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.

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. $13 \sqrt{3}$
$3 \sqrt{3}+10 \sqrt{3}=13 \sqrt{3}$

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.

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 $0.1,0.2,0.3, \ldots, 0.9$

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 $\mathrm{q} \neq 0$.

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.

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.

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.

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 $\mathrm{a} / \mathrm{b}$

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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,
$\sqrt{3}, \sqrt{99}$