Answer
405.6k+ views
Hint: A natural number is something that we can count by hands, these are also called the counting numbers. The set of natural numbers is denoted as $ \mathbb{N} $ and is defined as $ \mathbb{N}: = \{ 1,2,3,...\} $ .
A set is a collection of well-defined distinct objects, and the cardinal number of a set is the total number of elements in it. We will expand the given set and list the elements in it and count the cardinal number of the set. Then we will match the correct option to our answer.
Complete step-by-step answer:
The given set is $ A = \left\{ {x|x \in \mathbb{N},x = \dfrac{{4{n^2} + 5n + 10}}{n}} \right\} $
We have an element of the set $ x = \dfrac{{4{n^2} + 5n + 10}}{n} $
Now we will analyze what can be the values of this.
$ x = \dfrac{{4{n^2} + 5n + 10}}{n} $
$ \Rightarrow x = \dfrac{{4{n^2}}}{n} + \dfrac{{5n}}{n} + \dfrac{{10}}{n} $
$ \Rightarrow x = 4n + 5 + \dfrac{{10}}{n} $
Clearly, we have $ (4n + 5) \in \mathbb{N} $ while $ \dfrac{{10}}{n} \in \mathbb{N} $ only for $ n = 1,2,5,10 $ .
This is because $ \dfrac{{10}}{n} $ is a decimal number when $ n \ne 1,2,5,10 $ and decimal numbers are not counted in natural numbers.
Therefore the only possibility that $ x = \dfrac{{4{n^2} + 5n + 10}}{n} \in \mathbb{N} $ is that $ n = 1,2,5,10 $
Thus, the set we have now is $ A = \left\{ {x|n = 1,2,5,10,x = \dfrac{{4{n^2} + 5n + 10}}{n}} \right\} $
Now, we don’t need to list the elements as we have found that there can be only four values of $ n $ . So, the set contains only four elements.
Therefore, the cardinal number of set $ A $ is $ 4 $ and the second option is correct.
So, the correct answer is “Option 2”.
Note: As the value of $ x $ depends on the addition of certain terms namely, $ 4n + 5 + \dfrac{{10}}{n} $ , and we found out that this is a natural number only when $ n = 1,2,5,10 $ . Since, natural numbers are non-zero and positive we can say that for different values of $ n $ we will get different values of $ x $ . Also there is no repetition when our numbers $ n = 1,2,5,10 $ are chosen. So, we can conclude that there are only four possibilities.
A set is a collection of well-defined distinct objects, and the cardinal number of a set is the total number of elements in it. We will expand the given set and list the elements in it and count the cardinal number of the set. Then we will match the correct option to our answer.
Complete step-by-step answer:
The given set is $ A = \left\{ {x|x \in \mathbb{N},x = \dfrac{{4{n^2} + 5n + 10}}{n}} \right\} $
We have an element of the set $ x = \dfrac{{4{n^2} + 5n + 10}}{n} $
Now we will analyze what can be the values of this.
$ x = \dfrac{{4{n^2} + 5n + 10}}{n} $
$ \Rightarrow x = \dfrac{{4{n^2}}}{n} + \dfrac{{5n}}{n} + \dfrac{{10}}{n} $
$ \Rightarrow x = 4n + 5 + \dfrac{{10}}{n} $
Clearly, we have $ (4n + 5) \in \mathbb{N} $ while $ \dfrac{{10}}{n} \in \mathbb{N} $ only for $ n = 1,2,5,10 $ .
This is because $ \dfrac{{10}}{n} $ is a decimal number when $ n \ne 1,2,5,10 $ and decimal numbers are not counted in natural numbers.
Therefore the only possibility that $ x = \dfrac{{4{n^2} + 5n + 10}}{n} \in \mathbb{N} $ is that $ n = 1,2,5,10 $
Thus, the set we have now is $ A = \left\{ {x|n = 1,2,5,10,x = \dfrac{{4{n^2} + 5n + 10}}{n}} \right\} $
Now, we don’t need to list the elements as we have found that there can be only four values of $ n $ . So, the set contains only four elements.
Therefore, the cardinal number of set $ A $ is $ 4 $ and the second option is correct.
So, the correct answer is “Option 2”.
Note: As the value of $ x $ depends on the addition of certain terms namely, $ 4n + 5 + \dfrac{{10}}{n} $ , and we found out that this is a natural number only when $ n = 1,2,5,10 $ . Since, natural numbers are non-zero and positive we can say that for different values of $ n $ we will get different values of $ x $ . Also there is no repetition when our numbers $ n = 1,2,5,10 $ are chosen. So, we can conclude that there are only four possibilities.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)