Tag: properties of irrational numbers

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

The number $\displaystyle\frac{3-\sqrt{3}}{3+\sqrt{3}}$ is 

  1. Rational

  2. Irrational

  3. Both

  4. Can't say
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$Here,\quad we\quad will\quad carry\quad out\quad rationalization.\quad \ \frac { 3-\sqrt { 3 }  }{ 3+\sqrt { 3 }  } =\frac { 3-\sqrt { 3 }  }{ 3+\sqrt { 3 }  } x\frac { 3-\sqrt { 3 }  }{ 3-\sqrt { 3 }  } =\frac { { (3-\sqrt { 3 } ) }^{ 2 } }{ (3+\sqrt { 3) } (3-\sqrt { 3 } ) } =\frac { 9+3-6\sqrt { 3 }  }{ 9-3 } =\frac { 12-6\sqrt { 3 }  }{ 6 } =\frac { 2-\sqrt { 3 }  }{ 1 } \ Since\quad \sqrt { 3 } is\quad irrational\quad number\quad and\quad subtraction\quad of\quad rational\quad and\quad irrational\quad is\quad irrational.\ The\quad given\quad expression\quad is\quad irrational.\ \quad $

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

Give an example of two irrational numbers, whose sum is a rational number

  1. $4 +\sqrt{5},-\sqrt{5}$
  2. $4 +\sqrt{5},\sqrt{5}$
  3. $4 -\sqrt{5},-\sqrt{5}$
  4. $ 2+\sqrt{5},2+\sqrt{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let be the Number are $\sqrt{5}  and  -\sqrt{5}$
Sum of Number  $\left(\sqrt{5}\right) + \left(-\sqrt{5}\right)$
$\sqrt{5}-\sqrt{5} = 0$
Which is a rational number

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

Give an example of two irrational numbers, whose difference is an irrational number.

  1. $4\sqrt{3},2\sqrt{3}$
  2. $\sqrt{3},\sqrt{3}$
  3. $2\sqrt{3},2\sqrt{3}$
  4. $4\sqrt{3},4\sqrt{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let be the Number are $4\sqrt{3}  and  2\sqrt{3}$
Difference of Number  $4\sqrt{3} - 2\sqrt{3} = 2\sqrt{3}$
Which is a irrational number

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

Give an example of two irrational numbers, whose quotient is an irrational number.

  1. $\sqrt{15},\sqrt{5}$
  2. $\sqrt{45},\sqrt{5}$
  3. $\sqrt{20},\sqrt{5}$
  4. $\sqrt{80},\sqrt{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let be the Number are $\sqrt{15}  and  \sqrt{5}$
Quotient of Numbers  $\frac{\sqrt{15}}{\sqrt{5}} = \sqrt{\frac{15}{5}} = \sqrt{3} $
Which is a irrational number

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

Give an example of two irrational numbers, whose sum is an irrational number.

  1. $2\sqrt{5},3\sqrt{5}$
  2. $2\sqrt{5},-2\sqrt{5}$
  3. $2+\sqrt{5},2-\sqrt{5}$
  4. $2+\sqrt{5},3-\sqrt{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let be the Number are $2\sqrt{5}  and  3\sqrt{5}$
Sum of Number  $2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$
Which is a irrational number

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

Give an example of two irrational numbers, whose quotient is a rational number.

  1. $\sqrt{5},\sqrt{2}$
  2. $\sqrt{8},\sqrt{2}$
  3. $\sqrt{3},\sqrt{2}$
  4. $\sqrt{7},\sqrt{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let be the Number are $\sqrt{8}  and  \sqrt{2}$
Quotient of Numbers  $\frac{\sqrt{8}}{\sqrt{2}} = \sqrt{\frac{8}{2}} = \sqrt{4} = 2 $
Which is a rational number

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

Give an example of two irrational numbers, whose product is a rational number.

  1. $\sqrt{8},\sqrt{2}$
  2. $\sqrt{5},\sqrt{2}$
  3. $2+\sqrt{8},\sqrt{2}$
  4. $\sqrt{8},2+\sqrt{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let be the Number are $\sqrt{8}  and  \sqrt{2}$
Product of Numbers  $\sqrt{8}\times \sqrt{2} = \sqrt{16} = 4$
Which is a rational number

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

Give an example of two irrational numbers, whose product is an irrational number.

  1. $\sqrt{3},\sqrt{3}$
  2. $\sqrt{2},\sqrt{2}$
  3. $\sqrt{2},-\sqrt{2}$
  4. $\sqrt{2},\sqrt{3}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let be the Number are $\sqrt{2}  and  \sqrt{3}$
Product of Numbers  $\sqrt{2}\times \sqrt{3} = \sqrt{6} $
Which is a irrational number

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

$\displaystyle log _{4}18$ is 

  1. an irrational number

  2. a rational number

  3. natural number

  4. whole number
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$  Log AB = log A + log B $
Also, $ log a^b = b log a $

So, $ log _{4} 18 = \frac { log 2 \times 3^2}{log 4}  = \frac { log 2 \times 3^2}{log 2^2}  = \frac { log 2}{2log 2} + \frac {2 log 3}{2 log 2}  = \frac {1}{2} +  \frac {log 3}{log 2}   $

As both $ log 2 $ and $ log 3 $ are irrational numbers, $ log _{x} 18 $ is an irrational number too. 

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

Number of integers lying between $1 $ to $102$  which are divisible by all $\displaystyle \sqrt{2},\sqrt{3},\sqrt{6}, $ is 

  1. $16$
  2. $17$
  3. $15$
  4. $0$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a number to be divisible by $\sqrt { 2 } $, it must be an irrational number. An integer is not an irrational,

so  there are no  numbers between  $ 1$ to  $102$ which are divisible by all  $\sqrt{2},\sqrt{3},\sqrt{6}$.