NEXUS SPM MATHEMATICS FORM 4 & 5 Page 41 (More Cloned SPM Questions)

Supplementary Materials for

CHAPTER 3: SETS Cloned SPM Question (2006, Paper 1) Given the universal set and set R = {x : x is a number such that the sum of its two digits is an even number}, find R'.

{ :11 21, is an integer}x x xξ = ≤ <

A {11, 13, 15, 17, 19, 20} C {12, 14, 16, 18} B {12, 14, 16, 18, 21} D {12, 14, 16} Solution ξ = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} R = {11, 13 15, 17, 19, 20} R' = {12, 14, 16, 18} Answer: C Pointers • Be careful with the definition of the universal set, ξ. The integer 21 is not an element of ξ

because of the inequality < 21. • R' is the complement of R. That is it contains elements which are not in R, but in ξ.

©Sasbadi Sdn. Bhd. i

NEXUS SPM MATHEMATICS FORM 4 & 5 Page 41 (More Cloned SPM Questions)

Supplementary Materials for

Cloned SPM Question (2006, Paper 1) The Venn diagram below shows the relationship between the universal set ξ, set P and set Q.

ξ P Q

The region (P Q)' is equivalent to the region ∪A P' ∩ Q' C P ∩ Q' B P' ∪ Q' D (P ∩ Q)' Solution (P ∪ Q)' can be represented by the shaded region as shown in the Venn diagram below.

P Qξ

The shaded region is equivalent to P' ∩ Q'. Answer: A Pointers • (P ∪ Q)' means the complement of (P ∪ Q), that is the region outside (P ∪ Q) or the

shaded region as shown in the Venn diagram. • Since the shaded region also represents the overlap of the regions outside P and outside Q,

the region is also equivalent to P' ∩ Q'.

©Sasbadi Sdn. Bhd. ii

NEXUS SPM MATHEMATICS FORM 4 & 5 Page 41 (More Cloned SPM Questions)

Supplementary Materials for

Cloned SPM Question (2006, Paper 1) The table on the left below shows the data obtained from a survey of 120 television viewers. The diagram on the right is a Venn diagram that represents part of the information shown in the table.

Programmes Watched Number of Viewers

Entertainment 70 Sports 64 Sports and Entertainment only 16 News and Entertainment only 13 News only 19 Sports only 21

News

Sports

13

Entertainment 16

19

Find the number of viewers who watched sports or news and also entertainment programmes. A 40 B 43 C 46 D 64 Solution Let the number of viewers who watched sports and entertainment and also news programmes be x. By using all the information in the table, the number of viewers in each region of the Venn diagram can be filled as follows.

Pointers • The region which represents the viewers who watched

sports or news and also entertainment programmes is the shaded region shown in the Venn diagram.

• By finding the number of viewers represented by the intersection of the three sets, the

problem can be solved.

Total number viewers = 120 21 + 16 + x + 27 – x + 19 + 13 + 41 – x = 120 137 – x = 120 x = 17

News

Sports Entertainment

19

13

16 21

x

41 – x

27 – x

Hence, the number of viewers who watched sports or news and also entertainment programmes = 16 + x + 13 = 16 + 17 + 13 = 46 Answer: C

Entertainment

19

13

16 Sports

News

©Sasbadi Sdn. Bhd. iii

NEXUS SPM MATHEMATICS FORM 4 & 5 Page 41 (More Cloned SPM Questions)

Supplementary Materials for

Cloned SPM Question (2006, Paper 2) The Venn diagram below shows the sets P, Q and R such that the universal set ξ = P Q∪R. ∪

P Q R On separate diagrams, shade (a) the set P' ∩ Q, (b) the set (P' ∪ Q') ∩ R. Solution (a) P' ∩ Q

©Sasbadi Sdn. Bhd. iv

Pointers • P' ∩ Q means the complement of P intersects with Q, that is the region outside P and

overlapping with Q. • (P' ∪ Q') ∩ R can be considered as the region of (P' ∪ Q') intersecting or overlapping

with R. • The region of (P' ∪ Q') can be represented by the shaded region as shown in the Venn

diagram below.

As a result, (P' ∪ Q') ∩ R is R.

(b) (P' ∪ Q') ∩ R

P Q R

P Q R

P Q