Tag: proofs of irrationality

Questions Related to proofs of irrationality

State whether the given statement is True or False :

The number $6+\sqrt { 2 } $ is irrational.
maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

$Here \  6 \text{ is a  rational number and }$$\sqrt2$ is a $irrational$ number


And $\text{the addition of rational and irrational is always an irrational number}$

So that $(6+\sqrt2)$ is an irrational number.

hence option A is correct

State whether the given statement is True or False :
If $p,  q $ are prime positive integers, then $\sqrt { p } +\sqrt { q } $ is an irrational number.
maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:
$\sqrt{p}+\sqrt{q}$ is rational ........ assumption
$\sqrt{p}+\sqrt{q}=\dfrac{a}{b}$
squaring, we get
$p+q+2\sqrt{pq}=\left(\dfrac{a}{b}\right)^{2}$

$\sqrt{pq}=\dfrac{1}{2}\left[\left(\dfrac{a}{b}\right)^{2}-p-q\right] - (i)$

Now, $p$ & $q$ are prime positive numbers so, $\sqrt{p}$ and $\sqrt{q}$ is irrational also $\sqrt{pq}$ 

so in (i)
Irrational $=$ rational $\Rightarrow$ which is a contradiction

$\Rightarrow\ \sqrt p+\sqrt q$ is a irrational number if $p,q$ are prime positive numbers

Prove following equation as irrational 

maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. $2+\sqrt {3}$

  2. $2-\sqrt {3}$

  3. $3\sqrt {2}+\sqrt {3}$

  4. $\dfrac {1}{\sqrt {2}}$

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

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Correct Option: A

$\sqrt { 7 }$ is a

maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. an integer

  2. an irrational number

  3. a rational number

  4. none of these

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Correct Option: B
Explanation:

$\sqrt 7 $ is a rational number


State True or False:

$4\, - \,5\sqrt 2 $ is irrational if $\sqrt 2 $ is irrational.

maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

We know that difference of a rational and irrational number is also irrational, 


So if $\sqrt{2}$ is irrational 4-5$\sqrt{2}$ is also irrational. 

GCF of $99$ and $100$ is __________

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  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
Correct Option: A
Explanation:

when we divide $100$ with $99$ we will get remainder $1$ and hence next we divide 99 with $1$ which gives us remainder $0$.


therefore $1$ is the HCF or GCF of $99 $ and $100$