Tag: real numbers (rational and irrational numbers)

Questions Related to real numbers (rational and irrational numbers)

The value of x on simplifying $x -2 |x|=-3$ is

absolute value real numbers (rational and irrational numbers) real numbers basic algebra maths

  1. -1 or 3

  2. 1 or -3

  3. -1 or -3

  4. 1, 3

Show answer
Correct Option: A
Explanation:

$x-2|x| = -3$

There two cases possible either $x$ is positive or$x$ is negative
We have to solve both cases
Firstly taking x as positive
$x-2x=-3$
$x=3$
Now taking $x$ as negative
$x-2(-x)=-3$
$3x=-3$
$x=-1$
So we get two values of $x$ that is -1 and 3
 So correct answer will be option A

Simplification of $-|-48|$ is

absolute value real numbers (rational and irrational numbers) real numbers basic algebra maths

  1. $48$

  2. $-48$

  3. $0$

  4. $-47$

Show answer
Correct Option: B
Explanation:
Since $|-x| = x$
$-|-48| = -(48) = -48$
So, option $B$ is correct.

If $a$ and $b$ are any real numbers, then which of the following expressions is always positive?

absolute value real numbers (rational and irrational numbers) real numbers basic algebra maths

  1. $\left| a \right| $

  2. $\left| a+b \right| $

  3. $\left| a-b \right| +1/2$

  4. ${ a }^{ 2 }+{ b }^{ 2 }$

  5. ${ \left( a+b \right) }^{ 2 }$

Show answer
Correct Option: C
Explanation:

Lets check each option one by one.

A. |a| cab be 0 if a=0 
B. |a+b| can be 0 if a+b=0
C. |a-b|+ 1/2  will always positive because |a+b| is either 0 or positive 
D. ${a}^{2}+{b}^{2}$ can be 0 if both $a$ and $b$ becomes 0
E ${(a+b)}^{2}$ can equal to 0 if $a+b$ = 0
So correct answer will be option C 

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 $0.150150015000...$