Let \(S=\){\(n∈NIn^3+3n^2+5n+3\) is not divisible by \(3\)}.Then, which of the following statements is true about \(S\)
Let \(S=\){\(n∈NIn^3+3n^2+5n+3\) is not divisible by \(3\)}.Then, which of the following statements is true about \(S\)
\(S=ϕ\)
\(|S|≥2\) and \(|S|\)\(\) is a multiple of \(5\)
\(|S|\) is infinite
\(S\) is non empty and \(|S|\) is a multiple of \(3\)
\(S\) is non empty and \(|S|\) is a multiple of \(3\).
The Correct Option is
Solution and Explanation
Given that:
\(S=\){\(n∈NIn^3+3n^2+5n+3\) is not divisible by \(3\)}.
Here as \(n∈N\)
let us test:
take, \(n=1\)
\(⇒1^3+3.1^2+5.1+3=12\)
take \(n=2\)
\(⇒2^3+3.2^2+5.2+3=33\)
take, \(n=3\)
\(⇒3^3+3.3^2+5.3+3=72\)
All these above cases proves that S is divisible by \(3\)
Hence,\(S={n∈N : n^3+3n^2+5n+3}\) is divisible by \(3\)
So, S is non empty and \(|S|\)is a multiple of \(3\) (_Ans.)
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".