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Test: Set Notations - JEE MCQ


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10 Questions MCQ Test Mathematics (Maths) for JEE Main & Advanced - Test: Set Notations

Test: Set Notations for JEE 2024 is part of Mathematics (Maths) for JEE Main & Advanced preparation. The Test: Set Notations questions and answers have been prepared according to the JEE exam syllabus.The Test: Set Notations MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Set Notations below.
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Test: Set Notations - Question 1

Let X = {x | x = 2 + 4k, where k = 0, 1, 2, 3,...24}. Let S be a subset of X such that the sum of no two elements of S is 100. What is the maximum possible number of elements in S ?  

Detailed Solution for Test: Set Notations - Question 1

Calculation:

The set X is given by

{2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98}.

We want to find the maximum size of a subset S of X such that no two

elements sum to 100.

The pairs in X that sum to 100 are

(2, 98), (6, 94), (10, 90), (14, 86), (18, 82), (22, 78), (26, 74), (30, 70), (34,

66), (38, 62), (42, 58), (46, 54).

Therefore, 

S = {2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50}

∴ The maximum possible number of elements in S be 13.

Test: Set Notations - Question 2

In a party of 150 persons, 75 persons take tea, 60 persons take coffee and 50 persons take milk. 15 of them take both tea and coffee, but no one taking milk takes tea. If each person in the party takes at least one drink, then what is the number of persons taking milk only ?

Detailed Solution for Test: Set Notations - Question 2

Let, x number of persons taking milk only.

F2 Savita Defence 31-5-23 Sachin K D28

According to the question

60 + 15 + (x - 5) + (50 - x) + x = 150

120 + x = 150

x = 150 - 120 = 30

∴ The required value is 30.

Test: Set Notations - Question 3

Which of the following is a set?

Detailed Solution for Test: Set Notations - Question 3

(a) The collection of all months of a year is a well-defined collection of objects because one can definitely identify a month that belongs to this collection. Hence, this collection is a set.

(b) A collection of novels is not a well-defined collection because one cannot identify a book that belongs to this collection. Hence, this collection is not a set.

(c) A collection of top rich persons can not be defined. Hence, this collection is not a set.

(d) The collection of the ten most talented writers of India is not a well-defined collection because the criteria for determining a writer's talent may vary from person to person. Hence, this collection is not a set.

Test: Set Notations - Question 4

If Q = {x : x = 1 / y, where y ∈ N}, then:

Detailed Solution for Test: Set Notations - Question 4

As y ∈ N, y can be 1, 2, 3, 4...
∴ x will be 1, 1/2, 1/3, 1/4...
1 ∈ Q

Test: Set Notations - Question 5

Write A = {1, 4, 9, 16, 25} in set builder form.

Detailed Solution for Test: Set Notations - Question 5
  • We know that, 12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25
  • Therefore the set A = {1, 4, 9, 16, 25...} can be written in set builder form as: 
    A = {x: x is the square of a natural number}
Test: Set Notations - Question 6

A = { x : x ≠ x } represents:

Detailed Solution for Test: Set Notations - Question 6
  • According to this, x should be a number which is not equal to the number itself and there is no number which is not equal to the number itself.
  • Therefore A is an empty set.
Test: Set Notations - Question 7

If A = {3, 6, 9, 12} and B = {6, 8, 9} then intersection of A and B is:

Detailed Solution for Test: Set Notations - Question 7
  • An intersection is the collection of all the elements that are common to all the sets under consideration.
  • Here element 6 & 9 are common in both the sets. So option C is correct. 
Test: Set Notations - Question 8

Which of the following is a finite set?

Detailed Solution for Test: Set Notations - Question 8
  • finite set is a set that has a finite number of elements.
  • Since  x2 – 25 = 0 has finite number of elements. So it is a finite set. 
Test: Set Notations - Question 9

Consider the following statements:

I: If A = {x: x is an even natural number} and B = {y: y is a natural number}, A subset B.

II: Number of subsets for the given set A = {5, 6, 7, 8} is 12.

III: Number of proper subsets for the given set A = {5, 6, 7, 8} is 15.

Which of the following statement(s) is/are correct?

Detailed Solution for Test: Set Notations - Question 9

Concept:

The null set is a subset of every set. (ϕ ⊆ A)

Every set is a subset of itself. (A ⊆ A)

The number of subsets of a set with n elements is 2n.

The number of proper subsets of a given set is 2n - 1

Calculation:

Statement I: If A = {x: x is an even natural number} and B = {y: y is a natural number}, A subset B.

A = {2, 4, 6, 8, 10, 12, ...} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...}.

It is clear that all the elements of set A are included in set B.

So, set A is the subset of set B.

Statement I is correct.

Statement II:  Number of subsets for the given set A = {5, 6, 7, 8} is 16.

Given: A = {5, 6, 7, 8}

The number of elements in the set is 4

We know that,

The formula to calculate the number of subsets of a given set is 2n

 = 2= 16

Number of subsets is 16

Statement II is incorrect.

Statement III: Number of proper subsets for the given set A = {5, 6, 7, 8} is 15.

The formula to calculate the number of proper subsets of a given set is 2n - 1

 = 2- 1

 = 16 - 1 = 15

The number of proper subsets is 15.

Statement III is correct.

∴ Statements I and III are correct.

Test: Set Notations - Question 10

Which of the following has only one subset?

Detailed Solution for Test: Set Notations - Question 10
  • Every set has the empty set as a subset. So if a set has 1 element, like {0}, then it will have 2 subsets: itself and the empty set, which is denoted by { }.
  • So, if a set has only one subset, then this set must be the empty set.
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