The value of 5√3-3+√12+2√75 on simplifying is
The value of 5√3-3+√12+2√75 on simplifying is :

Question : The value of 5√3-3+√12+2√75 on simplifying is :

  1. $5 \sqrt{3}$
  2. $6 \sqrt{3}$
  3. $\sqrt{3}$
  4. $9 \sqrt{3}$
Solution: 
Correct option is 1. $9 \sqrt{3}$
The given expression is
$5 \sqrt{3}-3 \sqrt{12}+2 \sqrt{75}$
$=5 \sqrt{3}-6 \sqrt{3}+10 \sqrt{3}$
$=9 \sqrt{3}$
Therefore, option 1 is correct.
Simplify 3√3+10√3
Simplify 3√3+10√3

Question : Simplify  3√3+10√3

  1. $13 \sqrt{3}$
  2. $10 \sqrt{3}$
  3. $12 \sqrt{3}$
  4. $11 \sqrt{3}$
Solution:
Correct option is 1. $13 \sqrt{3}$
$3 \sqrt{3}+10 \sqrt{3}=13 \sqrt{3}$
5√3+2√3 = 7√6 enter 1 for true and 0 for false
5√3+2√3 = 7√6 enter 1 for true and 0 for false

Question : 5√3+2√3 = 7√6
enter 1 for true and 0 for false

Solution :
Correct answer is  0
$5 \sqrt{3}+2 \sqrt{3}=7 \sqrt{3}$
Therefore: False.
Find the nine rational numbers between 0 and 1
Find the nine rational numbers between 0 and 1

Question : Find the nine rational numbers between 0 and 1.

  1. $0.1,0.2,0.3, \ldots, 0.9$
  2. $1.1,0.2,10.3, \ldots, 0.9$
  3. $0.1,0.2,0.3, \ldots, 20.9$
  4. $0.1,0.2,10.3, \ldots, 0.9$
Solution :
Correct option is 1. $0.1,0.2,10.3, \ldots, 0.9$
$0<(0+0.1)=0.1<(0.1+0.1)=0.2<(0.2+0.1)$
$=0.3<\ldots<(0.8+1)=0.9<(0.9+0.1)=1$
$0<0.1<0.2<0.3<\ldots<0.9<1$
$\therefore$ The nine rational numbers between 0 and 1 are $0.1,0.2,0.3, \ldots, 0.9$
Which of the following numbers are rational ?
Which of the following numbers are rational ?

Question : Which of the following numbers are rational ?

  1. $1$
  2. $-6$
  3. $3 \frac{1}{2}$
  4. All above are rational
Solution :
The correct option is 4. All above are rational
None of the number is irrational as every number can be expressed in the form of $\frac{\mathrm{p}}{\mathrm{q}}$, where $\mathrm{q} \neq 0$.
Two rational numbers between 1/5 and 4/5 are
What are two rational numbers between $\frac{1}{5}$ and $\frac{4}{5}$ ?

Question : Two rational numbers between $\frac{1}{5}$ and $\frac{4}{5}$ are :

  1. 1 and $\frac{3}{5}$
  2. $\frac{2}{5}$ and $\frac{3}{5}$
  3. $\frac{1}{2}$ and $\frac{2}{1}$
  4. $\frac{3}{5}$ and $\frac{6}{5}$
Solution :
The correct option is 2. $\frac{2}{5}$ and $\frac{3}{5}$
Since the denominator of both rational numbers are same. So, for getting the rational numbers between the given rational numbers, we only have to consider the numerators of the rational numbers.
Two numbers between 1 & 4 are 2 and 3.
So, two rational numbers between the given rational numbers will be $\frac{2}{5}$ and $\frac{3}{5}$
So, the correct answer is option 2.
A rational number can always be written in a fraction a/b​, where a and b are not integers (b≠0).
A rational number can always be written in a fraction $\frac{a}{b}$, where a and $b$ are not integers $(b \neq 0)$

Question : State True or False. A rational number can always be written in a fraction $\frac{\mathrm{a}}{\mathrm{b}}$, where a and $\mathrm{b}$ are not integers $(b \neq 0)$.

  1. True
  2. False
Solution :
The correct option is 2. False
A number that can always be written in the form of $\mathrm{p} / \mathrm{q}$, where $\mathrm{p}$ is any integer and $\mathrm{q}$ is a non-zero integer, is a rational number.
The given statement is false.
1/0 is not rational

Question : Say true or false. $\frac{1}{0}$ is not rational.

  1. True
  2. False
Solution :
The correct option is 1. True
A rational number is a number that can be defined in the form of $\frac{p}{q}$, where $q$ is nonzero.
Now, if $\mathrm{q}$ is 0 , although an integer, the solution will not be a rational number. It will give an undefined result, so the statement is true.
Prove that root 5 is irrational number
Prove that root 5 is irrational number

Question : $\sqrt{5}$ is an irrational number.

  1. True
  2. False
Solution :
The correct option is 1. True
An irrational number is any real number that cannot be expressed as a ratio
$\mathrm{a} / \mathrm{b}$, where a and $\mathrm{b}$ are integers and $\mathrm{b}$ is non-zero.
$\sqrt{5}$ is irrational as it can never be expressed in the form $\mathrm{a} / \mathrm{b}$
What is the value of (6+√27)−(3+√3)+(1−2√3) when simplified
What is the value of (6+√27)−(3+√3)+(1−2√3) when simplified ?

Quetion : The value of $(6+\sqrt{27})-(3+\sqrt{3})+(1-2 \sqrt{3})$ when simplified is :

  1. positive and irrational
  2. negative and rational
  3. positive and rational
  4. negative and irrational
Solution :
The correct option is 3. positive and rational

$6+\sqrt{27}-(3+\sqrt{3})+(1-2 \sqrt{3})=6+3 \sqrt{3}-3-\sqrt{3}+1-2 \sqrt{3} = 4$

4 is a positive rational number.
Hence, correct answer is option 3.
Find the Value of the Following: (3−1 + 4−1 + 5−1)0
Find the value of $\left(3^{1}+4^{1}+5^{1}\right)^{0}$

Question : Find the value of $\left(3^{1}+4^{1}+5^{1}\right)^{0}$.

Solution :
The correct answer is 1
Any number with a power of zero is equal to one.
What is the rationalizing factor of (a+√b)
What is the rationalizing factor of (a+√b)

Question : The rationalizing factor of $(\mathrm{a}+\sqrt{\mathrm{b}})$ is

  1. $a-\sqrt{b}$
  2. $\sqrt{a}-b$
  3. $\sqrt{a}-\sqrt{b}$
  4. None of these
Solution :
Correct option is 1. $a-\sqrt{b}$
The rationalizing factor of $a + \sqrt{b}$ is $a – \sqrt{b}$ as the product of these two expressions give a rational number.
The decimal expansion of π is
The decimal expansion of π is

Question : The decimal expansion of π is :

  1. terminating
  2. non-terminating and non-recurring
  3. non-terminating and recurring
  4. doesn’t exist
Solution :
The correct option is 2. non-terminating and non-recurring
We know that $\pi$ is an irrational number and Irrational numbers have decimal
expansions that neither terminate nor become periodic.
So, correct answer is option 2.
How many rational numbers Between any two rational numbers ?
Between any two rational numbers, there are infinitely many rational numbers

Question : Between any two rational numbers

  1. there is no rational number
  2. there is exactly one rational number
  3. there are infinitely many rational numbers
  4. there are only rational numbers and no irrational numbers
Solution :
The correct option is 3. there are infinitely many rational numbers
Recall that to find a rational number between r and s, you can add
$\mathrm{r}$ and $\mathrm{s}$ and divide the sum by 2 , that is $\frac{\mathrm{r}+\mathrm{s}}{2}$ lies between $\mathrm{r}$ and $\mathrm{s}$.
For example, $\frac{5}{2}$ is a number between 2 and 3
We can proceed in this manner to find many more rational numbers between 2 and 3.
Hence, we can conclude that there are infinitely many rational numbers between any two given rational numbers.
Find five rational numbers between -3/2 and 5/3
Find any five rational numbers between -3/2 and 5/3

Question : State true or false :
Five rational numbers between $\frac{-3}{2}$ and $\frac{5}{3}$ are $\frac{-8}{6}, \frac{-7}{6}, 0, \frac{1}{6}, \frac{2}{6}$

  1. True
  2. False
Solution :

The Correct option is 1. True

To get the rational numbers between $\frac{-3}{2}$ and $\frac{5}{3}$

Take an LCM of these two numbers: $\frac{-9}{6}$ and $\frac{10}{6}$

All the numbers between $\frac{-9}{6}$ and $\frac{10}{6}$ form the answer

Some of these numbers are $\frac{-8}{6}, \frac{-7}{6}, 0, \frac{1}{6}, \frac{2}{6}$

Hence the statement is true.
five rational numbers which are smaller than 2
five rational numbers which are smaller than 2

Question : Following are the five rational numbers that are smaller than 2 $\Rightarrow 1, \frac{1}{2}, 0,-1, \frac{-1}{2}$
If true then enter 1 and if false then enter 0

Solution :
Correct option is 1
Any number in the form of $\frac{p}{q}$ which is less than 2 will form the answer.
So given numbers are $1, \frac{1}{2}, 0,-1, \frac{-1}{2}$ rational number which are smaller than 2
So the statement is true.
Find five rational numbers between:2/3 and 4/5

Question : State true or false:

Five rational numbers between

$\frac{2}{3}$ and $\frac{4}{5}$ are $\frac{41}{60}, \frac{42}{60}, \frac{43}{60}, \frac{44}{60}, \frac{45}{60}$

  1. True
  2. False
Solution :

Correct option is 1.True

To get the rational numbers between $\frac{2}{3}$ and $\frac{4}{5}$

Take an LCM of these two numbers: $\frac{10}{15}$ and $\frac{12}{15}$

Multiply numerator and denominator by $4: \frac{40}{60}$ and $\frac{48}{60}$

All the numbers between $\frac{40}{60}$ and $\frac{48}{60}$ form the answer

Some of these numbers are $\frac{41}{60}, \frac{42}{60}, \frac{43}{60}, \frac{44}{60}, \frac{45}{60}$

Hence the statement is true.
Every rational number is
Every rational number is a real number

Question : Every rational number is

  1. A natural number
  2. An integer
  3. A real number
  4. A whole number
Solution : 
The correct option is 3. A real number
Real number is a value that represents a quantity along the number line.
Real number includes all rational and irrational numbers.
Rational numbers are numbers that can be represented in the form $\frac{p}{q}$ where,
$\mathrm{q} \neq 0$ and $\mathrm{p}, \mathrm{q}$ are integers.
Therefore, a rational number is a subset of a real number.
We know that rational and irrational numbers taken together are known as real numbers. Therefore, every real number is either a rational number or an irrational number. Hence, every rational number is a real number. Therefore, (3) is the correct answer.
A number is an irrational if and only if its decimal representation is

Question : A number is irrational if and only if its decimal representation is :

  1. Non-terminating
  2. Non-terminating and repeating
  3. Non-terminating and non-repeating
  4. Terminating
Solution :
The correct option is 3. non-terminating and non-repeating
According to the definition of an irrational number, If written in decimal notation,
an irrational number would have an infinite number of digits to the right
of the decimal point, without repetition.
Hence, a number having non terminating and non-repeating decimal
representation is an irrational number.
So, option 3. is correct.
There are numbers which cannot be written in the form p/q ​ , where q ≠ 0 and both p, q are integers

Question : State true or false:
There are numbers which cannot be written in the form p/q ​ , where q ≠ 0 and both pq are integers.

  1. True
  2. False
Solution :
The correct option is 1. True
The statement is true as there are Irrational numbers which don’t satisfy the
condition of rational numbers i.e irrational numbers cannot be written in the
form of $_{\mathrm{q}}^{\mathrm{p}} $ , $\mathrm{q} \neq 0$, where $\mathrm{p}, \mathrm{q}$ are integers.
Example,
$\sqrt{3}, \sqrt{99}$