Sets Solutions

Here are the detailed explanations of the Sets Questions. The solutions are easily understandable so that the candidates can easily learn the topic and ace their CAT exam.

Test Your Sets & Venn Diagrams Basics Skills-Solutions

1. Option (d)

S = 110+80+120+70 = 380; X needs to be maximized.

I = 80.

S = I + 2II + 3III + 4 IV= 380.

To maximize X one needs to minimize the overlapped regions.

Assume III= 0, IV = 0.

So that 2II = 380-80 = 300. II = 150. X max = 80 + 150 = 230.

 

2. Option (c)

Given that S= 310 (i.e.100+ 70+ 140).

II +3III= 40+30+60 = 130

III=10 =>II = 100

S-X=II+2III

X=310-120= 190 => No. of candidates having none of the three = 200-190=10.

 

3. Option (c)

S = 25+60+50=135

X=100 (given)

S-X= II+2III

35= II+2III

The least value II can take is 1, as the number of people has to be a whole number

2III= 34 =>III(max) = 17.

 

4. Option (d)

In this case, the minima of all is that part of the venn diagram which has 5 overlapping regions. This can be found using a shortcut technique.

To find the minimum number of spectators who are fans of all 5 teams, take the difference of each from X (i.e.the value of atleast 1), add it up and call the sum Y; then take the difference of Y from 100 again to get the value Here X=100,

Step 1- Taking the difference of each from X

100-90=10

100-85=15

100-70=30

100-75=25

100-80=20

Total(Y) ∑=100

Step 2- Take the difference of Y from 100, 100-100=0 Minimum number of spectators who are fans of all 5 teams = 0.

 

5. Option (b)

S= 90+80+80=250

X= 100

Using the Maxima of all shortcut

(S-X)/(N-1) = 150/2= 75.

 

6. Option (b)

Using the venn diagram approach

Sets Solutions

II +III= 35,

a+b=20

Hence percentage playing only Football = 45-10-20 =15%.

 

Questions 7 & 8:

Soln:

Sets Solutions

n (E + M) = n(E) + n(M) – n(E ∩ M)

500 = 350 + 250 – n (E ∩ M)

n (E ∩ M) = 100

 

7. Option (a)

Since100 are both Engineers and MBAs.

 

8. Option (d)

People who are only Engineer = 250

People who are only MBA = 150

No. of people who are either only MBA or only Engineer = 250 + 150 = 400

 

9. Option (a)

X = I + II + III = 200

S = I + 2II + 3III = 140 + 150 + 160 = 450

S – X = II + 2III = 450 – 200 = 250

For III to be the minimum, II has to be the maximum. Now, II can take the maximum value of 200.

So, minimum value of III = 250 – 200= 50.  

 

Questions 10 & 11:

Sets Solutions

Consider Maths:

It is given that 20 students failed only in Maths.

So, x + z + y + 20 = 30 ➜x + y + z = 10

Also x + z = 8 y + z = 6 x + y + 2 z = 14 ➜ z = 14 – 10 = 4, then x = 4 and y = 2

So, redrawing the Venn diagram

Sets Solutions

 

10. Option (d)

Number of students who appeared in the examination = 18 + 20 + 16 + 6 + 4 + 2 + 4 = 70.

 

11. Option (d)

The number of students who failed in English and Science but not Mathematics = 6. 

 

Questions 12-14:

Sets Solutions

Number of students = 278 = x + y + z + 11 + 12 + 14 + 9

Let us consider the total number of seats: x + y + z + 2(11 + 12 + 14) +

(3×9) = [x + y + z + 11 + 12 + 14 + 9] + [11 + 12 + 14 + 9 + 9] = 278 + 55 = 333.

So, number of seats in each class = 333/3 = 111

x = 111 – (11 + 9 + 12) = 79

z = 111 – (12 + 9 + 14) = 76

y = 111 – (11 + 9 + 14) = 77

 

12. Option (a)

No. of students who have occupied only one seat = x + y + z = 79 + 76 + 77 = 232. 

 

13. Option (d)

No. of students who have occupied seat in Violin Class or Guitar Class, but not in Tabla Class = 77 + 14 + 76 = 167. 

 

14. Option (b)

No. of students who have occupied a seat in Tabla and Violin Class, but not in Guitar Class = 20 – 9 = 11. 

 

Questions 15-17:

18 married workers are engineers out of which

9 are men ➜ s + q = 18,

s = 9 ➜ q = 9

12 men are engineers ➜ r = 12.

12 men are married ➜ p = 12.

Sets Solutions

 

15. Option (d)

No. of unmarried engineer in the company = 5 + 12 = 17. 

 

16. Option (b)

No. of unmarried women = 5 {unmarried women: not men + not married i.e. z). 

 

17. Option (a)

No. of engineer men not married = r = 12. 

 

Questions 18-20:

From the data given in the question, one can fill up the Venn diagram as follows

Sets Solutions

From the information given, a+b+c+d=60, b =0.

 

18. Option (c)

The number of people who opted for only football= 150-60=90. 

 

19. Option (d)

The number of people who opted only for hockey cannot be determined (d).

 

20. Option (c)

The number of people who opted for only football and only cricket = 0+90=90.

 

21. Option (c)

x- number who play all 3 games

80= 125 –(14+12+20) +x

x= 1, S=125, X=80

S-X=45=II+2III II+III=46.

So III=1 The Venn diagram can be represented as

Sets Solutions

Answer = 15:4

 

22. Option (c)

Sets Solutions

So, the % of students failed in at least one subject = 19 + 19 + 14 = 52 %

So, the % of students who didn’t fail even in one subject = 100 – 52 = 48% % of students failed in Science = 38%  % of students who passed in Science = 100 – 38 = 62% % of students failed in Maths = 33%  % of students who passed in Mathematics = 100 – 33 = 67% Now, consider the % of students who passed:

Sets Solutions

So, as we see from the Venn diagram, % of students passed in only Science = 14%. If the total no. of students who appeared = x, then .14x = 700  x = 5000. 

 

23. Option (a)

Sets Solutions

be odd always .option c and d is out Hence x= 2. option a = 65 is possible.

 

24. Option (b)

Sets Solutions

From the Venn diagram given, it’s clear that no. of people who like C and not B are 30 – 25 = 5. People who like D are 100 – 50 = 50. Therefore, 5/50× 100 = 10%.

 

25. Option (d)

Since the question is about percentages, assume a base value 100 (total number of volunteers)

No. of volunteers receiving < 6 hours of sleep = 80

No. of volunteers who receive > 6 hours of sleep = 20 70 % of the above report no feeling of tiredness =14

Number of volunteers with > 6 hours of sleep and report tired = 6

Volunteers who receive < 6 hours of sleep and report feeling tired = 75                      

Number of volunteers who receive < 6 hours of sleep and are not tired = 80-75= 5

Total percentage of volunteers who report no feeling of tiredness during shifts = 14+5 = 19%.

So, the total percentage of volunteers who report no feeling of tiredness during shifts = 14+5 = 19%.

 

Stay tuned with Byju’s to get more important CAT Practice Questions with detailed solutions. Also, get updated with the latest CAT syllabus and pattern to prepare properly for the exam.