Test: Sets - JEE MCQ

# Test: Sets - JEE MCQ

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## 25 Questions MCQ Test Mathematics (Maths) for JEE Main & Advanced - Test: Sets

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Test: Sets - Question 1

### 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 15. 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: Sets - Question 1

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 A = {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 15.

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: Sets - Question 2

### Let A and B be two finite sets having m and n elements, respectively. Then the total number of mapping from A to B is

Detailed Solution for Test: Sets - Question 2

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Test: Sets - Question 3

### Consider the following relations: (i) A - B = A - (A ∩ B) (ii) A = (A ∩ B) ∪ (A - B) (iii) A - (B ∪ C) = (A - B) ∪ (A - C) Which of these is/are correct?

Detailed Solution for Test: Sets - Question 3

Using the Venn Diagram,
(i) A - B = A - (A ∩ B) is true

(ii) A = (A ∩ B) ∪ (A - B) is true

(iii) A - (B ∪ C) = (A - B) ∪ (A - C) is false

Test: Sets - Question 4

Two sets, A and B, are as under:
A = {(a, b) ∈ R × R : |a − 5| < 1 and |b − 5| < 1};
B = {a, b) ∈ R × R : 4 (a − 6)2 + 9(b−5) 2 ≤ 36}. Then,

Detailed Solution for Test: Sets - Question 4

A = {(a, b) ∈ R × R : |a − 5| < 1 and |b − 5| < 1}

⇒ a ∈ (4, 6) and b ∈ (4, 6)

Therefore, A = {(a, b): a ∈ (4, 6) and b ∈ (4, 6)}

Now, B = {a, b) ∈ R × R : 4 (a − 6) 2 + 9(b−5) 2 ≤ 36}

From the conditions above for set A, the maximum value of B is:

4 (a − 6) 2 + 9(b−5) 2 = 4 (4 − 6)2 + 9(6−5)2 = 25 ≤ 36

We can check for other values as well.

Elements of set A satisfy the conditions in Set B.

Hence, A is a subset of B.

Test: Sets - Question 5

Let A and B be two sets containing 4 and 2 elements, respectively. Then the number of subsets of the set AxB, each having at least 3 elements, is

Detailed Solution for Test: Sets - Question 5

n(A) = 4 and n(B) = 2

Therefore, n(A x B) = 8

Number of subsets of A x B having atleast 3 elements = 28 – 8C0 – 8C1 – 8C2

= 256 – 1 – 8 – 28

= 219

Test: Sets - Question 6

In a city, 20 per cent of the population travels by car, 50 per cent travels by bus and 10 per cent travels by both car and bus. Then persons travelling by car or bus is:

Detailed Solution for Test: Sets - Question 6

► n(Persons travelling by car) = 20 = n(C)
► n(Persons travelling by bus) = 50 = n(B)
► n(Persons travelling by both car and bus) = 10 = n(C ∩ B)
∴ n(C ∪ B) = n(C) + n(B) - n(C ∩ B)
= 20 + 50 - 10
= 60

Test: Sets - Question 7

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 5}, B = {6, 7}. Then A ∩ B’ is:

Detailed Solution for Test: Sets - Question 7

The correct answer is a) A

• B' gives us all the elements in U other than 6 and 7 i.e., B' = {1, 2, 3, 4, 5, 8, 9, 10}
• The intersection of this set with A will be the common elements in both of these (A and B') i.e., = {1, 2, 5} which is set A itself.
Test: Sets - Question 8

In the rule method, the null set is represented by:

Detailed Solution for Test: Sets - Question 8
• This method involves specifying a rule or condition which can be used to decide whether an object can belong to the set.
• This rule is written inside a pair of curly braces and can be written either as a statement or expressed symbolically or written using a combination of statements and symbols.
• {x :  x ≠ x}
This implies a null set.
Test: Sets - Question 9

If A and B are two sets, then A ∪ B = A ∩ B if:

Detailed Solution for Test: Sets - Question 9

If A ∪ B = A ∩ B then A = B.
Example: Let A = {1, 2, 3, 4} & B = {1, 2, 3, 4}
(AUB) = (A⋂B)
⇒{1,2,3,4} = {1,2,3,4}

Test: Sets - Question 10

If  A = {2, 3, 4, 8, 10}, B = {3, 4, 5, 10, 12}, C = {4, 5, 6, 12, 14} then (A ∩ B) ∪ (A ∩ C) is equal to:

Detailed Solution for Test: Sets - Question 10

A = {2, 3, 4, 8, 10}
B = {3, 4, 5, 10, 12}
C = {4, 5, 6, 12, 14}
► (A ∩ B) = {3,4,10}
► (A ∩ C) = {4}
► (A ∩ B) U (A ∩ C) = {3,4,10}

Test: Sets - Question 11

The shaded region in the given figure is:

Detailed Solution for Test: Sets - Question 11
• (A - B) is A but not B which means A excluding the intersection of A and B.
• Similarly, in the figure, the shaded region is A but not C and B which means A excluding B and C. But if B and C both are excluded from A then the intersection of A, B and C get excluded two times.
• So, we take B ∪ C to be excluded from A. Therefore the answer is A - (B ∪ C).
Test: Sets - Question 12

Let A and B be two sets containing four and two elements respectively.Then the number of subsets of the set A × B,each having atleast three elements is:

Detailed Solution for Test: Sets - Question 12

The correct answer is a) 219

⇒ Number of elements in Set A = 4
⇒ Number of elements in Set B = 2
∴ Number of elements in set (A × B) = 8
∴ Total number of subsets of (A×B) = 28 = 256
⇒ Number of subsets having 0 elements = 8C0 = 1
⇒ Number of subsets having 1 element = 8C1 = 8
⇒ Number of subsets having 2 elements = 8C2 = 28
∴ Number of subsets having at least 3 elements = 256 - 1 - 8 - 28 = 219

Test: Sets - Question 13

Let A and B be two sets then (A ∪ B)′ ∪ (A′ ∩ B) is equal to:

Detailed Solution for Test: Sets - Question 13

From Venn-Euler's Diagram

∴ (A ∪ B)′ ∪ (A′ ∩ B) = A′

Test: Sets - Question 14

The number of elements in the set {(ab) : 2a2 + 3b2 = 35, a∈ Z}, where Z is the set of all integers, is:

Detailed Solution for Test: Sets - Question 14

The correct answer is b) 8

2a2 + 3b2 = 35

3b2 = 35 – 2a2

The maximum value on RHS is 35.

3b2 ≤ 35

b2 ≤ 11.66

b = 0, ± 1, ± 2, ± 3….

3b2 = 35 – 2a2

Any number multiplied by 2 gives an even number.

3b2 is odd.

If b = 0, then 3b2 is not odd.

If b = ± 1, then 3bis odd.

If b = ± 2, then 3bis even.

If b = ± 3, then 3bis odd.

3b2 = 35 – 2a2

If b = ± 1, then b2 = 1,

3 = 35 – 2a2

2a2 = 32

a2 = 16

a = ± 4

If b = ± 3, then b2 = 9

27 = 35 – 2a2

2a2 = 8

a2 = 4

a = ± 2

[(4, 1), (4, -1), (-4, 1), (-4, -1), (2, -3), (2, 3), (-2, 3), (-2, -3)]

Therefore there are 8 elements in total.

Test: Sets - Question 15

Given n(U) = 20, n(A) = 12, n(B) = 9, n(A ∩ B) = 4, where U is the universal set, A and B are subsets of U, then n((A ∪ B)c) =

Detailed Solution for Test: Sets - Question 15
• n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 12 + 9 − 4 = 17
• Now, n((A ∪ B)c) = n(U) − n(A ∪ B) = 20 − 17 = 3
Test: Sets - Question 16

In a certain town, 25% families own a cell phone,15% families own a scooter and 65% families own neither a cell phone nor a scooter. If 1500 families own both a cell phone and a scooter, then the total number of families in the town is:

Detailed Solution for Test: Sets - Question 16

Step-by-step Explanation:
Since we have given that,
Percentage of families own a cell phone = 25%
Percentage of families own a scooter = 15%
Percentage of families neither own = 65%
So, it becomes,
P(C ∪ S)= 1 - P(C ∪ S)
0.65 = 1 - P(C ∪ S)
1 - 0.65 = P(C ∪ S)
P(C ∪ S) = 0.35
So, it becomes,
P(C ∪ S) = P(C) + P(S) - P(C ∩ S)
0.35 = 0.25 + 0.15 - x
0.35 = 0.40 - x
x = 0.05
Thus, 0.05 x Total number of families in town = 1500
Total number of families in the town =
1500 / 0.05 = 30,000
Hence, the total number of families in the town is 30,000.

Test: Sets - Question 17

If A = {2, 3, 4, 8, 10}, B = {3, 4, 5, 10, 12} and C = {4, 5, 6, 12, 14}, then  (A∩B)∪(A∩C) is equal to:

Detailed Solution for Test: Sets - Question 17

The correct answer is d)

A={2,3,4,8,10},B={3,4,5,10,12},C={4,5,6,12,14}

To find : (A∩B) ∪(A∩C)
A∩B = elements common in sets A and B ⟹A∩B={3,4,10}
A∩C = elements common in sets A and C ⟹A∩C={4}
Now,
(A∩B)∪(A∩C)=elements in (A∩B) and (A∩C)
(A∩B)∪(A∩C)={3,4,10}∪{4}
(A∩B)∪(A∩C)={3,4,10}.

Test: Sets - Question 18

If A = {2, 4, 5}, B = {7, 8, 9}, then n(A × B) is equal to:

Detailed Solution for Test: Sets - Question 18

The correct answer is c) 9

Given A=[2,4,5],B=[7,8,9]

To find: n(A×B)=?

A×B={(2,7),(2,8),(2,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9)}

∴n(A×B)=9

Test: Sets - Question 19

If A and B are disjoint, then n (A ∪ B) is equal to:

Detailed Solution for Test: Sets - Question 19

As n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
But in case of disjoint sets, n(A ∩ B) = 0
∴ n(A ∪ B) = n(A) + n(B)

Test: Sets - Question 20

The number of non-empty subsets of the set {1, 2, 3, 4} is:

Detailed Solution for Test: Sets - Question 20

The number of non-empty subsets = 2n − 1 = 24 − 1 = 16 − 1 = 15
The subsets are:
[{1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,4}, {1,2,3}, {1,3,4} ,{2,3,4}, {1,2,3,4}]

Test: Sets - Question 21

The set of intelligent students in a class is:

Detailed Solution for Test: Sets - Question 21

As the opinions of different person is different about the intelligent student. So, we can't exactly have the same student's name in a set, hence it is not a well-defined collection.

Test: Sets - Question 22

If A and B are two sets, then A ∩ (B − A) is:

Detailed Solution for Test: Sets - Question 22

A ∩ (B − A) = ψ,[∵ x ∈ B − A ⇒ x∉ A]

Test: Sets - Question 23

If n(A) = 3, n(B) = 6 and A ⊆ B. Then the number of elements in A ∪ B is equal to:

Detailed Solution for Test: Sets - Question 23

Since A ⊆ B
∴ A ∪ B = B
So, n(A ∪ B) = n(B) = 6

Test: Sets - Question 24

In a class of 100 students, 55 students have passed in Mathematics and 67 students have passed in Physics. Then the number of students who have passed in Physics only is:

Detailed Solution for Test: Sets - Question 24

Let A be the set of students who have passed in Mathematics, and B be the set of students who have passed in Physics.
We are given that |A| = 55 and |B| = 67, where |A| and |B| represent the number of students in sets A and B, respectively.
We are also given that there are 100 students in total. We need to find the number of students who have passed in Physics only, which means we need to find |B - A|.
First, let's find the number of students who have passed in both Mathematics and Physics, which can be represented by |A ∩ B|.
Using the principle of inclusion-exclusion, we have:
|A ∪ B| = |A| + |B| - |A ∩ B|
Since there are 100 students in total, we can say that |A ∪ B| = 100.
Now, we can find |A ∩ B|:
100 = 55 + 67 - |A ∩ B|
100 = 122 - |A ∩ B|
|A ∩ B| = 22
Now, we can find the number of students who have passed in Physics only, which is |B - A|:
|B - A| = |B| - |A ∩ B|
|B - A| = 67 - 22
|B - A| = 45
so, the correct answer is 45.

Test: Sets - Question 25

If A = {1, 2, 3, 4}, B = {4, 5, 6, 7}, then A - B = ?

Detailed Solution for Test: Sets - Question 25
• (A - B) is nothing but the elements which are present in set A but not present in set B.
• In other words, it is elements of set A excluding the elements which are in set B.
• Here A = {1, 2, 3, 4} and B = {4, 5, 6, 7}
So, (A - B) = {1, 2, 3}

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## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests