Chapter-1 (Sets) Important Questions

Sets: Important Questions

Mathematics • Grade 10 • Cardinality of Two and Three Sets — with answers

How to solve a Venn-diagram question

• Two sets: fill “both” first, then “only A” = n(A) − both, “only B” = n(B) − both, then “neither”.
• Three sets: fill “all three” → “exactly two” (pair − all three) → “only one” → “none”.
• If a value is unknown, call it x, add every region and set the total equal to n(U).
• Percentages: convert to numbers using the total (or call the total x).

A. Very Short Questions

1 mark1What is the cardinality of a set? Write the formula for n(A∪B) when A and B are overlapping sets.

Answer

Cardinality n(A) is the number of elements in A.

n(A∪B) = n(A) + n(B) − n(A∩B)

1 mark2If A ⊂ B, write the values of n(A∪B) and n(A∩B).

Answer

n(A∪B) = n(B), n(A∩B) = n(A)

1 mark3If A and B have 12 and 8 elements, find the minimum possible value of n(A∪B).

Answer

The union is smallest when B ⊂ A. Minimum = 12

1 mark4If n(A) = 35 and n(A′) = 25, find n(U).

Answer

n(U) = 35 + 25 = 60

2 marks5For sets P and Q, n(P) = 40, n(P∪Q) = 60 and n(P∩Q) = 10. Find n(Q).

Answer

60 = 40 + n(Q) − 10

n(Q) = 30

2 marks6If n(U) = 350, n(A) = 200, n(B) = 220 and n(A∩B) = 120, find n(A∪B) and n(A∪B)′.

Answer

n(A∪B) = 200 + 220 − 120 = 300n(A∪B)′ = 350 − 300 = 50

300 and 50

2 marks7If n(U) = 200, n_o(M) = 2x, n_o(E) = 3x, n(M∩E) = 60 and n(M∪E)′ = 40, find x.

Answer

2x + 60 + 3x + 40 = 200 ⇒ 5x = 100

x = 20

B. Short Questions — Two Sets

4 marks8In a survey of 180 students, 45 like Nepali only, 60 like English only and 15 like neither. (i) Show the information in a Venn diagram. (ii) How many like both? (iii) How many like at least one subject?

Answer

Students liking at least one subject = 180 − 15 = 165.

45 + x + 60 = 165 ⇒ x = 60

Both = 60 At least one = 165

4 marks9Among 900 people, 525 read Madhupark, 450 read Yubamanch and 75 read neither. (i) Draw the Venn diagram. (ii) How many read both papers? (iii) How many read only one paper?

Answer

At least one = 900 − 75 = 825.

525 + 450 − n(M∩Y) = 825 ⇒ n(M∩Y) = 150

Only one = 825 − 150 = 675.

Both = 150 Only one = 675

4 marks10In a survey of 150 people, 90 like modern songs, 70 like folk songs and 30 like neither. Find the number who like both and the number who like only modern songs.

Answer

Union = 150 − 30 = 120.

90 + 70 − x = 120 ⇒ x = 40

Only modern = 90 − 40 = 50.

Both = 40 Only modern = 50

4 marks11Among 360 players, 210 like volleyball, 180 like football and 30 like neither. Find how many like both games and how many like only one game.

Answer

Union = 360 − 30 = 330.

210 + 180 − x = 330 ⇒ x = 60

Only one = 330 − 60 = 270.

Both = 60 Only one = 270

5 marks12In an examination, 70% passed English, 60% passed Mathematics, 20% failed both subjects, and 550 students passed both. (i) Draw the Venn diagram. (ii) Find the total number of students. (iii) How many passed English only?

Answer

Passed at least one = 80%.

70% + 60% − both = 80% ⇒ both = 50%50% of total = 550 ⇒ total = 1100

English only = (70 − 50)% of 1100 = 220.

Total = 1100 English only = 220

5 marks13In a survey of 200 people, the ratio of those who use a laptop only to those who use a mobile only is 2 : 3. 30% use both and 15% use neither. (i) Draw the Venn diagram. (ii) How many use a laptop? (iii) How many use at most one gadget?

Answer

Both = 60, neither = 30. Let laptop only = 2x, mobile only = 3x.

2x + 60 + 3x + 30 = 200 ⇒ x = 22

Laptop users = 2(22) + 60 = 104. At most one = 200 − 60 = 140.

Laptop = 104 At most one = 140

5 marks14Among 300 foreign tourists, the ratio of those who visited Pokhara to those who visited Lumbini is 2 : 3. 90 visited both and 60 visited neither. Find the number who visited only one place and the number who visited at least one.

Answer

Let n(P) = 2x, n(L) = 3x.

(2x − 90) + 90 + (3x − 90) + 60 = 300 ⇒ 5x = 330 ⇒ x = 66

Only P = 42, only L = 108.

Only one = 150 At least one = 240

5 marks15In a survey of 75% students interested in science, 55% interested in staff nurse and 5% giving no answer, 21 students are interested in both. Find the total students surveyed and the number interested in staff nurse only.

Answer

Let total = x.

(0.75x − 21) + 21 + (0.55x − 21) + 0.05x = x ⇒ 0.35x = 21 ⇒ x = 60

Only staff nurse = 0.55(60) − 21 = 12.

Total = 60 Only staff nurse = 12

5 marks16Of 300 players, one-third play volleyball only. Of the remaining players, 60% play football only. 60 players play neither game. Find the ratio of the number of volleyball players to football players.

Answer

Volleyball only = 100. Remaining = 200, football only = 60% of 200 = 120.

Both = 300 − 100 − 120 − 60 = 20.

Volleyball = 120, football = 140.

Ratio = 6 : 7

5 marks17Among 65 players, 11 play volleyball only and 33 play cricket only. The number of cricket players is double the number of volleyball players. Find how many play both and how many play neither.

Answer

Let both = x.

33 + x = 2(11 + x) ⇒ x = 11

Neither = 65 − (11 + 11 + 33) = 10.

Both = 11 Neither = 10

5 marks18In a survey of 93 women, 80 work in agriculture, 71 do sewing and 10 do other jobs. (i) Draw the Venn diagram. (ii) How many do both agriculture and sewing? (iii) How many times is “agriculture only” more than “sewing only”?

Answer

At least one = 93 − 10 = 83.

80 + 71 − x = 83 ⇒ x = 68

Agriculture only = 12, sewing only = 3.

Both = 68 12 ÷ 3 = 4 times

5 marks19Among 1000 farmers, 800 grow potatoes, 500 grow tomatoes and 50 grow other crops. Find the number who grow both, and the ratio of potato-only to tomato-only farmers.

Answer

At least one = 950.

800 + 500 − x = 950 ⇒ x = 350

Potato only = 450, tomato only = 150.

Both = 350 Ratio = 3 : 1

C. Short Questions — Three Sets

3 marks20If n(U) = 120, n(A) = 48, n(B) = 51, n(C) = 40, n(A∩B) = 11, n(B∩C) = 10, n(A∩C) = 9 and n(A∩B∩C) = 4, find n(A∪B∪C) and n(A∪B∪C)′.

Answer

n(A∪B∪C) = 48 + 51 + 40 − 11 − 10 − 9 + 4 = 113n(A∪B∪C)′ = 120 − 113 = 7

113 and 7

3 marks21If n(U) = 100, n(M) = 45, n(E) = 50, n(S) = 35, n(M∩E) = 20, n(E∩S) = 20, n(S∩M) = 15 and n(M∩E∩S) = 5, find the number of elements outside all three sets.

Answer

n(M∪E∪S) = 45 + 50 + 35 − 20 − 20 − 15 + 5 = 80

Outside = 100 − 80 = 20

3 marks22If n(U) = 120, n(M) = 50, n(E) = 40, n(S) = 45, n(M∩E) = n(E∩S) = n(S∩M) = 15 and n(M∪E∪S)′ = 15, find n(M∩E∩S).

Answer

Union = 120 − 15 = 105.

105 = 50 + 40 + 45 − 45 + x ⇒ x = 15

n(M∩E∩S) = 15

3 marks23If n(A∪B∪C) = 105, n_o(A) = 25, n_o(B) = 25, n_o(C) = 15, n_o(A∩B) = 15, n_o(A∩C) = 10 and n(A∩B∩C) = 10, find n_o(B∩C).

Answer

25 + 25 + 15 + 15 + 10 + 10 + n_o(B∩C) = 105

n_o(B∩C) = 5

5 marks24In a class, 40 students like orange, 35 like mango and 50 like banana; 15 like orange and mango, 20 like mango and banana, 25 like orange and banana; 5 like all three and 30 like none. Draw the Venn diagram and find the total number of students.

Answer

O&M only = 10, M&B only = 15, O&B only = 20. O only = 5, M only = 5, B only = 10.

5 + 10 + 15 + 20 + 5 + 5 + 10 + 30 = 100

Total = 100

5 marks25Of 180 students, 86 passed Science, 80 passed Maths and 76 passed Nepali; 26 passed Science & Maths, 36 passed Maths & Nepali, 32 passed Science & Nepali; 20 passed none. Draw the Venn diagram and find how many passed all three.

Answer

180 = 86 + 80 + 76 − 26 − 36 − 32 + x + 20180 = 168 + x

x = 12

5 marks2636 students got medals in dance, 12 in drama and 18 in music. 45 students got at least one medal and 4 got medals in all three. Find the number who got medals in exactly two events.

Answer

45 = 36 + 12 + 18 − (sum of pair overlaps) + 4, so the sum of pair overlaps = 25.

Exactly two = 25 − 3 × 4 = 13

13 students

5 marks27Among 60 students, 23 play volleyball, 15 play basketball, 20 play cricket; 7 play volleyball & basketball, 5 play basketball & cricket, 4 play volleyball & cricket; 15 play none. (i) How many play all three? (ii) How many play volleyball and cricket only?

Answer

Union = 60 − 15 = 45.

45 = 23 + 15 + 20 − 7 − 5 − 4 + x = 42 + x ⇒ x = 3

Volleyball & cricket only = 4 − 3 = 1.

All three = 3 V & C only = 1

5 marks28In a survey, 60 like tea, 45 like coffee, 30 like milk; 25 like coffee & tea, 20 like milk & tea, 15 like coffee & milk; 10 like all three. Draw the Venn diagram and find how many people were surveyed.

Answer

60 + 45 + 30 − 25 − 20 − 15 + 10 = 85

Surveyed = 85 (everyone likes at least one drink)

5 marks29In an examination, 40% passed Science, 45% Maths and 50% Nepali; 10% passed Science & Maths, 20% Maths & Nepali, 15% Science & Nepali; 5% failed all three. Find the percentage who passed (i) all three, (ii) only two subjects, (iii) only one subject, (iv) at least one.

Answer

(iv) At least one = 100 − 5 = 95%.

95 = 40 + 45 + 50 − 10 − 20 − 15 + x ⇒ x = 5

(ii) Only two = (10 + 20 + 15) − 3(5) = 30%.

(iii) Only one = 95 − 30 − 5 = 60%.

All three 5% Two 30% One 60% At least one 95%

6 marks30In a survey of newspapers: 30 read Yubamanch, 25 read Madhupark, 15 read Yubamanch & Muna, 12 read Yubamanch & Madhupark; 9 read Madhupark only, 11 read Muna only, 5 read Yubamanch & Madhupark only, and 10 read none. Find (a) the total surveyed, (b) those who read exactly two newspapers, (c) those who read Muna.

Answer

All three = 12 − 5 = 7. Y&Mu only = 15 − 7 = 8. Madhupark: 25 = 9 + 5 + 7 + (M&Mu only) ⇒ M&Mu only = 4. Yubamanch only = 30 − 5 − 8 − 7 = 10.

Total = 10 + 5 + 8 + 7 + 9 + 4 + 11 + 10 = 64.

(a) 64 (b) 5 + 8 + 4 = 17 (c) 8 + 7 + 4 + 11 = 30

6 marks31Among 90 people, 48 like Nepali films, 40 like English, 31 like Hindi; 24 like Nepali & English, 19 like Hindi & English, 6 like all three and 21 like none. (a) How many like both Nepali and Hindi films? (b) How many like neither Nepali nor Hindi films?

Answer

At least one = 90 − 21 = 69.

69 = 48 + 40 + 31 − 24 − 19 − n(N∩H) + 6 ⇒ n(N∩H) = 13

n(N∪H) = 48 + 31 − 13 = 66, so neither = 90 − 66 = 24.

(a) 13 (b) 24

6 marks32In a survey of 45 people: 25 speak Nepal Bhasa, 23 speak Tamang and 15 speak Maithili; 12 speak Nepal Bhasa & Tamang, 5 speak Nepal Bhasa & Maithili, 10 speak Tamang & Maithili; 4 speak all three. Find (a) how many speak none of these, (b) how many speak only one language, (c) how many speak Nepal Bhasa and Tamang but not Maithili.

Answer

n(at least one) = 25 + 23 + 15 − 12 − 5 − 10 + 4 = 40

(a) None = 45 − 40 = 5.

Pairs only: NB&T = 8, NB&M = 1, T&M = 6. NB only = 12, T only = 5, M only = 4.

(a) 5 (b) 21 (c) 8

D. Algebra-Based Questions

5 marks33For overlapping sets A and B, n_o(A) = 16 + x, n_o(B) = 5x, n(A∩B) = y and n(A∪B)′ = x. (a) Show this in a Venn diagram. (b) If n(A) = n(B), find n(A∪B)′. (c) If n(U) = 50, find the ratio n(A∩B) : n(A∪B)′.

Answer

n(A) = 16 + x + y, n(B) = 5x + y.

16 + x + y = 5x + y ⇒ 4x = 16 ⇒ x = 4

(c) (16 + 4) + 20 + y + 4 = 50 ⇒ y = 6.

(b) 4 (c) 6 : 4 = 3 : 2

5 marks34n(U) = 100, n(A − B) = 32 + x, n(B − A) = 5x, n(A∩B) = x and n(A∪B)′ = y. (a) Draw the Venn diagram. (b) If n(A) = n(B), find n(A∩B). (c) Find n(A∪B)′.

Answer

n(A) = 32 + 2x and n(B) = 6x.

32 + 2x = 6x ⇒ x = 8

n(A∪B) = 40 + 40 + 8 = 88, so y = 100 − 88 = 12.

(b) 8 (c) 12

Last-minute check: after solving, add all regions of your Venn diagram — the total must equal n(U). If it doesn’t, one region is wrong.

Sets • Grade 10 Mathematics • Important Questions