Tag: rational numbers

Quizzes (14) Questions (65)

Questions Related to rational numbers

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

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

  1. An irrational number

  2. A non-integer rational number

  3. An odd number

  4. An even number
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let x = (2 + sqrt(3))^n. Let y = (2 - sqrt(3))^n. Since 0 < 2 - sqrt(3) < 1, y is between 0 and 1. x + y is an integer, so f = x - [x] = 1 - y. Then f^2 / (1 - f) = (1 - y)^2 / y. This simplifies to an even integer.

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

The product of two irrational numbers is 

  1. Always irrational

  2. Always rational

  3. Can be both rational and irrational

  4. always an integer
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers
State whether the given statement is True or False :

$4-5\sqrt { 2 } $ is an irrational number.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers
State whether the given statement is True or False :

$5-2\sqrt { 3 } $ is an irrational number.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers
State whether the given statement is True or False :

$3+\sqrt { 2 } $ is an irrational number.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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

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

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

  1. no root

  2. one real root

  3. one real root and one imaginary root

  4. two imaginary roots

  5. two real roots
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

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

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 } $

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers
State whether the given statement is True or False :

$2-3\sqrt { 5 }$ is an irrational number.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers
State whether the given statement is True or False :

$\sqrt { 3 } +\sqrt { 4 } $ is an irrational number.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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.