Tag: maths

Questions Related to maths

 $\sqrt2 + \sqrt3$ is 

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

  1. irrational

  2. rational

  3. natural

  4. None

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Correct Option: A
Explanation:

$\cfrac{m}{n} = \sqrt{2} + \sqrt{3} $
Square both sides:
$\cfrac{m^2}{ n^2} = 5 + 2\sqrt{6} $

"Solve" for $\sqrt{6}$
$\sqrt{6} = \cfrac{\left(m^{2} - 5n^{2}\right)}{\left(2n^{2}\right)} $
so if  $\sqrt{2} + \sqrt{3} $ is  rational,  then  so  is $ \sqrt{6}$
Let a and b be the integers with gcd(a,b) = 1 such that
$\cfrac{a}{b} = \sqrt{6}$
Square both sides and multiply by $b^2$:
$a^2 = 6b^2 $
Now, the right side is divisible by 2, so $a^2$ is divisible by 2, which
then implies that a is divisible by 2 (since 2 is prime).
Therefore we  can write a=2k for some integer k:
$4k^{2} = \left(2k \right)^{2} = 6b^{2} $
Divide by 2:
$2k^{2} = 3b^{2} $
Now the left side is divisible by 2, so $3b^{2}$ is divisible by 2, from which it follows that b is divisible by 2.
However, this would mean that 2 divides gcd(a,b) = 1. Contradiction.
$\therefore  \sqrt{6} $ is  irrational
,  and  $\therefore \sqrt{2} + \sqrt{3} $ is  also irrational.


Every surd is

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

  1. a natural number

  2. an irrational number

  3. a whole number

  4. a rational number

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Correct Option: B
Explanation:
When a number cannot be simplified further to remove a square root then it is a surd.  
A surd is an irrational number.

For. eg: square root of 2 cannot be simplified. thus it is a surd.

By definition, a surd is an irrational root of a rational number. So we know that surds are always irrational and they are always roots.

For eg, $\sqrt2$ is a surd since 2 is rational and $\sqrt 2$ is irrational.

Similarly, the cube root of 9 is also a surd since 9 is rational and the cube root of 9 is irrational.

On the other hand, $\sqrtπ$ is not a surd even though $\sqrtπ$ is irrational because π is not rational.

Thus, to answer the question, every surd is an irrational number, though an irrational number may or may not be a surd.

The answer is Option B

Which of the following is irrational?

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

  1. $\displaystyle\frac{1}{3}$

  2. $\displaystyle\frac{48}{5}$

  3. $0.7777\dots$

  4. $1.73202002\dots$

Show answer
Correct Option: D
Explanation:

$1.73202002$ is the irrational number because it can not  be expressed as a fraction

$0.\overline{35}$ is equal to

maths number systems existence of irrational numbers irrational numbers properties of irrational numbers

  1. $\displaystyle\frac{35}{66}$

  2. $\displaystyle\frac{35}{77}$

  3. $\displaystyle\frac{35}{99}$

  4. none of these

Show answer
Correct Option: C
Explanation:
$X=0.35353535$   -- i
Multiplying equation i with 100,

$100x=35.353535353$   --ii 
Subtracting equation i from ii 

$ 100x-x = 35.3535 - 0.3535 $
$99x=35$
$ x = \dfrac{35}{99}$

$3.\overline{25}$ is equal to

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

  1. $\displaystyle\frac{320}{99}$

  2. $\displaystyle\frac{321}{99}$

  3. $\displaystyle\frac{322}{99}$

  4. $\displaystyle\frac{323}{99}$

Show answer
Correct Option: C
Explanation:

Given that,$3.\overline{25}$.


Let,

$x=3.\overline{25}$

 $x=3.252525.....$


Multiply by 100 both sides,

  $ 100x=100\times 3.252525..... $

 $ 100x=325.2525..... $

 $ 100x=322+3.2525..... $

 $ 100x=322+x $

 $ 99x=322 $

 $ x=\dfrac{322}{99} $


Hence, this is the answer.

$0.\overline{05}$ is equal to

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

  1. $\displaystyle\frac{3}{99}$

  2. $\displaystyle\frac{4}{99}$

  3. $\displaystyle\frac{5}{99}$

  4. none of these

Show answer
Correct Option: C
Explanation:

Given that,$0.\overline{05}$

Let,

  $ x=0.\overline{05} $

 $ x=0.05050505..... $

Multiply by $100$ both sides,

 $ 100x=100\times 0.05050505..... $

 $ 100x=5.050505..... $

 $ 100x=5+0.050505..... $

 $ 100x=5+x $

 $ 99x=5 $

 $ x=\dfrac{5}{99} $


Hence, this is the answer.

Which statement is true?

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

  1. $ \displaystyle \frac{-8}{12} $= $ \displaystyle \frac{10}{-15} $

  2. $ \displaystyle \sqrt{3} $ is not a real number

  3. Additive identity of 5 is -5

  4. $ \displaystyle \frac{2}{5} $>$ \displaystyle \frac{4}{5} $

Show answer
Correct Option: A

Irrational number is defined as 

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

  1. a real number that cannot be made by dividing two integers.

  2. a real number that can be made by dividing two integer.

  3. a number that can be made derived after multiplying two integers.

  4. a real number that can be written as whole number.

Show answer
Correct Option: A
Explanation:
An irrational is any real number that cannot be expressed as a ratio of integers.

Therefore, $A$ is the correct answer.

Which of the following is an irrational number?
maths rational and irrational numbers existence of irrational numbers irrational numbers properties of irrational numbers

  1. $\dfrac{11}{2}$

  2. $\sqrt{16}$

  3. $\sqrt{9}$

  4. $\sqrt{11}$

Show answer
Correct Option: D
Explanation:
An irrational is any real number that cannot be expressed as a ratio of integers.
Option $A$ is a rational number.
Option $B$ and $C$ are $\sqrt{16}$ and $\sqrt{9}$, i.e. $4$ and $3$ respectively.
$D$ cannot be expressed as a ratio of integers.
$D$ is the correct answer.

The square root of any prime number is 

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

  1. rational

  2. irrational

  3. co-prime

  4. composite

Show answer
Correct Option: B
Explanation:

The square root of any prime number is irrational.
Example: $\sqrt{2}$ is a irrational number.