**Question 5:**

Diagram below shows a Venn diagram with the number of elements of set

*P*, set*Q*and set*R.*

It is given that the universal set,
$\xi =P\cup Q\cup R\text{and}n\left(Q\u2018\right)=n\left(Q\cap R\right).$

Find the value of

*x*.

Solution:Solution:

*n*(

*Q*)’ =

*n*(

*Q*∩

*R*)

3 + 8 + 5 =

*x*– 3 + 916 =

*x*+ 6

*x***= 10**

**Question 6:**

Diagram below is a Venn diagram showing the number of quiz participants in set

*P*, set*Q*and set*R.*It is given that the universal set,
$\xi =P\cup Q\cup R$
, set

*P*= {Science quiz participants}, set*Q*= {Mathematics quiz participants} and set*R*= {History quiz participants}.If the number of participants who participate in only one quiz is 76, find the total number of the participants.

Solution:Solution:

Number of participants who participate in only one quiz = 76

(5

*x*– 2) + (*x*+ 6) + (2*x*+ 8) = 76 8

8

*x*+ 12 = 768

*x*= 64*x*= 8Total number of the participants

= 76 + 7 + 4 + 5 + 3(8)

**= 116**

**Question 7:**

Diagram below is a Venn diagram showing the number of students in set

*K*, set*L*and set*M.*It is given that the universal set,
$\xi =K\cup L\cup M$
, set

*K*= {Karate Club}, set*L*= {Life Guards Club} and set*M*= {Martial Arts Club}.If the number of students who join both the Life Guards Club and the Martial Arts Club is 8, find the number of students who join only two clubs.

Solution:Solution:

Number of students who join both the Life Guards Club and the Martial Arts Club =

*n*(*L*∩*M*) = 2 + 2*x*2 + 2

*x*= 82

*x*= 6*x*= 3

Number of students who join only two clubs

=

*x*+ 4 + 2*x*= 3 + 4 + 2(3)

=

**13**