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The cardinal number of the set $ A = \left\{ {x|x \in \mathbb{N},x = \dfrac{{4{n^2} + 5n + 10}}{n}} \right\} $ is
1) $ 3 $
2) $ 4 $
3) $ 2 $
4) $ 1 $

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Last updated date: 19th Jun 2024
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Answer
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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.