Number Systems Test

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Number Systems Test - 27

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Weekly Quiz Competition

Question 1
1 / -0

$$\surd 4+\surd 83,$$ the correct option is

A
Does not exists as quadratic surd

B
$$2+\surd 83$$

C
$$\surd 83-2$$

D
Does not exist as real numbers

Solution

$$\sqrt{4}+\sqrt{83}=\sqrt{2\times 2}+\sqrt{83}=2+\sqrt{83}$$
Question 2
1 / -0

$${ \left( \dfrac { 2 }{ 3 } \right) }^{ 0 }=?$$

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

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

C
$$1$$

D
$$0$$

Solution

Given $${ \left( \dfrac { 2 }{ 3 } \right) }^{ 0 }$$
By using tha law of exponents i.e., $$\left(\dfrac{a}{b}\right)^0=1$$
$$\therefore { \left( \dfrac { 2 }{ 3 } \right) }^{ 0 }=1$$
Question 3
1 / -0

A rational number is defined as a number that can be expressed in the form $$\dfrac{p}{q}$$ where $$p$$ and $$q$$ are integers and

A
$$q=0$$

B
$$q=1$$

C
$$q\neq 1$$

D
$$q\neq 0$$

Solution

According to the definition of a rational number, it can be expressed in the form of $$\dfrac{p}{q}$$, where $$p$$ and $$q$$ are integers and $$q\neq$$0
Question 4
1 / -0

If $$y$$ is any non-zero integer, then $$y^0$$ is equal to ....

A
$$1$$

B
$$0$$

C
$$-1$$

D
Not found

Solution

Using the law of exponents, $$a^0 =1$$

Therefore $$y^0=1$$
Question 5
1 / -0

Find the positive rational number out of the following.

A
$$\dfrac{-4}{9}$$

B
$$\dfrac{-16}{36}$$

C
$$\dfrac{-20}{-45}$$

D
$$\dfrac{28}{-63}$$

Solution

$$\dfrac{-20}{-45}=\dfrac{20}{45}$$ is the only positive rational number among all in the above-given questions . Therefore, the option (c) is the odd one.
Question 6
1 / -0

$$\sqrt 3$$ is

A
an irrational number

B
a rational number

C
a composite number

D
a prime number

Solution
Question 7
1 / -0

After rationalising the denominator of $$\dfrac{7}{3\sqrt{3}-2\sqrt{2}}$$, we get the denominator of the resultant equivalent expression as:

A
$$5$$

B
$$13$$

C
$$19$$

D
$$35$$

Solution

$$\dfrac { 7 }{ 3\sqrt { 3 } -2\sqrt { 2 } } =\dfrac { 7 }{ 3\sqrt { 3 } -2\sqrt { 2 } } \times \left( \dfrac { 3\sqrt { 3 } +2\sqrt { 2 } }{ 3\sqrt { 3 } +2\sqrt { 2 } } \right) =\dfrac { 7\left( 3\sqrt { 3 } +2\sqrt { 2 } \right) }{ 27-8 } =\dfrac { 7\left( 3\sqrt { 3 } +2\sqrt { 2 } \right) }{ 19 }$$. Hence, the denominator of the equivalent expression is $$19.$$
Question 8
1 / -0

Which of the following is irrational number?

A
$$\sqrt{\dfrac{8}{18}}$$

B
$$\sqrt{\dfrac{12}{3}}$$

C
$$\sqrt{\dfrac{28}{8}}$$

D
$$\sqrt{81}$$

Solution

All numbers that can be written in the form of $$\dfrac{p}{q} $$, where $$p$$ and $$q$$ are integers are rational numbers.

Option $$A :$$
$$\sqrt{\dfrac{8}{18}} $$ $$= \sqrt{\dfrac{4}{9}} $$

$$= \dfrac{2}{3} $$
Hence, it is rational.

Option $$B :$$
$$\sqrt{\dfrac{12}{3}} $$ $$=\sqrt{4}$$
$$=2 $$
Hence, it is a rational number.

Option $$C :$$
$$\sqrt{\dfrac{28}{8}}$$ $$ =\sqrt{\dfrac{7}{2}} $$
This cannot be simplified further.
This is an irrational number.

Option $$D :$$
$$\sqrt{81} $$ $$=9$$
This is a rational number.

Hence, option $$C$$ is correct.
Question 9
1 / -0

The product of any two irrational numbers

A
Is always an irrational number

B
Is always a rational number

C
Is always an integer

D
Can be rational or irrational

Solution

The product of two irrational numbers can be rational or irrational depending on the two numbers.

For example, $$\sqrt{2}\times\sqrt{2}$$ is $${2}$$, which is a rational number whereas $$\sqrt{2}\times\sqrt{3}$$ is $$\sqrt{6}$$, which is an irrational number.

Hence, option $$D.$$
Question 10
1 / -0

The value of $$1.999...$$ in the form $$\dfrac {p}{q}$$, where $$p$$ and $$q$$ are integers and $$q\neq 0$$, is:

A
$$\dfrac {1}{9}$$

B
$$\dfrac {19}{10}$$

C
$$\dfrac {1999}{1000}$$

D
$$2$$

Solution

Let $$x=1.999...$$
Since, one digit is repeating, we multiply $$x$$ by $$10$$, we get
$$10x = 19.99...$$
So, $$10x=18+1.999...=18+x$$
Therefore, $$10x-x=18$$, i.e., $$9x=18$$
i.e., $$x=\dfrac { 18 }{ 9 } =\dfrac { 2 }{ 1 } =2$$

Hence, option $$D$$ is correct answer.

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Re-Attempt Weekly Quiz Competition

Number Systems Test - 27

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Number Systems Test - 27

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SHARING IS CARING

Question 1

Does not exists as quadratic surd

$$2+\surd 83$$

$$\surd 83-2$$

Does not exist as real numbers

Solution

Question 2

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

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

$$1$$

$$0$$

Solution

Question 3

$$q=0$$

$$q=1$$

$$q\neq 1$$

$$q\neq 0$$

Solution

Question 4

$$1$$

$$0$$

$$-1$$

Not found

Solution

Question 5

$$\dfrac{-4}{9}$$

$$\dfrac{-16}{36}$$

$$\dfrac{-20}{-45}$$

$$\dfrac{28}{-63}$$

Solution

Question 6

an irrational number

a rational number

a composite number

a prime number

Solution

Question 7

$$5$$

$$13$$

$$19$$

$$35$$

Solution

Question 8

$$\sqrt{\dfrac{8}{18}}$$

$$\sqrt{\dfrac{12}{3}}$$

$$\sqrt{\dfrac{28}{8}}$$

$$\sqrt{81}$$

Solution

Question 9

Is always an irrational number

Is always a rational number

Is always an integer

Can be rational or irrational

Solution

Question 10

$$\dfrac {1}{9}$$

$$\dfrac {19}{10}$$

$$\dfrac {1999}{1000}$$

$$2$$

Solution

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