Tag: existence of irrational numbers

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

Which of the following is irrational

  1. $\sqrt {\dfrac{4}{9}} $
  2. $\dfrac{4}{5}$
  3. $\sqrt 7 $
  4. $\sqrt {81} $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
A $=\sqrt{\dfrac{4}{9}}=\dfrac{2}{3}$         Rational

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

C $=\sqrt7$                     Irrational

D $=\sqrt{81}=9$          Rational
Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

Which of the following rational number represents a terminating decimal expansion?

  1. $

    \dfrac { 77 } { 210 }

    $
  2. $

    \dfrac { 13 } { 125 }

    $
  3. $

    \dfrac { 2 } { 15 }

    $
  4. $

    \dfrac { 17 } { 18 }

    $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Any rational number its denominator is in the form of $2^m\times 5^n$, where $m,n$ are positive integer s are terminating decimals.

Solution is $B$ as $A$ is non terminating decimals.
$A =\dfrac{77}{210}= 0.366......$

$B =\dfrac{13}{125}= 0.104$

$C =\dfrac{2}{15}= 0.133.....$

$D =\dfrac{17}{18}=  0. 9444....$
Multiple choice maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

Read out each of the following numbers carefully and specify the natural numbers in it.
$87, 54, 0, -13, -4.7, \sqrt{7}, 2{1}{7}, \sqrt{15}, -{8}{7}, 3\sqrt{7}, 4.807, 0.002, \sqrt{16}$ and $2+\sqrt{3}.$

  1. $0,87,54,\sqrt{16}$
  2. $87, 54,$ $\sqrt{16}$, $217$
  3. $0, -13, -4,7, 217, 54, 87$
  4. $\sqrt{7}$, $\sqrt{15}$, $3 \sqrt{7}$, $\sqrt{16}$, $2 + \sqrt{3}$,
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Natural numbers from the given list are 87, 54,  $\sqrt { 16 } =4$ and 217

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

There can be a pair of irrational numbers whose sum is irrational 

Such as: $\displaystyle \sqrt{3}+2$ and $\displaystyle 5+\sqrt{2}$

  1. True

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

To get the sum as irrational, the numbers need to have an irrational part as well which are different from each other.

Example, the pair of numbers $ \sqrt{3} + 2 $ and $ 5 + \sqrt {2} $ have the sum $ \sqrt{3} + 2 + 5 + \sqrt {2} = 7 + \sqrt {2} + \sqrt {3} $ which is an irrational number too.

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

Simplify : 

$\displaystyle \sqrt{2}\times \sqrt[3]{3} \times \sqrt[4]{4}$.

  1. $\sqrt[3]{12}$
  2. $\sqrt[3]{24}$
  3. $\sqrt[3]{20}$
  4. $\sqrt[3]{25}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$ \sqrt{2} \times \sqrt[3] {3} \times \sqrt[4]{4}$
$=2^{ \frac { 1 }{ 2 }  } \times 3^{ \frac { 1 }{ 3 }  }\times 2^{ \frac { 2 }{ 4 }  }$
$=2^{ \frac { 1 }{ 2 }  } \times 2^{ \frac { 1 }{ 2 }  }\times 3^{ \frac { 1 }{ 3 }  }$
$=2  \times3^{ \frac { 1 }{ 3 }  }$
$=2^{ \frac { 3 }{ 3 }  }\times3^{ \frac { 1 }{ 3 }  }  $
$=\sqrt [ 3 ]{ 2^{ 3 } }\times\sqrt[3]{3}$
$=\sqrt[3]{8\times3}$
$=\sqrt[3]{24}$

Multiple choice maths number systems existence of irrational numbers irrational numbers properties of irrational numbers
The value of  $\displaystyle \pi $ upto $50$ decimal places is $:\:314159265358979323846264338327950288419716939937510$
Which are the least occurring digits?
  1. $0$
  2. $1$
  3. $2$
  4. $3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We will be considering the digits only after the decimal point.


Digit after decimal point frequency
0 2
1 5
2 5
3 8
4 4
5 5
6 4
7 4
8 5
9 8
Total 50

The maximum occurring digits are 3 and 9. 
The least occurring digit is 0.

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

Which of the following is irrational?

  1. $\dfrac {22}{7}$
  2. $3.141592$
  3. $2.78181818$
  4. $0.123223222322223.......$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

An irrational number is any real number that cannot be expressed as a ratio of integers. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals.
Among all the options only $(D)$ $0.123223222322223$...... is non terminating and non repeating decimal.Therefore, it is a irrational number.