Rational Numbers Test

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Rational Numbers Test - 27

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Question 1
1 / -0

Addition of rational numbers does not satisfy which of the following property?

A
Commutative

B
Associative

C
Closure

D
None

Solution

Addition of rational numbers satisfy commutative, associative and closure properties.
Question 2
1 / -0

If A : Rational numbers are always closed under division and
R : Division by Zero is not defined, then which of the following statement is correct?

A
A is True and R is the correct explanation of A

B
A is False and R is the correct explanation of A

C
A is True and R is False

D
None of these

Solution

Let us take two rational numbers $$\displaystyle \frac{3}{5}$$ and $$\displaystyle \frac{0}{3}$$
Then the division $$\dfrac{3}{5}$$ $$\div$$ $$\dfrac{0}{3}$$ is not a rational number.
So division is not closure because division by zero is not defined.
$$\therefore$$ option $$B$$ is correct
Question 3
1 / -0

Which of the following alternatives is wrong? Given that
(i) difference of two rational numbers is a rational number
(ii) subtraction is commutative on rational numbers
(iii) addition is not commutative on rational numbers.

A
(ii) and (iii)

B
(i) only

C
(i) and (iii)

D
All the above

Solution

1. When a rational number is multiplied, add or subtracted by any rational number, result is a rational number. For division, denominator must not be zero.
2. Commutative property is only followed by addition and multiplication.

Hence, statements (ii) and (iii) are wrong.
Question 4
1 / -0

Which of the following statements is true?

A
The reciprocals $$1$$ and $$-1$$ are themselves

B
Zero has no reciprocal

C
The product of two rational numbers is a rational number

D
All the above

Solution

A. Reciprocal of a number is the number divided by $$1$$. So $$1$$ and $$-1$$ when divided by $$1$$ give the same result.
Hence, True.

B. When $$1$$ is divided by $$0$$, the result is not defined. Hence, no reciprocal of zero.

C. When a rational number is multiplied, add or subtracted by any rational number, result is a rational number. For division, denominator must not be zero.

Hence, all the above statements are correct.
Question 5
1 / -0

Name the property of multiplication illustrated by
$$\displaystyle \frac{-4}{3} \times \left(\displaystyle \frac{6}{5} + \frac{8}{7} \right) = \left(\displaystyle \frac{-4}{3} \times \frac{6}{5} \right) + \left(\displaystyle \frac{-4}{3} \times \frac{8}{7} \right)$$

A
Associative property

B
Commutative property

C
Distributive property

D
None of these

Solution

We know that for $$a , b , c $$ in $$ R$$ ;the distributive property is given by, $$a ( b + c ) = a b + a c$$.
$$ \therefore - \dfrac{4}{3} \times \left ( \dfrac{6}{5} + \dfrac{8}{7} \right) = \left ( \dfrac{-4}{3} \times \dfrac{6}{5} \right ) + \left ( -\dfrac{4}{3} \times \dfrac{8}{7} \right) $$
can be illustrated by distributive property considering $$ a = - \dfrac{4}{3} ; \, b = \dfrac{6}{5} ; \, c = \dfrac{8}{7}$$
Hence, option C distributive property is correct answer.
Question 6
1 / -0

Find the multiplicative inverse of the following:
$$-13$$, $$\displaystyle-\frac{13}{19}$$ and $$-\displaystyle\frac{5}{8}\times\displaystyle-\frac{3}{7}$$

A
$$-\displaystyle\frac{1}{13}\;;\;\displaystyle-\frac{19}{13}\;;\;\displaystyle\frac{56}{15}$$

B
$$-\displaystyle\frac{1}{13}\;;\;\displaystyle\frac{19}{13}\;;\;\displaystyle\frac{56}{15}$$

C
$$\displaystyle\frac{1}{13}\;;\;-\displaystyle\frac{19}{13}\;;\;\displaystyle\frac{56}{15}$$

D
$$\displaystyle\frac{1}{13}\;;\;\displaystyle\frac{19}{13}\;;\;\displaystyle\frac{56}{15}$$

Solution

The multiplicative inverse of $$-13$$ is $$(-13)^{-1}=\displaystyle\frac{1}{-13}$$

The multiplicative inverse of
$$\;\;\;\;\displaystyle\frac{-13}{19}$$

is

$$\begin{pmatrix}\displaystyle\frac{-13}{19}\end{pmatrix}^{-1}=\displaystyle\frac{19}{-13}$$.

We

have,

$$\displaystyle\frac{-5}{8}\times\displaystyle\frac{-3}{7}=\displaystyle\frac{-5\times-3}{8\times7}=\displaystyle\frac{15}{56}$$
$$\;\;\;\;$$ The multiplicative inverse of $$\displaystyle\frac{15}{56}$$ is
$$\;\;\;\;\begin{pmatrix}\displaystyle\frac{15}{56}\end{pmatrix}^{-1}=\displaystyle\frac{56}{15}$$
Question 7
1 / -0

The multiplicative inverse of $$\displaystyle \left ( \frac{1}{3} \right )^{-2}$$ is

A
$$9$$

B
$$\displaystyle \frac{1}{9}$$

C
$$\displaystyle \frac{1}{4}$$

D
None of these

Solution

Given rational number is $$\left( \cfrac { 1 }{ 3 }\right ) ^{ -2}$$
To find out,
The multiplicative inverse of the given rational number.

Let the multiplicate inverse be $$x$$.
We know that, the multiplicative inverse of a number is another number which when multiplied by the original number yields $$1$$ as a product.
We also know that, $$\left( \cfrac { a }{ b }\right ) ^{ -m}=\left( \cfrac { b}{ a }\right ) ^{ m}$$
$$\therefore\ \left( \cfrac { 1 }{ 3 }\right ) ^{ -2} =3^{ 2 }$$
$$ = 9$$
Now, $$9\times x=1$$
$$\therefore \ x=\dfrac19$$

Hence, the multiplicative inverse of $$\left( \cfrac { 1 }{ 3 }\right ) ^{ -2}$$ is $$ \cfrac 19$$.
Question 8
1 / -0

Choose the rational number which does not lie between rational numbers $$ \displaystyle \frac{3}{5} $$ and $$ \displaystyle \frac{2}{3} $$ :

A
$$ \displaystyle \frac{46}{75} $$

B
$$ \displaystyle \frac{47}{75} $$

C
$$ \displaystyle \frac{49}{75} $$

D
$$ \displaystyle \frac{50}{75} $$

Solution

For $$\dfrac{3}{5}$$ multiply numerator and denominator by $$15$$ to make denominator $$75$$ that comes into $$\dfrac{45}{75}.$$
Similarly doing for second then we have $$\dfrac{50}{75}.$$
Now question is asking about rational lying between them.
So, we need to check the numerator only that lies in between $$45$$ and $$50$$ or not.
Clearly $$D$$ is correct.
Question 9
1 / -0

Choose the rational number which does not lie between rational numbers $$\displaystyle\frac{3}{5}$$ and $$\displaystyle\frac{2}{3}$$

A
$$\displaystyle\frac{46}{75}$$

B
$$\displaystyle\frac{47}{75}$$

C
$$\displaystyle\frac{49}{75}$$

D
$$\displaystyle\frac{50}{75}$$

Solution

Consider the given rational numbers,

$$\dfrac{3}{5}$$ and $$\dfrac{2}{3}$$

Now,

$$\dfrac{3\times 15}{5\times 15}$$ and $$\dfrac{2\times 25}{3\times 25}$$

$$\dfrac{45}{75}$$ and $$\dfrac{50}{75}$$

Now, rational number which does not lie between thes rational numbers is,

$$\dfrac{50}{75}$$

Hence, this is the answer.

.
Question 10
1 / -0

Write the multiplicative inverse of each of the following rational numbers:
$$7$$; $$-11$$; $$\displaystyle\frac{2}{5}$$; $$\displaystyle\frac{-7}{15}$$

A
$$\displaystyle\frac{-1}{7};\;\displaystyle\frac{1}{-11};\;\displaystyle\frac{5}{2};\;\displaystyle\frac{15}{-7}$$

B
$$\displaystyle\frac{1}{7};\;\displaystyle\frac{1}{-11};\;\displaystyle\frac{5}{2};\;\displaystyle\frac{15}{-7}$$

C
$$\displaystyle\frac{1}{7};\;\displaystyle\frac{1}{11};\;\displaystyle\frac{5}{2};\;\displaystyle\frac{15}{-7}$$

D
$$\displaystyle\frac{1}{7};\;\displaystyle\frac{1}{-11};\;\displaystyle\frac{-5}{2};\;\displaystyle\frac{15}{-7}$$

Solution

Multiplicative inverse of $$7$$ is $$\displaystyle\frac{1}{7}$$ i.e. $$7^{-1}=\displaystyle\frac{1}{7}$$

Multiplicative inverse of $$-11$$ is $$\displaystyle\frac{1}{-11}$$ i.e. $$(-11)^{-1}=\displaystyle\frac{1}{-11}$$

Multiplicative inverse of $$\displaystyle\frac{2}{5}$$ is $$\displaystyle\frac{5}{2}$$ i.e. $$\begin{pmatrix}\displaystyle\frac{2}{5}\end{pmatrix}^{-1}=\displaystyle\frac{5}{2}$$

Multiplicative inverse of $$\displaystyle\frac{-7}{15}$$ is $$\displaystyle\frac{15}{-7}$$ i.e $$\begin{pmatrix}\displaystyle\frac{-7}{15}\end{pmatrix}^{-1}=\displaystyle\frac{15}{-7}$$

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Rational Numbers Test - 27

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Rational Numbers Test - 27

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Question 1

Commutative

Associative

Closure

None

Solution

Question 2

A is True and R is the correct explanation of A

A is False and R is the correct explanation of A

A is True and R is False

None of these

Solution

Question 3

(ii) and (iii)

(i) only

(i) and (iii)

All the above

Solution

Question 4

The reciprocals $$1$$ and $$-1$$ are themselves

Zero has no reciprocal

The product of two rational numbers is a rational number

All the above

Solution

Question 5

Associative property

Commutative property

Distributive property

None of these

Solution

Question 6

$$-\displaystyle\frac{1}{13}\;;\;\displaystyle-\frac{19}{13}\;;\;\displaystyle\frac{56}{15}$$

$$-\displaystyle\frac{1}{13}\;;\;\displaystyle\frac{19}{13}\;;\;\displaystyle\frac{56}{15}$$

$$\displaystyle\frac{1}{13}\;;\;-\displaystyle\frac{19}{13}\;;\;\displaystyle\frac{56}{15}$$

$$\displaystyle\frac{1}{13}\;;\;\displaystyle\frac{19}{13}\;;\;\displaystyle\frac{56}{15}$$

Solution

Question 7

$$9$$

$$\displaystyle \frac{1}{9}$$

$$\displaystyle \frac{1}{4}$$

None of these

Solution

Question 8

$$ \displaystyle \frac{46}{75} $$

$$ \displaystyle \frac{47}{75} $$

$$ \displaystyle \frac{49}{75} $$

$$ \displaystyle \frac{50}{75} $$

Solution

Question 9

$$\displaystyle\frac{46}{75}$$

$$\displaystyle\frac{47}{75}$$

$$\displaystyle\frac{49}{75}$$

$$\displaystyle\frac{50}{75}$$

Solution

Question 10

$$\displaystyle\frac{-1}{7};\;\displaystyle\frac{1}{-11};\;\displaystyle\frac{5}{2};\;\displaystyle\frac{15}{-7}$$

$$\displaystyle\frac{1}{7};\;\displaystyle\frac{1}{-11};\;\displaystyle\frac{5}{2};\;\displaystyle\frac{15}{-7}$$

$$\displaystyle\frac{1}{7};\;\displaystyle\frac{1}{11};\;\displaystyle\frac{5}{2};\;\displaystyle\frac{15}{-7}$$

$$\displaystyle\frac{1}{7};\;\displaystyle\frac{1}{-11};\;\displaystyle\frac{-5}{2};\;\displaystyle\frac{15}{-7}$$

Solution

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