Q1.
Which of the following Venn diagrams best represents the relationship between "Months", "Weeks", and "Days"?
- Three concentric circles (Days inside Weeks, Weeks inside Months).
- Three separate circles.
- Two overlapping circles (Months, Weeks) and a separate circle (Days).
- Three overlapping circles.
Show answer
Answer: A. Three concentric circles (Days inside Weeks, Weeks inside Months).
Days are part of Weeks, and Weeks are part of Months. This is a hierarchical relationship where one category is entirely contained within another, which is best represented by concentric circles.
Q2.
In a class of 50 students, 30 students like Cricket, and 25 students like Football. If 10 students like both Cricket and Football, how many students like only Cricket?
- 20
- 15
- 25
- 30
Show answer
Answer: A. 20
To find the number of students who like only Cricket, subtract the number of students who like both Cricket and Football from the total number of students who like Cricket.
Number of students who like only Cricket = (Students who like Cricket) - (Students who like both)
= 30 - 10 = 20.
Q3.
In a group, 40 people speak Hindi, and 30 people speak English. If 15 people speak both Hindi and English, how many people are there in the group (assuming everyone speaks at least one language)?
- 55
- 70
- 45
- 60
Show answer
Answer: A. 55
To find the total number of people in the group, use the formula for the union of two sets: n(A U B) = n(A) + n(B) - n(A ∩ B).
Total people = (People who speak Hindi) + (People who speak English) - (People who speak both)
= 40 + 30 - 15
= 70 - 15 = 55.
Q4.
In a survey of 100 students, 40 students passed in Math, 50 in Science, and 30 in English. 10 students passed in Math and Science, 8 in Science and English, 12 in Math and English. If 5 students passed in all three subjects, how many students passed in only one subject?
- 75
- 60
- 65
- 58
Show answer
Answer: A. 75
Let M, S, E be the sets of students who passed in Math, Science, and English respectively.
Given:
n(M) = 40, n(S) = 50, n(E) = 30
n(M∩S) = 10, n(S∩E) = 8, n(M∩E) = 12
n(M∩S∩E) = 5
Number of students who passed in only Math = n(M) - n(M∩S) - n(M∩E) + n(M∩S∩E)
= 40 - 10 - 12 + 5 = 23.
Number of students who passed in only Science = n(S) - n(M∩S) - n(S∩E) + n(M∩S∩E)
= 50 - 10 - 8 + 5 = 37.
Number of students who passed in only English = n(E) - n(M∩E) - n(S∩E) + n(M∩S∩E)
= 30 - 12 - 8 + 5 = 15.
Total students who passed in only one subject = 23 + 37 + 15 = 75.
Q5.
Using the data from the previous question (Q4): In a survey of 100 students, 40 students passed in Math, 50 in Science, and 30 in English. 10 students passed in Math and Science, 8 in Science and English, 12 in Math and English. If 5 students passed in all three subjects, how many students passed in at least one subject?
- 80
- 85
- 90
- 95
Show answer
Answer: D. 95
Let M, S, E be the sets of students who passed in Math, Science, and English respectively.
Given:
n(M) = 40, n(S) = 50, n(E) = 30
n(M∩S) = 10, n(S∩E) = 8, n(M∩E) = 12
n(M∩S∩E) = 5
To find the number of students who passed in at least one subject, we use the Principle of Inclusion-Exclusion for three sets:
n(M U S U E) = n(M) + n(S) + n(E) - n(M∩S) - n(S∩E) - n(M∩E) + n(M∩S∩E)
= 40 + 50 + 30 - 10 - 8 - 12 + 5
= 120 - 30 + 5
= 90 + 5 = 95.
Q6.
In a town, 50% of the population reads newspaper A, and 40% reads newspaper B. If 20% of the population reads both newspapers, what percentage of the population reads neither newspaper?
- 10%
- 20%
- 30%
- 40%
Show answer
Answer: C. 30%
Let A be the set of people who read newspaper A, and B be the set of people who read newspaper B.
Given:
Percentage reading newspaper A, n(A) = 50%
Percentage reading newspaper B, n(B) = 40%
Percentage reading both, n(A∩B) = 20%
Percentage of people who read at least one newspaper = n(A U B) = n(A) + n(B) - n(A∩B)
= 50% + 40% - 20% = 90% - 20% = 70%.
Percentage of people who read neither newspaper = Total population - n(A U B)
= 100% - 70% = 30%.
Q7.
Consider the following statements:
1. All engineers are graduates.
2. Some graduates are managers.
Which of the following conclusions logically follows from the given statements?
- All engineers are managers.
- Some managers are engineers.
- Some graduates are engineers.
- No engineer is a manager.
Show answer
Answer: C. Some graduates are engineers.
Let E = Engineers, G = Graduates, M = Managers.
Statement 1: All E are G (E is a subset of G).
Statement 2: Some G are M (There is an overlap between G and M).
From statement 1, if all engineers are graduates, it directly implies that some graduates are engineers. This is a fundamental property of set inclusion (If A ⊆ B, then B contains A, meaning some elements of B are also elements of A).
Options 1, 2, and 4 cannot be definitively concluded. For example, managers might only be among graduates who are not engineers, so 'All engineers are managers' and 'Some managers are engineers' are not necessarily true. 'No engineer is a manager' is also not necessarily true, as there could be an overlap.
Q8.
In a group of 80 people, 45 like coffee, 35 like tea, and 15 like both coffee and tea. How many people like only tea?
- 20
- 25
- 30
- 35
Show answer
Answer: A. 20
Let C = people who like coffee, T = people who like tea.
Given:
n(C) = 45
n(T) = 35
n(C∩T) = 15
Number of people who like only tea = (Total people who like tea) - (People who like both coffee and tea)
= n(T) - n(C∩T)
= 35 - 15 = 20.
Q9.
In a coaching center, 200 students are enrolled. 30% of students study Math, 25% study Physics, and 20% study Chemistry. 10% study Math and Physics, 8% study Physics and Chemistry, and 7% study Math and Chemistry. If 30% of students study none of the three subjects, how many students study all three subjects?
- 10
- 20
- 30
- 40
Show answer
Answer: D. 40
Total students = 200.
Students studying none of the three subjects = 30% of 200 = 60 students.
Students studying at least one subject = Total students - Students studying none = 200 - 60 = 140 students.
So, n(M U P U C) = 140.
Convert percentages to actual numbers of students:
n(M) = 30% of 200 = 60
n(P) = 25% of 200 = 50
n(C) = 20% of 200 = 40
n(M∩P) = 10% of 200 = 20
n(P∩C) = 8% of 200 = 16
n(M∩C) = 7% of 200 = 14
Using the Principle of Inclusion-Exclusion for three sets:
n(M U P U C) = n(M) + n(P) + n(C) - n(M∩P) - n(P∩C) - n(M∩C) + n(M∩P∩C)
Substitute the known values:
140 = (60 + 50 + 40) - (20 + 16 + 14) + n(M∩P∩C)
140 = 150 - 50 + n(M∩P∩C)
140 = 100 + n(M∩P∩C)
n(M∩P∩C) = 140 - 100 = 40.
Therefore, 40 students study all three subjects.
Q10.
In a group of 120 people, some like Coffee (C), some like Tea (T), and some like Juice (J).
1. All people who like Coffee also like Tea.
2. No one who likes Juice also likes Coffee.
3. The number of people who like only Tea is 30.
4. The number of people who like Juice is 40.
5. The number of people who like Tea and Juice but not Coffee is 15.
How many people like only Coffee?
- 0
- 10
- 15
- 20
Show answer
Answer: A. 0
Let's analyze the given statements:
1. "All people who like Coffee also like Tea." This statement implies that the set of people who like Coffee (C) is a subset of the set of people who like Tea (T). In set notation, C ⊆ T.
If C is a subset of T, it means that every person who likes Coffee must also like Tea. Therefore, there cannot be any person who likes 'only Coffee' (i.e., likes Coffee but not Tea). The region representing 'only Coffee' in a Venn diagram would be empty.
The other statements (2, 3, 4, 5) provide additional information about other parts of the Venn diagram but do not alter the conclusion derived from statement 1 regarding 'only Coffee'.
Thus, the number of people who like only Coffee is 0.
