfrac{-4}{7} \right ) \right \} = \left [ \dfrac{1}{4} \times \dfrac{2}{3} \right ] - \left ( \dfrac{-4}{7} \right )$$

C
$$\dfrac{-1}{4} \times \left \{ \dfrac{2}{3} + \left ( \dfrac{-4}{7} \right ) \right \} = \dfrac{2}{3} + \left ( -\dfrac{1}{4} \right ) \times \left ( \dfrac{-4}{7} \right )$$

D
$$\dfrac{-1}{4} \times \left \{ \dfrac{2}{3} + \left ( \dfrac{-4}{7} \right ) \right \} = \left \{ \dfrac{2}{3} + \left ( -\dfrac{4}{7} \times \right ) \right \} - \dfrac{1}{4}$$

Solution

Distributive property states that
$$a \times (b + c) = (a \times b) + (a \times c)$$

Let $$a=\dfrac{-1}{4} , b= \dfrac{2}{3}, c= \dfrac{-4}{7} $$
Then $$\dfrac{-1}{4} \times \left \{ \dfrac{2}{3} + \left ( \dfrac{-4}{7} \right ) \right \} = \left [ -\dfrac{1}{4} \times \dfrac{2}{3} \right ] + \left [ -\dfrac{1}{4} \times \left ( \dfrac{-4}{7} \right ) \right ]$$

Hence option A is the correct answer.
Question 6
1 / -0

To get the product 1, we should multiply $$\dfrac{8}{21}$$ by

A
$$\dfrac{8}{21}$$

B
$$\dfrac{-8}{21}$$

C
$$\dfrac{21}{8}$$

D
$$\dfrac{-21}{8}$$

Solution

Let the number be $$x$$
Then $$x \times \dfrac{8}{21} = 1$$
$$\Rightarrow x = \dfrac{21}{8}$$

Therefore, we should multiply with $$\dfrac{21}{8}$$ to get $$1.$$

$$\therefore$$ Option C is the correct answer.
Question 7
1 / -0

$$\dfrac{-3}{8} + \dfrac{1}{7} = \dfrac{1}{7} + \left(\dfrac{-3}{8}\right)$$ is an example to show that

A
Addition of rational numbers is commutative.

B
Rational numbers are closed under addition.

C
Addition of rational number is associative.

D
Rational numbers are distributive under addition.

Solution

Commutative Property states that $$a+b=b+a$$

Here, $$\dfrac{-3}8+\dfrac17=\dfrac{-13}{56}$$

and $$\dfrac17+\left(\dfrac{-3}8\right)=\dfrac{-13}{56}$$

$$\Rightarrow$$ Addition of rational numbers is commutative.

$$\therefore$$ Option A is the correct answer.
Question 8
1 / -0

Multiplication inverse of a negative rational number is

A
A positive rational number.

B
A negative rational number

C
Both positive and negative rational numbers

D
None of the above

Solution

We know that
Product of any rational number and its multiplicative inverse is $$1$$.

Hence the multiplicative inverse of a negative rational number must be $$\text {negative}$$ so that their product is a positive rational number $$1$$.

Hence option B is the correct answer.
Question 9
1 / -0

$$\dfrac{x + y}{2}$$ is a rational number

A
Between x and y

B
Less than x and y both.

C
Greater than x and y both.

D
Less than x but greater than y.

Solution

$$\dfrac{x+ y}{2}$$ lies in between x and y.

Hence, option A is the correct answer.
Question 10
1 / -0

Which of the following expressions shows that rational numbers are associative under multiplication ?

A
$$\dfrac{2}{3} \times (\dfrac{-6}{7} \times \dfrac{3}{5}) = (\dfrac{2}{3} \times \dfrac{-6}{7}) \times \dfrac{3}{5}$$

B
$$\dfrac{2}{3} \times (\dfrac{-6}{7} \times \dfrac{3}{5}) = \dfrac{2}{3} \times (\dfrac{3}{5} \times \dfrac{-6}{7})$$

C
$$\dfrac{2}{3} \times (\dfrac{-6}{7} \times \dfrac{3}{5}) = (\dfrac{3}{5} \times \dfrac{2}{3}) \times \dfrac{-6}{7}$$

D
$$(\dfrac{2}{3} \times \dfrac{-6}{7}) \times \dfrac{3}{5} = (\dfrac{-6}{7} \times \dfrac{3}{5}) \times \dfrac{2}{3}$$

Solution

For associative multiplication property-
$$a \times (b \times c) = (a \times b) \times c$$

let $$a = \dfrac{2}{3}, b = \dfrac{-6}{7}, c = \dfrac{3}{5}$$
Then, $$\dfrac{2}{3} \times (\dfrac{-6}{7} \times \dfrac{3}{5}) = (\dfrac{2}{3} \times \dfrac{-6}{7}) \times \dfrac{3}{5}$$

Hence option A is the correct answer.

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Re-Attempt Weekly Quiz Competition

Rational Numbers Test - 25

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Rational Numbers Test - 25

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Question 1

$$0$$

$$1$$

$$-1$$

$$2$$

Solution

Question 2

$$\dfrac {2}{8}$$

$$\dfrac {3}{8}$$

$$\dfrac {1}{8}$$

$$\dfrac {2}{3}$$

Solution

Question 3

No rational number

Only one rational number

Infinite numbers of rational numbers

No irrational number

Solution

Question 4

$$1$$

$$0$$

unlimited

$$100$$

Solution

Question 5

$$\dfrac{-1}{4} \times \left \{ \dfrac{2}{3} + \left ( \dfrac{-4}{7} \right ) \right \} = \left [ -\dfrac{1}{4} \times \dfrac{2}{3} \right ] + \left [ -\dfrac{1}{4} \times \left ( \dfrac{-4}{7} \right ) \right ]$$

$$\dfrac{-1}{4} \times \left \{ \dfrac{2}{3} + \left ( \dfrac{-4}{7} \right ) \right \} = \left [ \dfrac{1}{4} \times \dfrac{2}{3} \right ] - \left ( \dfrac{-4}{7} \right )$$

$$\dfrac{-1}{4} \times \left \{ \dfrac{2}{3} + \left ( \dfrac{-4}{7} \right ) \right \} = \dfrac{2}{3} + \left ( -\dfrac{1}{4} \right ) \times \left ( \dfrac{-4}{7} \right )$$

$$\dfrac{-1}{4} \times \left \{ \dfrac{2}{3} + \left ( \dfrac{-4}{7} \right ) \right \} = \left \{ \dfrac{2}{3} + \left ( -\dfrac{4}{7} \times \right ) \right \} - \dfrac{1}{4}$$

Solution

Question 6

$$\dfrac{8}{21}$$

$$\dfrac{-8}{21}$$

$$\dfrac{21}{8}$$

$$\dfrac{-21}{8}$$

Solution

Question 7

Addition of rational numbers is commutative.

Rational numbers are closed under addition.

Addition of rational number is associative.

Rational numbers are distributive under addition.

Solution

Question 8

A positive rational number.

A negative rational number

Both positive and negative rational numbers

None of the above

Solution

Question 9

Between x and y

Less than x and y both.

Greater than x and y both.

Less than x but greater than y.

Solution

Question 10

$$\dfrac{2}{3} \times (\dfrac{-6}{7} \times \dfrac{3}{5}) = (\dfrac{2}{3} \times \dfrac{-6}{7}) \times \dfrac{3}{5}$$

$$\dfrac{2}{3} \times (\dfrac{-6}{7} \times \dfrac{3}{5}) = \dfrac{2}{3} \times (\dfrac{3}{5} \times \dfrac{-6}{7})$$

$$\dfrac{2}{3} \times (\dfrac{-6}{7} \times \dfrac{3}{5}) = (\dfrac{3}{5} \times \dfrac{2}{3}) \times \dfrac{-6}{7}$$

$$(\dfrac{2}{3} \times \dfrac{-6}{7}) \times \dfrac{3}{5} = (\dfrac{-6}{7} \times \dfrac{3}{5}) \times \dfrac{2}{3}$$

Solution

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