Tag: existence of irrational numbers

Questions Related to existence of irrational numbers

$\sqrt {5}$ is a\an ......... number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. rational

  2. whole

  3. integer

  4. irrational

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Correct Option: D
Explanation:

$\sqrt {5} = \dfrac {a}{b}$

$b\sqrt {5} = a$ $(a$ and $b$ are co-prime i.e. they have no common factors$) ...(1)$ 
$5b^{2} = a^{2}$ (squaring both sides)
Therefore $5$ divides $a^{2}$
As per Fundamental Theorem of Arithmetic, $5$ divides $a.$
Let's take it as $a = 5c,$
$5b^{2} = 25 c^{2}$
$b^{2} = 5c^{2}$
As per Fundamental Theorem of Arithmetic, $5$ divides $a.$
So $a$ and $b$ have $5$ as a common factor but $a$ and $b$ have only $1$ common factor $1$ from equation $(1),$ so it is not rational.
So, we conclude that $\sqrt {5}$ is irrational.
Therefore, $D$ is the correct answer.

How many irrational numbers are there between $2$ and $6$?

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. $1$

  2. $3$

  3. $4$

  4. $10$

  5. Infinitely many

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Correct Option: E
Explanation:

There can be infinite number of irrational numbers between any two rational numbers. Hence the answer is infinitely many.

$\sqrt{21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15}}=$...........

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. $\sqrt{5}-2+2\sqrt{3}$

  2. $\sqrt{5}-\sqrt{4}-\sqrt{12}$

  3. $-\sqrt{5}+\sqrt{4}+\sqrt{12}$

  4. $-\sqrt{5}-\sqrt{4}+\sqrt{12}$

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Correct Option: C

State whether the following statements are true or false. 
$\sqrt {n}$ is not irrational if n is a perfect square

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

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Correct Option: A
Explanation:

False ,

$\sqrt{4}=2$ where 2 is a rational number.Here n is perfect square the  $\sqrt{n}$ is rational number 
$\sqrt{5}=2.236..$ is not rational  number But it is irrational number . here n is not a perfect square the  $\sqrt{n}$ is  irrational  number
So $\sqrt{n}$ is not irrational number if n is perfect square

If $p$ is prime, then $\sqrt {p}$ is:

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. Composite number

  2. Rational number

  3. Positive integer

  4. Irrational number

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Correct Option: D
Explanation:

SInce, we know that prime numbers are those which are never perfect square and not divisible by any other number except by itself.
which are $2,3,5,7,...$
Clearly, if $p$ is prime then $\sqrt p $ is irrational number.
Option $D$ is correct. 

State the following statement is true or false

$3\sqrt{18}$ is an irrational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

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Correct Option: A
Explanation:

$3\sqrt{18}=9\sqrt{2},$ which is the product of a rational and an irrational number and so an irrational number.

$6+\sqrt{2}$ is a rational number.

maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. True

  2. False

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Correct Option: B
Explanation:

Let's assume that $6+\sqrt2$ is rational..... 

then 

$6+\sqrt2 = p/q $

$\sqrt2 =( p-6q)/(q) $ 

now take $p-6q$ to be P and $q$ to be Q........where P and Q are integers 

which means, $\sqrt2= P/Q$...... 

But this contradicts the fact that $\sqrt2$ is rational 

So our assumption is wrong and $6+\sqrt2$ is irrational.