Tag: proofs of irrationality

Questions Related to proofs of irrationality

If $ x = ( 2 + \sqrt3)^n , n \epsilon N $ and $ f = x - [x],$ then $ \dfrac {f^2}{1-f} $ is :

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

  1. An irrational number

  2. A non-integer rational number

  3. An odd number

  4. An even number

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Correct Option: D

The product of two irrational numbers is 

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

  1. Always irrational

  2. Always rational

  3. Can be both rational and irrational

  4. always an integer

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Correct Option: C
Explanation:

Let $p=\sqrt 3$ and $q=\sqrt 3$ be two irrational numbers 
$pq=\sqrt 3\times \sqrt 3=3$
which is rational
Now let $p=\sqrt 3$ and $q=\sqrt 2$
$pq=\sqrt 3\times \sqrt 2=\sqrt 6$
which is an irrational number
So the product can be both rational and irrational .
Option $C$ is correct.

Which of the following irrational number lies between 20 and 21

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

  1. $\sqrt442$

  2. $\sqrt440$

  3. $\sqrt443$

  4. $\sqrt444$

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Correct Option: B
Explanation:

$20^2=400, 21^2=441\ \Rightarrow 400<440<441\ \Rightarrow 20<\sqrt{440}<21 $

State whether the given statement is True or False :

$4-5\sqrt { 2 } $ is an irrational number.
maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. True

  2. False

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Correct Option: A
Explanation:

$\text{Here 4 is a rational number and }$$5\sqrt2$ is a $irrational$ number


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

So that $(4-5\sqrt2)$ is an irrational number.

hence option A is correct.

State whether the given statement is True or False :

$5-2\sqrt { 3 } $ is an irrational number.
maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. True

  2. False

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Correct Option: A
Explanation:

$\text{Here 5 is a rational number and }$$2\sqrt3$ is a $irrational$ number


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

So that $(5 - 2\sqrt3)$ is an irrational number.

hence option A is correct.

State whether the given statement is True or False :

$3+\sqrt { 2 } $ is an irrational number.
maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

  1. True

  2. False

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Correct Option: A
Explanation:

$3 \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 $(3+\sqrt2)$ is an irrational number.

hence option A is correct

The equation $\sqrt{x+4}$- $\sqrt{x-3}$+ 1=0 has:

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

  1. no root

  2. one real root

  3. one real root and one imaginary root

  4. two imaginary roots

  5. two real roots

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Correct Option: A
Explanation:

$\Longrightarrow \sqrt { x+4 } -\sqrt { x-3 } +1=0\ \Longrightarrow \sqrt { x+4 } +1=\sqrt { x-3 } \ \Longrightarrow x+4+1+2\sqrt { x+4 } =x-3\ \Longrightarrow 2\sqrt { x+4 } =-8\ \Longrightarrow x+4=16\ \therefore x=12$

But x = 12 will not satisfy given equation.
$\therefore$ No roots for given equation.

State whether True or False :


All the following numbers are irrationals.
(i) $\dfrac { 2 }{ \sqrt { 7 }  } $ (ii) $\dfrac { 3 }{ 2\sqrt { 5 }  }$ (iii) $4+\sqrt { 2 } $ (iv) $5\sqrt { 2 } $

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

  1. True

  2. False

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Correct Option: A
Explanation:

In all of the above questions $\sqrt7,\sqrt5,\sqrt2$ are a $irrational$ numbers


And $\text{the addition, subtraction, division and product between  rational and irrational gives  an irrational number}$

So that all of the above are irrational numbers.

hence option A is correct.

State whether the given statement is True or False :

$2-3\sqrt { 5 }$ 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:

Here 2 is a rational number

and $3\sqrt5$ is an irrational number  and $\text{difference of rational and irrational is always an irrational number}$


So that $2 - 3\sqrt5 $  is an $irrational$ number

hence option A is correct.

State whether the given statement is True or False :

$\sqrt { 3 } +\sqrt { 4 } $ 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:

$\sqrt4= 2 \text{ is a  rational number and }$$\sqrt3$ is a $irrational$ number


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

So that $(\sqrt3+\sqrt4)$ is an irrational number.

hence option A is correct.