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Test: Sets- 2 - JEE MCQ


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25 Questions MCQ Test Chapter-wise Tests for JEE Main & Advanced - Test: Sets- 2

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Test: Sets- 2 - Question 1

If set A = {1, 3, 5, 7}, set B = {1, 4, 7}, find the value of A − B.

Detailed Solution for Test: Sets- 2 - Question 1

Concept use:

A - B = The element which are present in A but not in B

Calculations:

If set A = {1, 3, 5, 7}, set B = {1, 4, 7}, 

The Value of A - B = {3, 5} (The element which are present in A but not in B)

Test: Sets- 2 - Question 2

If the Cartesian product of two sets A and B(A × B) = {(3, 2), (3, 4), (5, 2), (5, 4)}, find set A.

Detailed Solution for Test: Sets- 2 - Question 2

Given:

Cartesian product of two sets A and B: A × B = {(3, 2), (3, 4), (5, 2), (5, 4)}

Concept used: 

In a Cartesian product A × B, the first element of each ordered pair belongs to set A, and the second element belongs to set B.

Calculation:

From the given Cartesian product A × B, we can extract the elements of set A:

A = {3, 5}

∴ The set A is {3, 5}.

Test: Sets- 2 - Question 3

A ∪ (B ∩ C) = ____________

Test: Sets- 2 - Question 4

If A = {2, 3, 4} and B = {3, 5} then A ∪ B will be:

Detailed Solution for Test: Sets- 2 - Question 4

Given that

A = {2, 3, 4}

B = {3, 5}

Union of A and B 

A ∪ B = {2, 3, 4, 5}

Therefore, A ∪ B = {2, 3, 4, 5}.

Test: Sets- 2 - Question 5

If A = {1, 3, 4} and B = {x : x ∈ R and x2 - 7x + 12 = 0} then which of the following is true ?

Detailed Solution for Test: Sets- 2 - Question 5

CONCEPT:

Let A and B be two sets then A is said to be proper subset of B, if A is a subset of B and A is not equal to B. It is denoted as A ⊂ B.

CALCULATION:

Given: A = {1, 3, 4} and B = {x : x ∈ R and x2 - 7x + 12 = 0}

First let's find the roaster form of set B

In order to do so we need to find the roots of the equation x2 - 7x + 12 = 0

⇒ x2 - 3x - 4x + 12 = 0

⇒ x(x - 3) - 4(x - 3) = 0

⇒ (x - 4) × (x - 3) = 0

⇒ x = 3, 4

⇒ B = {3, 4}

As we can clearly see that, all the elements of B are there in set A but A ≠ B i.e B ⊂ A

Hence, the correct option is 3.

Test: Sets- 2 - Question 6

Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then the number of subsets of A containing exactly two elements is

Detailed Solution for Test: Sets- 2 - Question 6

Concept:

Combination: Selecting r objects from given n objects.

  • The number of selections of r objects from the given n objects is denoted by  nCr

Calculation:

Number of elements in A = 10

Number of subsets of A containing exactly two elements = Number of ways we can select 2 elements from 10 elements

⇒ Number of ways we can select 2 elements from 10 elements = 10C2 = 45

∴ Number of subsets of A containing exactly two elements = 45

Test: Sets- 2 - Question 7

If A = {1, 2, 3, 4, 5, 7, 8, 9} and B = {2, 4, 6, 7, 9} then find the number of proper subsets of A ∩  B ?

Detailed Solution for Test: Sets- 2 - Question 7

CALCULATION:

Given: A = {1, 2, 3, 4, 5, 7, 8, 9} and B = {2, 4, 6, 7, 9}

As we know that, A ∩ B = {x : x ∈ A and x ∈ B}

⇒ A ∩ B = {2, 4, 7, 9}

As we can see that,

The number of elements present in A ∩ B = 4

i.e n(A ∩ B) = 4

As we know that;

If A is a non-empty set such that n(A) = m then

The numbers of proper subsets of A are given by 2m - 1.

So, The number of proper subsets of A ∩  B = 24 - 1 = 15

Hence, the correct option is A

Test: Sets- 2 - Question 8

If A = {1, 2, 5, 7} and B = {2, 4, 6} then find the number of proper subsets of A U B ?

Detailed Solution for Test: Sets- 2 - Question 8

CALCULATION:

Given: A = {1, 2, 5, 7} and B = {2, 4, 6}

As we know that,  A ∪ B = {x : x ∈ A or x ∈ B}.

⇒ A ∪ B = {1, 2, 4, 5, 6, 7}

As we can see that, the number of elements present in A U B = 6 i.e n(A U B) = 6

As we know that, if A is a non-empty set such that n(A) = m then numbers of proper subsets of A is given by 2m - 1.

So, the number of proper subsets of A Δ B = 26 - 1 = 63

Hence, the correct option is 3.

Test: Sets- 2 - Question 9

If A = {1, 3, 5} then find the cardinality of the power set of A ?

Detailed Solution for Test: Sets- 2 - Question 9

CALCULATION:

Given: A = {1, 3, 5}

Here, we have to find the cardinality of the power set of A i.e n (P(A))

As we know that if A is a finite set with m elements. Then the number of elements (cardinality) of the power set of A is given by: n (P(A)) = 2m.

Here, we can see that, the given A has 3 elements i.e n(A) = 3

So, the cardinality of the given set is n(P(A)) = 23 = 8

Hence, the correct option is 4.

Test: Sets- 2 - Question 10

If A = {x, y, z}, then the number of subsets in powerset of A is

Detailed Solution for Test: Sets- 2 - Question 10

Given, A = {x, y, z}. 

The power set (or powerset) of a Set A is defined as the set of all subsets of the Set A including the Set itself and the null or empty set.

Powerset of A = {ϕ,{x}, {y}, {z}, {x, y}, {y, z}, {x, z},{x, y, z}}.

Hence, the number of subsets in the powerset of A is 8.

Test: Sets- 2 - Question 11

In every (n + 1) - - elementic subset of the set (1, 2, 3, .......2n) which of the following is correct:

Detailed Solution for Test: Sets- 2 - Question 11

We divide the set into n classes {1, 2}, {3, 4},......{2n - 1, 2n}.

By the pigeonhole principle, given n +1 elements at least two of them will be in the same case {2k - 1, 2k} (1 ≤ k ≤ n). But 2k - 1 and 2k are relatively prime because their difference is 1. 

Test: Sets- 2 - Question 12

If A is a subset of B and B is a subset of C, then the cardinality of A ∪ B ∪ C is equal to:

Detailed Solution for Test: Sets- 2 - Question 12

Calculations:

Since, A ⊂ B and B ⊂ C, therefore A ∪ B ∪ C = C.

⇒ n(A ∪ B ∪ C) = n(C).

∴ The cardinality (number of elements) of A ∪ B ∪ C = cardinality (number of elements) of C.

Test: Sets- 2 - Question 13

In a survey of 1,000 consumers it is found that 720 consumers liked product A and 450 liked product B. What is the least number that must have liked both the products?

Detailed Solution for Test: Sets- 2 - Question 13

Given:

Total consumers = 1000

Consumers who like product A = 720 

Consumers who like product B = 450

Formula Used: 

n(A U B) = n(A) + n(B) - n(A ∩ B) 

Calculation:

Least number that like both the products are n(AꓵB)

⇒ 1000 = 720 + 450 - n(A ∩ B)   

⇒ 1000 = 1170 - n(A ∩ B)

⇒ n(A ∩ B) = 170

∴ 170 consumers like both the products A and B 

Test: Sets- 2 - Question 14

Consider the following statements:

1. A = {1, 3, 5} and B = {2, 4, 7} are equivalent sets.
2. A = {1, 5, 9} and B = {1, 5, 5, 9, 9} are equal sets.

Which of the above statements is/are correct?

Detailed Solution for Test: Sets- 2 - Question 14

Concept:

Equivalent set: Two sets A and B are said to be equivalent if they have the same number of elements (cardinality), regardless of what the elements are. , i.e n(A) = n(B).

Equal set: Two sets A and B are said to be equal if they have exactly the same elements. The repetition of elements does not matter in sets because a set is defined as a collection of distinct objects, so repeated elements are generally ignored.

If A = B, then n(A) = n(B) and for any x ∈ A, x ∈ B too.

Calculation:

From the definition of equivalent and equal set we can see that;

Statement 1 is true because the number of elements in A = Number of elements in B = 3

Statement 2 is true because {1, 5, 9} ∈ A and {1, 5, 9} ∈ B

Test: Sets- 2 - Question 15

The table above shows the research data gathered from a group of rural and urban residents. What is the number of rural residents who own a car?

(1) The number of rural residents in the group is 60% of the number of urban residents in the group.

(2) The difference between the number of urban residents who own a car and the number of rural residents who own a car is one-fifth of the number of rural and urban residents in the group.

Detailed Solution for Test: Sets- 2 - Question 15

Steps 1 & 2: Understand Question and Draw Inferences

Given: Data on two parameters are:

  • Car ownership – yes or no
  • Residence – urban or rural

To find: Number of rural residents who own a car

  • Let this number be Q
  • Let the number of urban residents without a car be P

The table can be filled in terms of P and Q as follows:

 

Step 3: Analyze Statement 1 independently

  • Number of rural residents = 60/100

*(Number of urban residents)

 

  • Linear equation with 2 unknowns. Not sufficient to find Q.

Step 4: Analyze Statement 2 independently

  • |Number of car-owning urban residents - Number of car-owning rural residents| = 1/5 (Total Number of rural and urban residents)
  • |15 - Q| = 15 ∗ (33 + Q)
    • Note: we have considered the absolute value of the difference here because we do not know yet which of the 2 values is greater, 15 or Q

Case 1: Q < 15

Case 2: Q > 15

2 values of Q. Not sufficient.

 

Step 5: Analyze Both Statements Together (if needed)

  • From Statement 1: 18 - P + Q = 0.6(15 + P). . . (I)
  • From Statement 2: Q = 7 or 27
  • Let’s put the 2 values of Q in Eq. I to see if both lead to sensible values of P:

Case 1 : Q = 7

Case 2: Q = 27

Since the number of people cannot be in fractions, Case 2 can be rejected

So, Q = 7.

Answer: Option C

Test: Sets- 2 - Question 16

Of the 48 students in a class, 16 like to study History. What percentage of the girls in the class do not like to study History?

(1) One-third of the boys in the class like to study History

(2) The number of girls who like History is 50 percent of the number of girls who do not like History

Detailed Solution for Test: Sets- 2 - Question 16

Steps 1 & 2: Understand Question and Draw Inferences

Given:

  • Let number of girls who do not like history = X
  • Total number of girls = Y
  • We need to find the value of X/Y
  •  to answer the question
  • Completing the above table in terms of X and Y:

Step 3: Analyze Statement 1 independently

One-third of the boys in the class like to study History

Sufficient

Step 4: Analyze Statement 2 independently

The number of girls who like History is 50 percent of the number of girls who do not like History

Sufficient

 

Step 5: Analyze Both Statements Together (if needed)

Since we’ve already arrived at a unique answer in Step 3 and Step 4, this step is not required

 

Answer: Option D

Test: Sets- 2 - Question 17

All the employees of Company X were appraised at the year end. 40% of the employees got a promotion, 50% of the employees got a salary hike and 30% of the employees neither got a promotion nor got a salary hike. If there were 40 employees who got both the promotion and the salary hike, what was the number of employees in the company?

Detailed Solution for Test: Sets- 2 - Question 17

Given

  • Percentage of employees who got a promotion = 40%
  • Percentage of employees who got a salary hike = 50%
  • Percentage of employees who neither got a promotion nor a salary hike = 30%
  • Number of employees who got both the promotion and the salary hike = 40

To Find: Number of employees in the company?


Approach

  1. To find the total number of employees in the company, we will try to express the number of employees who got both the promotion and the salary hike(i.e. 40), as a percentage of the total employees of the company.
     

Total employees in the company = Employees who got only salary hike + Employees who got only promotion + Employees who got salary hike and promotion + Employees who got neither salary hike nor promotion

  1. Adding “Employees who got salary hike and promotion on both sides” we have
  2. Total employees in the company + Employees who got salary hike and promotion = (Employees who got only salry hike + Employees who got salary hike and promotion)+ (Employees who got only promotion + Employees who got salary hike and promotion) + Employees who got neither salary hike nor promotion
  3. Total employees in the company + Employees who got salary hike and promotion = Employees who got a salary hike+  Employees who got a promotion  + Employees who got neither salary hike nor promotion
  4. From the above equation, we can find the percentage of employees who got salary hike and promotion

Working Out

  1. Total employees in the company + Employees who got salary hike and promotion = Employees who got a salary hike+  Employees who got a promotion  + Employees who got neither salary hike nor promotion
    a.  100% + Employees who got salary hike and promotion = 50% + 40% + 30%
    b.  Employees who got salary hike and promotion = 20%
     
  2. Now, we know that employees who got salary hike and promotion = 40
    1. So, if 20% is 40, 100% = 200

Hence, the total number of employees in the company = 200


Answer: D

Test: Sets- 2 - Question 18

The human resources manager of a company compiled the data that the company had on the academic qualifications of its 100 employees. He observed that employees who had an Engineering and an MBA degree were the most in number and employees who had neither an Engineering nor an MBA degree were the least in number. If there were 20 employees who had an Engineering degree, but not an MBA degree, which of the following cannot be the number of employees who had an MBA degree?

Detailed Solution for Test: Sets- 2 - Question 18

Given

  • Number of employees = 100
     
  • Let us assume:
    • Number of employees who only have an MBA degree = c.
    • Number of employees who have both an MBA and an Engineering degree = b.
    • Number of employees who do neither have an MBA nor an Engineering degree = d.
       
  • Employees having neither Engineering nor an MBA were the least in number.
    • So d is the least in the number out of 20,b, c & d.
       
  • Number of employees who had an Engineering degree and not an MBA degree = 20

To Find: Among the options, which of them cannot be the number of employees who had an MBA degree?


Approach

  1. Let’s first draw the venn diagram to understand the information given

 

2. We need to find the possible range of values of employees who had an MBA degree = b + c

  1. So, we need to find the maximum and minimum value of b + c

3. As we know that total number of employees is 100 and d is minimum number of employees possible, we will use this information to find out the range of values of b + c


Working Out

  1. As the total number of employees = 100
    1. 20 + b + c + d = 100
    2. b + c + d = 80
       
  2. Maximum value of b + c
    1. Maximum value of b + c is possible, when d is minimum
    2. As the minimum value of d = 0, maximum value of b + c = 80…….(1)
       
  3. Minimum value of b + c
    1. b + c will be minimum , when d is maximum
    2. Since we know the number of employees who had only an Engineering degree = 20, d < 20
    3. So, maximum value of d = 19 (remember that out of 20, b, c and d, we are given that d holds the minimum value. So, d must be less than 20)
    4. Hence, minimum value of b + c = 80 – 19 = 61…….(2)
       
  4. Using (1) and (2), we can write 61 ≤ b + c ≤ 80
     
  5. The only value in the options that does not lie in the range is 55, option A

 

Answer: A

Test: Sets- 2 - Question 19

A survey was conducted to find out the number of languages spoken by the 210 employees of a company. It was found that 60 employees did not speak English, 150 employees did not speak German and 170 employees did not speak French. If there were 20 employees who did not speak German or English and 20 employees who did not speak French or English, what was the maximum number of employees who spoke only English? Assume that each of the employees spoke at least one language and no employee spoke any language other than English, French and German.

Detailed Solution for Test: Sets- 2 - Question 19

Given

  • Number of employees = 210
  • Number of employees who did not speak English = 60
    • So, number of employees who spoke English = 210 – 60 = 150
  • Number of employees who did not speak German = 150
    • So, number of employees who spoke German = 210 – 150 = 60
  • Number of employees who did not speak French = 170
    • So, number of employees who spoke French = 210 – 170 = 40
  • Number of employees who did not speak German or English = 20
    • So, these employees must speak only French (because it’s given that each employee speaks at least one language)
  • Number of employees who did not speak French and English = 20
    • So, these employees must speak only German (because it’s given that each employee speaks at least one language)

To Find: Maximum number of employees who spoke only English

Approach

 

  1. To find the maximum number of employees who spoke only English, we will first analyse the relation between total number of employees and number of employees who speak only English.
    1. Total number of employees = number of employees (who only speak English + who only speak French + who only speak German + who speak both German & French + who speak both German & English + who speak both French & English + who speak all the three languages + who speak none of the languages)
    2. Number of employees who only speak English = (Total number of employees) – (number of employees (who only speak French + who only speak German + who speak both German & French + who speak both German & English + who speak both French & English + who speak all the three languages + who speak none of the languages)
    3. As we need to maximize the number of employees who spoke only English, we need to minimize the number of employees who spoke English as well as some other language.
    4. So, of the employees who speak more one language we would maximize the number of employees who spoke only French and German
       
  2. Now, we know the number of employees who spoke French and German and also the number of employees who spoke only  French and German.
     
  3. We will use this information to maximize the number of employees who spoke only French and German.

 

Working Out

  1. As there are 40 employees who spoke French and 20 employees who only spoke French,              
    1. Number of employees who spoke French and some other language = 40 – 20 = 20
  2. Similarly, as there are 60 employees who spoke German and 20 employees who spoke only German,
    1. Number of employees who spoke German and some other language = 60 – 20 = 40
  3. Total number of employees who spoke more than 1 language
    1. Number of employees who spoke English + Number of employees who spoke French + Number of employees who spoke German – Total number of employees = 150 + 60 + 40 – 210 = 40
  4. As there are 40 employees who spoke more than 1 language, we will try to distribute maximum number of these employees into the number of employees who spoke French and German only.
    1. As there are 20 employees who spoke French and some other language and there are 40 employees who spoke German and some other language, the maximum number of employees who spoke French and German = 20
  5. So, that still leaves us with 40 – 20 = 20 employees who spoke German and some other language.
    1. The only possibility is for these employees to speak German and English
  6. So, there are 20 employees who spoke English and some other language
  7. As there are 150 employees who spoke English, there can be a maximum of 150 – 20 = 130 employees who spoke only English.

 

Answer: C

Test: Sets- 2 - Question 20

In a certain group of women, 70 percent of the women were employed and 25 percent did not have a graduate degree. Which of the following statements cannot be true?

  1. No woman in the group who had a graduate degree was unemployed
  2. Less than half of the women in the group were employed and had a graduate degree
  3. The number of unemployed women with a graduate degree was 50 percent greater than the number of employed women without a graduate degree
Detailed Solution for Test: Sets- 2 - Question 20

Given:

  • The Women in the group are divided into groups based on 2 parameters:
    • Employment (Employed, Unemployed)
    • Education (Have a Graduate Degree, Don’t have Graduate Degree)
  • So, the given information can be represented in a table as follows:

  • Now, % of women Unemployed = 100% - 70% = 30%
  • And, % of Women with Graduate Degree = 100% - 25% = 75%

To find: Which of the 3 statements cannot be true?

  • That is, which of the 3 statements are definitely false

Approach:

  1. We’ll evaluate the 3 statements one by one to evaluate which of them are definitely false.

Working Out:

  • Evaluating Statement I
  • No woman in the group who had a graduate degree was unemployed
    • Let the Women with Graduate Degree who are Unemployed be X, and let the Women with Graduate Degree who are Employed be Y.
    • As per this Statement, X = 0%

  • Now, the maximum possible value of Y is 70% (happens when ALL employed women have graduate degrees)
  • So, the minimum possible value of X is 5% (since the sum of Y and X is 75%)
  • Therefore, Statement I is definitely false.
  • Evaluating Statement II
  • Less than half of the women in the group were employed and had a graduate degree
    • Again, Let the women who were employed and had Graduate Degree be Y
    • As per this Statement, Y < 50%

Remember that the question is asking if a Statement is definitely false (that is, false for all possible values of Y)

  • In our analysis of Statement I above, we saw that the maximum possible value of Y is 70%
    • Therefore, for the maximum value of Y, Statement II is false
    • But we cannot say at this point if Statement II is definitely false (that is, false for all possible values of Y)
  • So, let’s now evaluate is also false for the minimum value of Y.
    • The maximum number of Employed Women without Graduate Degree is 25% (happens when ALL women without Graduate Degree are Employed)

  • So, the minimum value of Y is 70% - 25% = 45%
  • We observe that for the minimum value of Y, Statement II is true.
  • Thus, Statement II is true for some values of Y and false for other values of Y.
     
  • So, Statement II is not definitely false.

 

  • Evaluating Statement III
  • The number of unemployed women with a graduate degree was 50 percent greater than the number of employed women without a graduate degree
    • Let the Women with Graduate Degree who are Unemployed be X, and let the Women without Graduate Degree who are Employed be Z.
    • As per Statement III, X = 1.5Z

  • Note that this Statement will be definitely false if no valid pair of (X,Z) satisfies the equation X = 1.5Z
  • We will first find the values of X and Z that will satisfy this equation as well as the table above. Then, we will check if these values are acceptable.

  • So, the equation that we can write to relate X and Z is:
    • X = 75% - (70% - Z)
      • = 5% + Z
    • Let’s now substitute X = 1.5Z in this equation:
      • 1.5Z = 5% + Z
      • So, 0.5Z = 5%
      • Therefore, Z = 10%
      • Corresponding value of X = 1.5*10% = 15%
         
  • So, X = 1.5Z is satisfied by X = 15% and Z = 10%
  • We now need to check if these values of X and Z are acceptable
    • The minimum possible value of Z is 0%.
    • In this case, the table will look as under:

The maximum possible value of Z is 25%. In this case, the table will look as under:

  • So, the values of Z range from 0% to 25% and correspondingly, the values of X range from 5% to 30%
  • So, the values X = 15% and Z = 10% do indeed fall in the acceptable range of values of X and Z.
  • So, Statement III is true for one particular pair of X and Z
  • Therefore, Statement III is not definitely false

 

  • Getting to the answer
    • We’ve seen above that out of Statements I, II and III, only Statement I is definitely false.

Looking at the answer choices, we see that the correct answer is Option A

Test: Sets- 2 - Question 21

A group of friends went to an ice-cream parlour and ordered only two types of ice-cream - chocolate and strawberry. Of the people in the group, at least one person ate only one type ice-cream, some people ate both types of ice-cream and at least one person did not eat any type of ice-cream. Did more people eat chocolate ice-cream than strawberry ice-cream?

1) The ratio of the number of people who ate chocolate ice-cream and people who ate strawberry ice-cream was greater than the ratio of the total number of people who went to ice-cream parlour and the number of people who did not eat any type of ice-cream.

2) The number of people who ate only one type of ice-cream is greater than the number of people who ate strawberry ice-cream.

Detailed Solution for Test: Sets- 2 - Question 21

Steps 1 & 2: Understand Question and Draw Inferences

Given:

  • Let us assume:
    • a be the number of people who ate only chocolate ice-cream
    • c be the number of people who ate only strawberry ice-cream
    • b be the number of people who ate both chocolare & strawberry ice-cream
    • d be the number of people who did not eat any of ice-creams
       
  • This is represented in the venn diagram below:

  • Some people ate both types of ice-cream
    • b > 0 
  • Atleast one person did not eat any type of ice-creamd
    • d > 0
  • Atleat one person ate only one type of ice-cream
    • Either of a or c or both > 0


To Find: If a + b > c + b ?

  • If a > c ?

Step 3: Analyze Statement 1 independently

The ratio of the numbers of people who ate chocolate ice-cream and people who ate strawberry ice-cream was greater than the ratio of the total number of people who went to ice-cream parlour and the number of people who did not eat any type of ice-cream.

 

  • Minimum value of a+b+c+d = a+b+d
    • ​Either of a or c >0. So let us take a case, when a>0 and c=0
  • ​​​
  • Using (1), we can write:  
  • a + b > b + c
  • a > c
  • Sufficient to answer

     

    Step 4: Analyze Statement 2 independently

    The number of people who ate only one type of ice-cream is greater than the number of people who ate strawberry ice-cream.

     

  • a + c > b + c
  • a > b
  • Does not tell us if a > c .
  • Insufficient to answer

     

    Step 5: Analyze Both Statements Together (if needed)

    As we have a unqiue answer from step-3, this step is not required.

    Answer: A

Test: Sets- 2 - Question 22

In College X in the United States, there are 200 students in a class, of which  80 are female. If 10% of the class consists of female non-American students and 3/5th of the class are American students, how many of the students in the class are male American students?

Detailed Solution for Test: Sets- 2 - Question 22

Given

  • Number of students in the class = 200
  • Number of female students = 80
    • So, number of male students = 200 – 80 = 120
  • Number of female non-American students = 10% of 200 = 20
  • Number of American students = 3/5 * 200 = 120

 

To Find: Number of students who are male Americans?


Approach

1. To find the number of male americans, we will use the relation:

  1. Number of American students = Number of female American students + Number of male American students             
  2. We know the number of American students, but we do not know the number of female American students.
  3. So, we need to find the number of female American students

2. Now, Number of female students = Number of female American students + Number of female non-American students

  1. As we know the number of female students and the number of female non-American students, we can calculate the the number of female American students

 

Working Out

  1. Number of female students = 80
    1. Number of female non-American students = 20
    2. Number of female American students = 80 – 20 = 60

 

  1. Number of American students = 120
    1. Number of female American students = 60 (calculated above)
    2. So, Number of male American students = 120 – 60 = 60

So, there are 60 male American students.

Answer: C

Test: Sets- 2 - Question 23

In a locality, for every person, who owns only a car, there are 3 people who own only a bike. The number of people who own both a car and a bike is half the number of people who either own only a car or only a bike. If the number of people who neither own a car nor a bike is equal to the number of people who own a bike, which of the following can be the total number of people in the locality?

Detailed Solution for Test: Sets- 2 - Question 23

Given

  • Let the number of people who own only a car be x
    • So, number of people who own only a bike = 3x
    • Also, since x denotes the number of people, x must be a non-negative integer
  • Number of people who own a car and a bike = ½ *(Number of people who own either a bike or a only car) = ½ * (x+3x) = 2x
  • Number of people who neither own a car nor a bike = Number of people who own a bike

To Find: Among the options, which can be the total people in the locality?

Approach

  1. We will try to express the number of people in terms of x through the venn diagram and then evaluate the possible options.

Working Out

 

  1. Number of people who own a bike = 3x + 2 x = 5x
    1. So, the number of people who own neither a bike nor a car = 5x
  2. Hence, total people in the locality = 3x + 2x + x + 5x = 11x
    1. So, the total people in the locality should be a multiple of 11
    2. The only multiple of 11 in the options is 66, option C

 

Answer: C

Test: Sets- 2 - Question 24

In a certain class of 66 students, 30 students study Science, 40 students study Humanities and 40 students study Dance. If all the students studied at least one of the subjects, then what is the maximum possible number of students who study all the three subjects?

Detailed Solution for Test: Sets- 2 - Question 24

Given:

  • Total number of students = 66
  • Students who take:
    • Science = 30
    • Humanities = 40
    • Dance = 40
    • None of these 3 subjects = 0
  • Maximum possible number of students who study all 3 subjects = x

 

To find: X – N = ?

Approach:

  1. To answer the question, we need to find the value of x
  2. We’ll visually represent the given information to analyze the maximum possible overlap for 3 subjects.

Working Out:

  • Analyzing the Maximum possible overlap for the 3 subject
  • To maximize x, i.e. to minimize the other overlap regions, among the sets, let’s consider the number of students who study only two subjects to be 0
    • So, number of students who study only Science = 30 – x
    • Number of students who study only Humanities = 40 -x
    • Number of students who study only Dance = 40 – x
  • So, number of students who study only 1 subject + Number of students who study all the 3 subjects = 66
  • (30-x) + (40-x) + (40- x) + x = 66
  • 2x = 44
  • x = 22
  • Getting to the answer
    • So, the maximum number of students who can study all the 3 subjects is 22

Looking at the answer choices, we see that the correct answer is Option D

Test: Sets- 2 - Question 25

For any set A, (A')' is equal to:

Detailed Solution for Test: Sets- 2 - Question 25

Calculation:

Complement of A = (A') = All elements not in A

Complement of A'

((A')') = All elements not in A', which are the elements in A.

Therefore, (A')' = A.

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