12

1 © Penerbitan Pelangi Sdn. Bhd.

Paper 1

1. { }

Answer: A

2. P = {2, 4, 6, 8}Answer: C

3. d Q (d is an element of set Q)

Answer: D

4.

b

a

e

dP

Answer: A

5. The number of elements in set R, n(R) = 3

Answer: B

6. The elements of set H = {a, b, c, d, e} are a, b, c, d and e.

Answer: D

7. {p, q} is a subset of M.{p, q} M

Answer: B

8. QP

P ⊂ QP is a subset of Q.Answer: C

9. Number of subsets of P = 24

= 16

Answer: C

10. P′ is the complement of set P.

ξP

Answer: A

11. E′ = {1, 9}\ Elements of E′ = 1, 9

Answer: D

12. R T= {a, b , c , d, e } { b , c , e , f}= {b, c, e}

Answer: B

13. ξ P

P � Q

Q

Answer: C

14. ξ P QP � Q

Answer: C

15. n(E F G)= number of elements in sets E, F and G= 6

Answer: A

12 Sets

2

Mathematics SPM Chapter 12

© Penerbitan Pelangi Sdn. Bhd.

16. = E F GE = {A, P, R, I, L}F = {M, A, Y}G = {J, U, L, Y}E F G = { }n(E F G) = 0

Answer: A

17. J = {R, E, A, D}K = {W, R, I, T, E}J K = {R, E, A, D, W, I, T}n(J K) = 7

Answer: C

18. ξ

b c

d

ak f

e

hg

P Q

(P Q)′ = {a, b, e, f, g, h, k}

Answer: D

Paper 2

1. (a) 1

2

3

P

(b) abcd

Q

2. (a) A = {2, 3, 5, 7} \ Elements of A = 2, 3, 5, 7.

(b) B = {a, b, d, k, p} \ Elements of B = a, b, d, k, p

3. PQ

c e

ab

4. A B

5. Number of possible subsets= 23

= 8

{ }, {h}, {i}, {j}, {h, i}, {h, j}, {i, j}, {h, i, j}

6. A

ξ

Complementof set A, i.e. A'

7. Q is a subset of P.P is a subset of ξ.

PQ

ξ

a

b

c

e

d

8. (a) A′ = {6, 12, 18, 21} Elements of A′ = 6, 12, 18, 21

(b) Number of elements in set A′, n(A′) = 4

9. (a) P = {a, b} \ Elements of P = a, b

(b) Q = {a, b, c} \ Elements of Q = a, b, c

(c) Q′ = {d, e} \ Elements of Q′ = d, e

10. (a) A B = {1, 3, 4, 5, 6, 7} \ Elements of A B = 1, 3, 4, 5, 6, 7

(b) A B = {7} \ Element of A B = 7

11. (a) P Q

(b) P Q

3



12. (a)

E F

E F

=

E F

(b) EF

EF

=

EF

13. (a) P

Q R

(b) P

Q R

Paper 1

1. n(P < R) = 5 + 6 + 4 + x + 2 + 2x = 17 + 3x

n(Q9) = 5 + 6 + 2x + (2x + 1) = 12 + 4x

n(P < R) = n(Q9) 17 + 3x = 12 + 4x 17 – 12 = 4x – 3x 5 = x

n( ) = 17 + 3x + (2x + 1) = 18 + 5x = 18 + 5(5) = 43Answer: B

2. ξL M

N

ξL M

N

L < M N′

=

ξL M

N

L < M > N′Answer: D

3. ξ

L

125 – 40 40 92 – 40

28

M

Total number of Form Five students= (125 − 40) + 40 + (92 − 40) + 28= 205Answer: D

4. QP

66 30 x

x + 30 + 66 = 164 x + 96 = 164 x = 164 – 96 = 68 n(Q) = 68 + 30 = 98Answer: C

5. a b

c

d

e

kg

h

P Q

R

f

R′ = {a, b, d, e}Answer: A

4



6. ξ EF

ξ EF

E F′

=

ξ E

F

E F′

Answer: B

7. ξ = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} J = {4, 6, 8, 10, 12} K = {3, 4, 6, 12} L = {3, 5, 7, 11, 13} J K = {4, 6, 12}(J K)′ = {3, 5, 7, 8, 9, 10, 11, 13}

(J K)′ L = {3, 5, 7, 8, 9, 10, 11, 13}\ Elements of (J K)′ L = 3, 5, 7, 8, 9, 10, 11, 13

Answer: D

8. {a}, {b}, {a, b}, { }

Answer: D

9. ξP Q

ab

cd

fg

e

Q′ = {a, b, f, g}

Answer: C

10. E , F and F ∩ G ≠ f

E G

F

Answer: A

11. F

E

Answer: A

12. EF G

EF G

E F′ E

F G

G′

=

EF G

E ∩ F′ ∩ G′

Answer: A

13. n(R) = n(P Q)′ 1 + 2 + (x − 5) + (x − 2) = x − 2 3 + 2x − 7 = x − 2 2x − x = −2 − 3 + 7 x = 2

Answer: B

14. S ∩ T= {a, d, e, h} ∩ {a, d, f, g, h}= {a, d, h}

R (S ∩ T) = {a, c, d, f} {a, d, h}= {a, c, d, f, h}

Answer: C

15. (3x + 1) + (x + 5) = 38 4x + 6 = 38 4x = 38 – 6 4x = 32 x = 8Number of players who play two games= 2x= 2(8)= 16

Answer: C

16. ξ = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} P = {8, 12, 16} Q = {6, 8, 10, 12, 14, 16}All the elements of set P are also elements of set Q.Hence, P , Q

Answer: B

5



17. ξ = {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} P = {16, 18, 21, 23, 25, 27} P′ = {17, 19, 20, 22, 24, 26}

Answer: D

18. ξF G

ξF G

F′ G′

=

ξF G

(F G)′

Answer: B

19.

8

1630 26

z

xy

Badminton

BowlingSoccer

x = 69 − (8 + 30 + 16) = 15y = 71 − (x + 16 + 26) = 71 − (15 + 16 + 26) = 14z = 120 − (8 + 30 + 16 + 26 + x + y) = 120 − (8 + 30 + 16 + 26 + 15 + 14) = 11

Number of pupils who played soccer or bowling and also badminton= 8 + x + y= 8 + 15 + 14= 37

Answer: C

Paper 2

1. (a) P Q

Q′

(b) A B

D

A B

D

B (A D)

=

A B

D

B (A D)

2. (a) A DB

A B

(b) A D

B A D

B

A (B D′)

=

A D

B

A (B D′)

3. (a) RQP

P R

(b) RQP

RQP

(P Q′) R

=

RQP

(P Q′) R

6



4. (a) A CB

A B

(b) A CB

A CB

(A′ B) C

=

A C

B

(A′ B) C

5. (a) PQ

R

P Q′

(b) PQ

R

PQ

R

(P Q) R′

=

P

QR

(P Q) R′

Paper 1

1. n(G′) = n(E F) (8 − x) + 5 + 6 = 5 + x 8 − x + 11 = 5 + x 8 + 11 − 5 = x + x 14 = 2x

x = 14–––2

= 7

Answer: C

2. ξ E

7 – 5 11 – 55

F

n(E′) = 11 − 5 = 6

Answer: B

3. ξM N

a be

cf

gd

h

(M N)′ = {d, g, h}

Answer: C

4. ξ = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12} E = {3, 5, 7, 9, 11} F = {3, 6, 9, 12}E F = {3, 5, 6, 7, 9, 11, 12}

n(E F) = 7

Answer: C

5. n(P Q R) = 32 4 + 6 + 2 + 3 + 5 + 1 + x = 32 21 + x = 32 x = 32 − 21 = 11

Answer: D

6. F

G H

ξ F

G H

ξ

(F G) H

=

F

G H

ξ

(F G) H

Answer: D

7



7.

P

Q Rξ

∩

P

Q Rξ

P′ (Q R)

=

P

Q Rξ

P′ (Q R)

Answer: B

8. M N = {B, L, U, E, P, I, N, K} L′ = {U, E, P, I, N}(M N) L′ = {U, E, P, I, N}

Answer: C

9. ξ X ZY

(X Y)

ξ X ZY

Z′

=

ξ X ZY

(X Y) Z′

Answer: A

10. ξP

Q

∩

ξP

Q

Q P′= φ (empty set)

Answer: D

Paper 2

1. (a) GFE

E F

(b) GFE

GFE

(E F)′ G

=

GFE

(E F)′ G

2. (a) P R

Q

P Q

(b) P R

Q

P RQ

(P Q) R

=

P R

Q

(P Q) R

3. (a) A B

C

A B′

8



(b) A BC

A BC

(A B) C′

=

A B

C

(A B) C′

4. (a) X Y

Z

X Y′

(b) X Y

Z

X Y

Z

(X Y) Z

=

X Y

Z

(X Y) Z

5. (a) Q

PR

P R′(b)

QP

R QP

R

(Q R) P′

=

Q

PR

(Q R) P′

6. (a) E GF

E′ G′

(b) E GF

E GF

(E F) (F G)

=

E GF

(E F) (F G)

7. (a) P

Q R

(P Q)′

(b) P

Q R

P

Q R (P Q)′ R

=

P

Q R

(P Q)′ R

8. (a) A

BC

B C′

9



(b) ABC

ABC

(A B′) C

=

A

BC

(A B′) C

9. (a) P Q R

P R′

(b) P Q R

P Q R

P (Q′ R)

=

P Q R

P (Q′ R)

10. (a) P Q

R

P R′

(b) P QR

P Q

R

(P′ Q′) R

=

P Q

R

(P′ Q′) R

12

Documents

elements of b

e f e f

p answer

elements of e

elements of p

b number of elements

ef ef e f

e f ge