Skip to content
Science Playground
Book a Visit

← Grade 9 Math

Exam practice

Every question from the end of every unit, with hints and answers to open. After you check an answer, mark how you did, and come back to the ones you didn’t get.

27 questions

Unit 1 · Sets

Exercise 1.1

Question 1

1.

In Figure 1.29, F is the set of students who like football and V the set who like volleyball. Rambahadur and Harkabahadur like neither game. Write these sets by the listing method:

U (11 students)FVChhiringDorjeHariMayaAashaGaneshBatuliDolmaSuntali
Figure 1.29 Exercise 1.1, question 1. Rambahadur and Harkabahadur are outside both circles.
  • a.F
  • b.V
  • c.F ∪ V
  • d.F ∩ V
  • e.U
Answer
  • a.F = {Chhiring, Dorje, Hari, Maya, Aasha, Ganesh}
  • b.V = {Batuli, Dolma, Suntali, Maya, Aasha, Ganesh}
  • c.F ∪ V = {Chhiring, Dorje, Hari, Batuli, Dolma, Suntali, Maya, Aasha, Ganesh}
  • d.F ∩ V = {Maya, Aasha, Ganesh}
  • e.U = {Chhiring, Dorje, Hari, Batuli, Dolma, Suntali, Maya, Aasha, Ganesh, Rambahadur, Harkabahadur}

Question 2

2.

If U = {a, b, c, d, e, f, g, h, i, j, k}, A = {a, c, e, f, g, i, k} and B = {b, d, i, j, k, h}, find these sets and show each in a Venn diagram:

  • a.A ∩ B
  • b.B ∪ A
  • c.A − B
  • d.B − A
Answer
UABa, c, ef, gi, kb, dh, j
Figure 1.30 Exercise 1.1, question 2. Every letter of U is in A or B.
  • a.A ∩ B = {i, k} (shade the overlap).
  • b.B ∪ A = {a, b, c, d, e, f, g, h, i, j, k} = U (shade both circles).
  • c.A − B = {a, c, e, f, g} (shade A outside B).
  • d.B − A = {b, d, h, j} (shade B outside A).

Question 3

3.

If U = {x:x is a whole number from 1 to 30}, A = {x:x is a multiple of 3 from 1 to 30}, B = {x:x is a multiple of 4 from 1 to 30} and C = {x:x is a multiple of 5 from 1 to 30}, write these sets by the listing method and show them in a Venn diagram:

  • a.A − B
  • b.B − A
  • c.A − C
  • d.B − C
  • e.A ∪ B
  • f.A ∪ B ∪ C
  • g.A ∩ B ∩ C
  • h.B ∪ C
Answer

A = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30}, B = {4, 8, 12, 16, 20, 24, 28}, C = {5, 10, 15, 20, 25, 30}.

U (1 to 30)ABC3, 6, 918, 21, 274, 816, 285, 102512, 24201530
Figure 1.31 Exercise 1.1, question 3. Outside the three circles are 1, 2, 7, 11, 13, 14, 17, 19, 22, 23, 26, 29. Each part is a choice of these regions.
  • a.A − B = {3, 6, 9, 15, 18, 21, 27, 30}
  • b.B − A = {4, 8, 16, 20, 28}
  • c.A − C = {3, 6, 9, 12, 18, 21, 24, 27}
  • d.B − C = {4, 8, 12, 16, 24, 28}
  • e.A ∪ B = {3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30}
  • f.A ∪ B ∪ C = {3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30}
  • g.A ∩ B ∩ C = {}: a multiple of 3, 4 and 5 is a multiple of 60.
  • h.B ∪ C = {4, 5, 8, 10, 12, 15, 16, 20, 24, 25, 28, 30}, so B ∪ C = {1, 2, 3, 6, 7, 9, 11, 13, 14, 17, 18, 19, 21, 22, 23, 26, 27, 29}.

Question 4

4.

If U = {a, b, c, d, e, f, g, h, i, j, k}, A = {a, c, d, f} and B = {g, h, i}:

  • a.Find these sets:
  • i.A
  • ii.B
  • iii.A ∪ B
  • iv.A ∩ B
  • v.A ∪ B
  • vi.A ∩ B
  • b.Which of the sets in part a are equal?
Answer
UABa, cd, fg, h, ib, e, j, k
Figure 1.32 Exercise 1.1, question 4: A and B are disjoint.
  • a.i. A = {b, e, g, h, i, j, k}
  • ii.B = {a, b, c, d, e, f, j, k}
  • iii.A ∪ B = {a, b, c, d, e, f, g, h, i, j, k} = U
  • iv.A ∩ B = {b, e, j, k}
  • v.A ∪ B = {a, c, d, f, g, h, i}, so A ∪ B = {b, e, j, k}
  • vi.A ∩ B = ∅, so A ∩ B = U
  • b.iv and v are equal, A ∩ B = A ∪ B, and iii and vi are equal, A ∪ B = A ∩ B = U. These are De Morgan’s laws, true for any two sets.

Question 5

5.

If U = {natural numbers from 1 to 12}, E = {even numbers from 1 to 12}, O = {odd numbers from 1 to 12} and P = {prime numbers from 1 to 12}, find these sets and show them in a Venn diagram:

  • a.E
  • b.O
  • c.P
  • d.E ∪ P
  • e.P ∩ O
  • f.P − O
  • g.P ∩ E
  • h.E ∪ O ∪ P
  • i.E ∩ O ∩ P
  • j.E ∪ P − (P ∩ O)
  • k.P ∪ (E ∩ O)
Answer

E = {2, 4, 6, 8, 10, 12}, O = {1, 3, 5, 7, 9, 11}, P = {2, 3, 5, 7, 11}.

U (1 to 12)EOP4, 6, 810, 121, 935, 7112
Figure 1.33 Exercise 1.1, question 5. No number is both even and odd, so every region inside E and O together is empty.
  • a.E = {1, 3, 5, 7, 9, 11}
  • b.O = {2, 4, 6, 8, 10, 12}
  • c.P = {1, 4, 6, 8, 9, 10, 12}
  • d.E ∪ P = {2, 3, 4, 5, 6, 7, 8, 10, 11, 12}, so E ∪ P = {1, 9}
  • e.P ∩ O = {3, 5, 7, 11}, so P ∩ O = {1, 2, 4, 6, 8, 9, 10, 12}
  • f.P − O = {2}, so P − O = {1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
  • g.P ∩ E = {2}: 2 is the only even prime.
  • h.E ∪ O ∪ P = U, so E ∪ O ∪ P = {}
  • i.E ∩ O ∩ P = {}, so E ∩ O ∩ P = U = {1, 2, 3, …, 12}
  • j.{1, 9} − {3, 5, 7, 11} = {1, 9}
  • k.E ∩ O = {}, so P ∪ (E ∩ O) = P = {1, 4, 6, 8, 9, 10, 12}

Question 6

6.

If U = {m, n, o, p, q, r, s, t, u, v}, A = {q, r, s, t, u, v}, B = {n, o, p, q, r} and C = {m, u, s, t, q, r}, find these sets and show them in Venn diagrams:

  • a.A ∩ B
  • b.(A ∪ B) ∩ C
  • c.A ∪ B ∪ C
  • d.A ∩ B ∩ C
  • e.A − B
  • f.A ∪ B ∪ C
  • g.A ∩ B
  • h.A
  • i.(A ∩ C) ∪ B
Answer
UABCvn, opmst, uq, r
Figure 1.34 Exercise 1.1, question 6. Every letter of U is in at least one set.
  • a.A ∩ B = {q, r}
  • b.A ∪ B = {n, o, p, q, r, s, t, u, v}, so (A ∪ B) ∩ C = {q, r, s, t, u}
  • c.A ∪ B ∪ C = {m, n, o, p, q, r, s, t, u, v} = U
  • d.A ∩ B ∩ C = {q, r}
  • e.A − B = {s, t, u, v}
  • f.A ∪ B ∪ C = {}
  • g.A ∩ B = {m, n, o, p, s, t, u, v}
  • h.A = {m, n, o, p}
  • i.A ∩ C = {q, r, s, t, u}, so (A ∩ C) ∪ B = {n, o, p, q, r, s, t, u}

For each Venn diagram, shade the regions of that part’s letters.

Question 7

a.

Write the set shaded in the Venn diagram in set notation.

UAB
Figure 1.35 Exercise 1.1, question 7a.
Answer

A − B (the part of A outside B).

b.

Write the set shaded in the Venn diagram in set notation.

UAB
Figure 1.36 Exercise 1.1, question 7b.
Answer

B − A.

c.

Write the set shaded in the Venn diagram in set notation.

UAB
Figure 1.37 Exercise 1.1, question 7c.
Answer

B, which here is also B − A, since the sets are disjoint.

d.

Write the set shaded in the Venn diagram in set notation.

UA
Figure 1.38 Exercise 1.1, question 7d.
Answer

A, that is U − A.

e.

Write the set shaded in the Venn diagram in set notation.

UABC
Figure 1.39 Exercise 1.1, question 7e.
Answer

A ∪ B ∪ C.

f.

Write the set shaded in the Venn diagram in set notation.

UABC
Figure 1.40 Exercise 1.1, question 7f.
Answer

A ∩ B ∩ C.

Question 8

8.

If P and Q are overlapping subsets of a universal set U, show these sets in Venn diagrams:

  • a.P − Q
  • b.Q − P
  • c.(P − Q) ∪ P
  • d.P ∩ (Q − P)
Answer

a. Shade the part of P outside Q:

UPQ
Figure 1.41 Question 8a: P − Q.

b. Shade the part of Q outside P:

UPQ
Figure 1.42 Question 8b: Q − P.

c. P − Q is already part of P, so (P − Q) ∪ P = P: shade the whole of P.

UPQ
Figure 1.43 Question 8c: (P − Q) ∪ P = P.

d. Q − P has nothing in common with P, so P ∩ (Q − P) = ∅: nothing is shaded.

UPQ
Figure 1.44 Question 8d: P ∩ (Q − P) = ∅, so no region is shaded.

Question 9

9.

Write a universal set U and two of its subsets X and Y. Then write these sets by the listing method:

  • a.X ∪ Y
  • b.X ∩ Y
  • c.X
  • d.X ∩ Y
Answer

Any correct example. For instance, U = {1, 2, 3, 4, 5, 6, 7, 8}, X = {1, 2, 3, 4}, Y = {3, 4, 5, 6}. Then X = {5, 6, 7, 8} and Y = {1, 2, 7, 8}.

  • a.X ∪ Y = {1, 2, 5, 6, 7, 8}
  • b.X ∩ Y = {1, 2} (the same as X − Y)
  • c.X = X = {1, 2, 3, 4}
  • d.X ∩ Y = {7, 8}

Check with De Morgan’s laws: X ∩ Y = {3, 4}, and its complement is a’s answer; X ∪ Y = {1, 2, 3, 4, 5, 6}, and its complement is d’s.

Question 10

10.

Make these sets from the students of your class: a. the set of all the students, b. the set of girls, c. the set of boys. Which is the universal set, and which are its subsets? Write them with the right symbols. Then find the complement of each set.

Answer

Your class’s names. For example, in a class of six: U = {Sneha, Tashi, Bibek, Rizwan, Junu, Nabin}, G = {Sneha, Tashi, Junu}, B = {Bibek, Rizwan, Nabin}.
U (all the students) is the universal set; G ⊂ U and B ⊂ U.
G = U − G = B, B = U − B = G, and U = ∅. In any class the girls and the boys are complements of each other, as long as everyone is one or the other.

Exercise 1.2

Question 1

a.

From Figure 1.53, find

  • i.n(S)
  • ii.n(T)
  • iii.n(U)
  • iv.n(S ∩ T)
  • v.n(R ∪ T)
  • vi.n(S ∩ R ∩ T)
  • vii.n(S ∪ R ∪ T)
  • viii.n(S ∪ R ∪ T)
  • ix.n₀(S)
  • x.n₀(S ∩ R)
  • xi.n(R)
USTR5, 67, 8910243111, 12
Figure 1.53 Exercise 1.2, question 1a.
Answer

S = {1, 2, 3, 5, 6}, T = {1, 2, 4, 7, 8, 9}, R = {1, 3, 4, 10}, U = {1, 2, …, 12}.

  • i.5
  • ii.6
  • iii.12
  • iv.2 (1, 2)
  • v.8 (1, 2, 3, 4, 7, 8, 9, 10)
  • vi.1 (1)
  • vii.10 (1 to 10)
  • viii.2 (11, 12)
  • ix.2 (5, 6)
  • x.1 (3)
  • xi.8 (all but 1, 3, 4, 10)
b.

If A and B are disjoint sets with n(A) = 30 and n(B) = 35, find n(A ∪ B).

Answer

Disjoint, so n(A ∪ B) = n(A) + n(B) = 30 + 35 = 65.

Question 2

a.

From Figure 1.54, find

  • i.n(A)
  • ii.n(B)
  • iii.n(A ∪ B)
  • iv.n(A ∩ B)
  • v.n₀(A)
  • vi.n₀(B)
UABd, ef, ga, bch, ijk, l, m
Figure 1.54 Exercise 1.2, question 2a.
Answer

A = {a, b, c, d, e, f, g} and B = {a, b, c, h, i, j}.

  • i.7
  • ii.6
  • iii.10
  • iv.3
  • v.4
  • vi.3

Check: 7 + 6 − 3 = 10. ✓

b.

If U = {a, b, c, d, e, f, g}, A = {c, d, e, f}, B = {a, b, e, f} and C = {d, e, f, g}, find

  • i.n(A − B)
  • ii.n(B − C)
  • iii.n(A − C)
  • iv.n(A)
  • v.n(A ∪ B)
  • vi.n{(A ∪ B) − (A ∩ B)}
  • vii.n{(A − B) ∪ (B − A)}
Answer
  • i.A − B = {c, d}: 2
  • ii.B − C = {a, b}: 2
  • iii.A − C = {c}: 1
  • iv.A = {a, b, g}: 3
  • v.A ∪ B = {a, b, c, d, e, f}, so A ∪ B = {g}: 1
  • vi.A ∩ B = {e, f}, so (A ∪ B) − (A ∩ B) = {a, b, c, d}: 4
  • vii.(A − B) ∪ (B − A) = {c, d} ∪ {a, b} = {a, b, c, d}: 4

vi and vii are the same set: the symmetric difference, the elements in exactly one of A and B.

Question 3

a.

If U = {natural numbers less than 20}, A = {even numbers less than 20}, B = {prime numbers less than 20} and C = {square numbers less than 20}, find

  • i.n(U)
  • ii.n(C)
  • iii.n(A ∩ B)
  • iv.n(B − C)
  • v.n(A)
  • vi.n(A ∪ C)
  • vii.n(A ∩ B)
Answer

U = {1, 2, …, 19}, A = {2, 4, 6, 8, 10, 12, 14, 16, 18}, B = {2, 3, 5, 7, 11, 13, 17, 19}, C = {1, 4, 9, 16}.

  • i.19
  • ii.4
  • iii.A ∩ B = {2}: 1
  • iv.No square is prime, so B − C = B: 8
  • v.n(A) = 19 − 9 = 10
  • vi.A ∩ C = {4, 16}, so n(A ∪ C) = 9 + 4 − 2 = 11
  • vii.n(A ∩ B) = 19 − 1 = 18
b.

If U = {x:x is a natural number from 1 to 20}, A = {y:y is a prime number}, B = {z:z is a factor of 18} and C = {p:p is a multiple of 3 less than 20}, show these sets in Venn diagrams and find

  • i.n(A ∪ B)
  • ii.n(B ∪ C)
  • iii.n(A ∪ B ∪ C)
  • iv.n(A ∩ B ∩ C)
  • v.n₀(A)
  • vi.n(A − B)
  • vii.n(A ∩ B)
  • viii.n(C)
Answer

In U: A = {2, 3, 5, 7, 11, 13, 17, 19}, B = {1, 2, 3, 6, 9, 18}, C = {3, 6, 9, 12, 15, 18}.

U (1 to 20)ABC5, 7, 1113, 17, 19112, 15269, 1834, 8, 1014, 16, 20
Figure 1.55 Exercise 1.2, question 3b. Each part shades some of these regions.
  • i.A ∩ B = {2, 3}, so n(A ∪ B) = 8 + 6 − 2 = 12
  • ii.B ∩ C = {3, 6, 9, 18}, so n(B ∪ C) = 6 + 6 − 4 = 8
  • iii.A ∪ B ∪ C = {1, 2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19}: 14
  • iv.A ∩ B ∩ C = {3}: 1
  • v.A only = {5, 7, 11, 13, 17, 19}: 6
  • vi.A − B = {5, 7, 11, 13, 17, 19}: 6 (here the same as A only, since no prime outside B is in C)
  • vii.n(A ∩ B) = 20 − 2 = 18
  • viii.n(C) = 20 − 6 = 14

(If U is taken as the natural numbers less than 20, as in part a, the last two are 17 and 13.)

Question 4

4.

From Figure 1.56, find

  • a.n(P ∪ Q)
  • b.n(P ∩ Q)
  • c.n(P ∪ Q)
  • d.n(P ∩ Q)
UPQm, n, op, q, rs, ta, bc, de, f
Figure 1.56 Exercise 1.2, question 4.
Answer

P and Q are disjoint, with nothing outside them: n(U) = 8 + 6 = 14.

  • a.n(P ∪ Q) = 8 + 6 = 14
  • b.P ∩ Q = ∅: 0
  • c.P ∪ Q = ∅: 0
  • d.n(P ∩ Q) = 14 − 0 = 14

Question 5

5.

Using Figure 1.57, find

  • a.n(S)
  • b.n(T)
  • c.n(S ∩ T)
  • d.n(S ∪ T)
  • e.n₀(T)
  • f.n₀(S)
  • g.n(T)
  • h.n(S ∪ T)
  • i.n(S ∩ T)
  • j.n(U)
UST2, 46, 8103512, 1416, 18207
Figure 1.57 Exercise 1.2, question 5.
Answer
  • a.7
  • b.7
  • c.2 (3, 5)
  • d.7 + 7 − 2 = 12
  • e.5
  • f.5
  • g.13 − 7 = 6
  • h.1 (7)
  • i.13 − 2 = 11
  • j.13

Question 6

6.

Look at Figure 1.58 and find

  • a.n(A)
  • b.n(B)
  • c.n(C)
  • d.n(A ∪ B)
  • e.n(A ∪ B ∪ C)
  • f.n(A ∩ B ∩ C)
  • g.n(A ∪ B ∪ C)
  • h.n₀(A)
  • i.n₀(C)
  • j.n(U)
UABC56724318
Figure 1.58 Exercise 1.2, question 6.
Answer

A = {1, 2, 3, 5}, B = {1, 2, 4, 6}, C = {1, 3, 4, 7}.

  • a.4
  • b.4
  • c.4
  • d.6 (1 to 6)
  • e.7
  • f.1
  • g.1 (8)
  • h.1 (5)
  • i.1 (7)
  • j.8

Question 7

7.

Use Figure 1.59 to verify these relations:

  • a.n(A) = n(U − A)
  • b.n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
  • c.n₀(A) = n(A − B)
  • d.n(A ∪ B) = n₀(A) + n(B)
  • e.n(A ∪ C) = n₀(A) + n(C)
  • f.n(B ∪ C) = n₀(B) + n(C)
  • g.A ∪ B ∪ C = U − (A ∪ B ∪ C)

In c to f, “only” is with respect to the two sets in the relation: in d, n₀(A) is A outside B; in e, A outside C.

UABC0, 16, 795823410, 1112
Figure 1.59 Exercise 1.2, question 7.
Answer

U = {0, 1, 2, …, 12}, n(U) = 13; A = {0, 1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {2, 3, 4, 8, 9}.

  • a.A = {6, 7, 8, 9, 10, 11, 12}, so n(A) = 7; U − A is the same set, and 13 − 6 = 7. ✓
  • b.A ∪ B = {0, 1, …, 8}: 9. n(A) + n(B) − n(A ∩ B) = 6 + 5 − 2 = 9. ✓
  • c.A outside B = {0, 1, 2, 3} = A − B: both 4. ✓
  • d.n₀(A) + n(B) = 4 + 5 = 9 = n(A ∪ B). ✓
  • e.A ∪ C = {0, 1, 2, 3, 4, 5, 8, 9}: 8. A outside C is {0, 1, 5}, and 3 + 5 = 8. ✓
  • f.B ∪ C = {2, 3, 4, 5, 6, 7, 8, 9}: 8. B outside C is {5, 6, 7}, and 3 + 5 = 8. ✓
  • g.A ∪ B ∪ C = {0, 1, …, 9}, and the elements of U outside it are {10, 11, 12}, which is A ∪ B ∪ C. ✓

Mixed exercise

Question 1

1.

Think of the two words SUSTAINABILITY and URBANIZATION as sets of letters.

  • a.Write the set of letters of each word, by the listing method and in set-builder form.
  • b.Show the two sets in a Venn diagram.
  • c.Write the set of letters that are in both words.
  • d.How many letters are in exactly one of the two words?
  • e.If U = {a, b, c, …, z}, how many letters are in neither word?
Answer

a. S = {x:x is a letter of SUSTAINABILITY} = {s, u, t, a, i, n, b, l, y}
R = {x:x is a letter of URBANIZATION} = {u, r, b, a, n, i, z, t, o}
Repeated letters are written once: SUSTAINABILITY has 14 letters but only 9 different ones.
b.

U (26 letters)SRs, lya, b, in, t, uo, rz
Figure 1.60 Mixed exercise, question 1. The other 14 letters of the alphabet are outside both circles: they are in neither word.
  • c.S ∩ R = {a, b, i, n, t, u}
  • d.S − R = {s, l, y} and R − S = {o, r, z}: 6 letters.
  • e.n(S ∪ R) = 9 + 9 − 6 = 12, so n(S ∪ R) = 26 − 12 = 14.

Question 2

2.

U = {whole numbers from 1 to 25}, A = {multiples of 3 from 1 to 25}, B = {multiples of 4 from 1 to 25} and C = {multiples of 5 from 1 to 25}.

  • a.Write all these sets by the listing method.
  • b.Show the three sets in a Venn diagram.
  • c.Write the set of numbers in all three sets.
  • d.How many numbers are in all three sets, and how many in none of them?
Answer

a. U = {1, 2, 3, …, 25}, A = {3, 6, 9, 12, 15, 18, 21, 24}, B = {4, 8, 12, 16, 20, 24}, C = {5, 10, 15, 20, 25}.
b.

U (1 to 25)ABC3, 6, 918, 214, 8165, 102512, 242015
Figure 1.61 Mixed exercise, question 2. Outside the three circles: 1, 2, 7, 11, 13, 14, 17, 19, 22, 23.
  • c.A ∩ B ∩ C = {}: the smallest number in all three is 60.
  • d.None is in all three. A ∪ B ∪ C has 5 + 3 + 3 + 2 + 1 + 1 = 15 numbers (count the regions), so 10 are in none: 1, 2, 7, 11, 13, 14, 17, 19, 22, 23.