Tag: rational and irrational numbers

Questions Related to rational and irrational numbers

If a and b are any two such real numbers that ab $ = 0 $ , then

decimal representation of rational numbers rational and irrational numbers maths

  1. $a = 0, b \leq 0$

  2. $b = 0, a \leq 0$

  3. a = 0 or b = 0 or both

  4. $a = b$ and $b = 0$

Show answer
Correct Option: C
Explanation:

if both number are real the either a or b or both should be zero.
then only ab will be 0.
if any real number is multiplied by 0 then result will be zero.
So, answer is
C a = 0 or b = 0 or both

If $f(x)-2f(1-x) = x^2+2$, then what is $f(x)$?

decimal representation of rational numbers rational and irrational numbers maths

  1. $f(x)=-x^2+\dfrac{4}{3}x-\dfrac{3}{8}$

  2. $f(x)=−x^2+\dfrac{4}{3}x−\dfrac{8}{3}$

  3. $f(x)=−x^2+\dfrac{8}{3}x−\dfrac{4}{3}$

  4. $f(x)=−x^2+\dfrac{3}{8}x−\dfrac{3}{4}$

Show answer
Correct Option: B
Explanation:
$f\left(x\right)-2f\left(1-x\right)={x}^{2}+2$      .......$(1)$

Setting $x=1-x$ then we get

$f\left(1-x\right)-2f\left(1-1+x\right)={\left(1-x\right)}^{2}+2$ 

$f\left(1-x\right)-2f\left(x\right)={x}^{2}-2x+3$ 

$2f\left(1-x\right)-4f\left(x\right)=2{x}^{2}-4x+6$    .......$(2)$

Adding $(1)$ and $(2)$ we get

$-3f\left(x\right)=3{x}^{2}-4x+8$ 

$\therefore f\left(x\right)=-{x}^{2}+\dfrac{4}{3}x-\dfrac{8}{3}$ 

State whether the following statement is True or False:
A rational and Irrational number between $2.357$ and $3.121$ is $3, 3.101101110 ...$

irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Any terminating decimal between 2.357 and 3.121 will be a rational number like, ${3}$

Any non-terminating and non-recurring decimal between  2.357 and 3.121 will be an irrational number like $3.101101110...$

State the following statement is True or False
A rational number between  $3.623623$ and  $0.484848 $ are $4$ and $4.909009000 .$..

irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

$4>3.623623$ and $4.90900900...>3.623623$

Since both the numbers do not lie between $0.4848$ and $3.623623.$, the given statement is false. 

State true or false:
The three rational numbers between $\displaystyle \sqrt{3}$ and $\displaystyle \sqrt{5}$ are 1.6, 1.8, 2.2

irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

We know that, $ \sqrt {3} = 1.732 $ and $ \sqrt {5} = 2.236 $





Hence three rational numbers between $ 1.732 $ and $ 2.236 $  can be $

1.8( = \frac {18}{10}) ; 2 $ and $ 2.2 (= \frac {22}{10}) $


State the following statement is true or false:
$\dfrac{5}{12}$  lies between $\cfrac{1}{3}$ and $\cfrac{1}{2}$.

irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

The average of the two numbers will be in between the two numbers.
$ \cfrac { \left( \dfrac { 1 }{ 3 }  \right) +\left( \dfrac { 1 }{ 2 }  \right)  }{ 2 } $

$= \cfrac { \left( \dfrac { 5 }{ 6 }  \right)  }{ 2 } $

$= \dfrac { 5 }{ 12 } $ is a rational number which lies between $\dfrac{1}{3}\ and\ \dfrac{1}{2}$. 




State whether the given statement is true/false.
An irrational number between two numbers $\dfrac{1}{7}$ and $\dfrac{2}{7}$ is $0.1501500 15000...$ .

irrational numbers irrational numbers and their operations rational and irrational numbers real numbers (rational and irrational numbers) maths

  1. True

  2. False

Show answer
Correct Option: A
Explanation:
Let us first find the decimal forms of the given numbers as follows: 
 
$\dfrac { 1 }{ 7 } =0.\overline { 142857 } ,\dfrac { 2 }{ 7 } =0.\overline { 285714 }$

We find a number which is non-terminating non-recurring lying between them.
So, we can find infinite many such numbers. For example, $0.150150015000...$ and $0.20200200020000....$

Hence, an irrational number between two numbers $\dfrac {1}{7}$ and $\dfrac {2}{7}$ is $0.150150015000...$

If $\sqrt{a}$ is an irrational number, what is a? 

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

  1. Rational

  2. Irrational

  3. $0$

  4. Real

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Correct Option: A
Explanation:

Consider the given irrational number$\sqrt{a}$ ,

Definition  of rational number- which number can be write in the form of $\dfrac{p}{q}$ but $q\ne 0$ is called rational number.

Hence, $a=\dfrac{a}{1}$

That why  $a$ is rational number

 

Hence, this is the answer.

Which of the following is irrational

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

  1. $\sqrt {\dfrac{4}{9}} $

  2. $\dfrac{4}{5}$

  3. $\sqrt 7 $

  4. $\sqrt {81} $

Show answer
Correct Option: C
Explanation:
A $=\sqrt{\dfrac{4}{9}}=\dfrac{2}{3}$         Rational

B $=\dfrac{4}{5}$                       Rational

C $=\sqrt7$                     Irrational

D $=\sqrt{81}=9$          Rational

The number $23+\sqrt{7}$ is

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

  1. Natural number

  2. Irrational number

  3. Rational number

  4. None of these

Show answer
Correct Option: B
Explanation:

As we've $\sqrt{7}$ is an irrational number and $23$ is a rational number then the sum of an irrational number and a rational number is again an irrational number.