Teknik Menjawab Matematik Spm

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Transcript of Teknik Menjawab Matematik Spm

P

Q

1 2 3 456 7R6 = 2 (1 )2 12 = 2 (2 )2 22 = 2 (3 )2 36 = 2 (4 )2 +4 +4 +4 +4

MATHEMATICS SPM Section A Paper 2( Selected Topics )D

4 7 10

T 7 Q P 14 21 O 14 60 14 6 cm E

G

F J H

K

8 cm

L 3 15 36 4 2 6

x 1 y = 6 60 Disediakan Oleh: EN. LEE MEOW CHOON ( SMK Bukit Gambir, Ledang )

Topic 1 2 3 4 5 6 7 Set Lines and planes in 3-dimensions Mathematical Reasoning - Induction Mathematical Reasoning - Argument Volume of solids Area and perimeter of sector Matrices . . .. . 2-6 7 - 10 11 12 - 14 15 - 18 19 - 21 22 - 25

1

1. [ Set ] 1.1 Basic set operation P={ 1 2 3 Q={ 3 P Q = { 1 2 3 P Q = {1234} {345} = {12345} P={ 1 Q={ 2 3 3 4 4 4 } } }

5 5

+

addition

4 4 4

5

} } }

x

only the common ones

P Q = { 3 P Q = {1234} {345} = {34} P={ 1 Q={ = P Q ={ 1 P ={ 1 P={ P = { 1234} = {5} 2 2 2 3 3 3 3

4 4 4 4

5 5 5

} } } } }

Not P, outside P

1.2 It is given that the universal set = { x : 1 < x < 12 , x is an integer} Set P = { 2 , 3 ,7 ,9 } Set Q = { x : x is a prime number } and Set R = { x : x is a multiple of 4 }. The elements of the set ( P R ) Q are A 5, 11 = {1 2 P={ 2 Q={ 2 R= { 3 3 3 B 1,5, 11 4 5 5 4 6 7 8 C 2,3 11 7 7 8 9 9

[ SPM2004/P1/Q32 ]

D 2,3,9 10 11 11 12 } } } 12 }

( P R ) Q ={ ( 2379) (48 12)} ( 2357 11) ( 2357 11) = { 234789 12 } = {1 56 10 11} ( 2357 11) = { 5 11} 1.3 It is given that the universal set. = { x : 19 < x < 31 , x is an integer } and [ SPM2006/P1/Q29 ] Set R = { x : x is a number such that the sum of its two digits is an even number } Find set R . A {20,22,24,26,28} C {19,21,23,25,27,29 } B { 21,23,25,27,29 } D {21,23,25,27,29,30 }

= { 19 ,20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 } R = { 19, 20, 22, 24, 26, 28, 31 } R={ 21, 23, 25, 27, 29, 30 }

2

1.4. Strategy in solving 1.4.1(a) Shade on the Venn diagram provided the region for (i) P Q R Steps ` P1.Assign each region with a number

1.4(a)(i)

1.4(a)(ii) P Q P

1.4(a)(iii)

Q

R 2.A > N

1 2 3 4 5

1 2 3 456 7R P Q R ={1245} {2356} {4567} = { 123456 } {4567} = {456} P Q

1Q

3

2 4R

P Q R ={12}{234}{45} ={ 1234 } {45} = {4}

P Q R = {12} {234} {4} = { 1234 } {4} = {4} P

3. ` P Shade the regions Q

R

1 2 3 4 5

1 2 3 456 7R

1Q

3

2R

4

1.4(b) Shade on the Venn diagram provided the region for P Q R Steps A> N 1.4(b)(i) P Q R ={12} {234} {45} = { 1234 } {123} = { 123 } P 1.4(b)(ii) P Q R ={1245} {2356} {4567} = { 123456 } {123} = { 123} 1.4(b)(iii) P Q R = {12} {234 } {4} = { 1234 } {123 } = { 123 }

3. Shade the regions

Q

R

1 2 3 4 5

P

Q

P

1 2 3 456 7R

1Q

3

2 4R

3

1.4(c) Shade the region defined by P ( Q R) Steps P1.Assign each regions with a number

1.4(c)(i) P

1.4(c)(ii) Q P

1.4(c)(iii)

Q Q R P R R

Q

R **

5 4 3 2 1

P

Q

P

1 6 5 274 3R P ( Q R ) = {1267} ({ 123} {2347}) = {1267} ( 23 ) = {12367 } P Q

1 2Q

3

R

4)

P ( Q R ) = { 45 } ({15} {12}) = {45 } ( 1 ) = { 145 } P

P ( Q R ={12} ( (1) (4) ) = {12} { } = {12} P

3. Shade the regions

Q Q R 3. Shade the regions P R R

5 4 3 2 1

P

Q

P

Q

1 6 5 274 3R

1 2 3

Q

R

R

4

4

1.4(d) Shade the region defined by P ( Q R) Steps A >N 1.4(d)(i) P ( Q R) 1.4(d)(ii) P ( Q R) 1.4(d)(iii) P ( Q R)

1&3. Assign and Shade the regions

P

P

Q

P

Q Q R R P ( Q R ) ={1267} ( {123} {2347)} = {1267} { 23 } = {1267} { 14567 } = { 124567} P Q P ( Q R ) = {12} ({1} {4}) = {12} ({}) = {12} { 1234} = {1234} P R

A >N

P ( Q R ) ={45} ( {15} {12}) = {45} ( {1}) = {45} {2345} = { 2345 } P

1&3. Assign and Shade the regions

5 4 3 2 1

Q

1 6 5 27 4 3R

1 2 3

Q

R

R

4

5

1.5. The Venn diagrams below shows the sets of P, Q and R. Given that = P Q R. In each diagram below, shade the region of (a) Q R P R (b) P R P R (c) P ( Q R) P R

Q

Q

Q

(a) Q R = { 34} {123} = {1234} P 5 4 Q 3 2

(b) P R = { 1} {123} = {1} P 5 4 Q3 2 R

R 1

1

(c) P ( Q R) = { 2345}({34} {123}) = { 2345}({1234}) = { 2345}({5}) = {5} P 5 4 R Q3 1 2

1.6. The diagrams below show the set P, Q and R. Given that the universal set = P Q R. Shade the region of (a) P R P R Q (b) P R (c) P R Q

P R

Q

P R

Q

(a) P R P R Q

(b)

P R

(c) P

R Q

P R

Q

P R

Q

6

1.7. The diagrams below show the Venn diagrams with the universal set = P Q R . Shade the region of (a) P Q P Q R (b) P Q R P Q R

1.8. For each of the following Venn diagrams, the universal set = A B C . Shade the region of (a) A B C A B A (b) B C B (c) ( A C ) B A B

C 1.9.[SPM2004] (a) A B C Shade A B '

C

C 1.10. [SPM 2006]

(a) Q P R ShadeP 'Q

(b) A B

Shade A B C '

(b) Q P R

Shade ( P Q' ) R

C

7

1.11 (a) S

Shade S T U U

(a)

1.12 Shade V W

T

V W

(b)

Shade S T U U T S

(b)

Shade (V W )

V W

8

A

1.[SPM2004] (a) Shade B CA B '

2. [SPM 2006] (a) Q P R ShadeP 'Q

A 1 2

B 3

C 4

p1

Q 1 P 3 2 4

R

p2

A= { 1,2 } B= { 1 , 4} = { 1 } A B' A B = ( 12 ) ( 14) = {1} A 1 2 B 3 C 4 Q

P = { 1, 3, 4 } Q = { 1,2 ,3 } 3 } P 'Q = { 1 , P Q = {134} {123} = {13} R 4 Shade ( P Q' ) R R 1 P 3 2 (b) Q P

p3

(b) Shade A B C ' A 1 B

p1

p2

C A 6 7 1 25 4 3 C A={ 1,2, B={ A B = { 1, 2, C ={ 1, A B C ' ={ 1, 5, 6 } 4, 5, 6 , 7 } 4, 5, 6, 7 } 6, 7 } 6, 7 } B

Q 1 P 3 2 4

R

p3

P={ 2 Q= { 4 P Q = { 2 4 R ={ 3, 4 4 ( P Q' ) R ={

} } } } }

= =

A B C = {1256} {4567} {167 } = { 124567 } {167 } = { 167 } A B

P Q R { 2 } {4} { 34} { 24 } { 34} = {4} R 4

Q 1 P 3 2

6 7 1 25 4 3

9

(a) S

3. Shade S T U U T

(a)

4. Shade V W

V W

S 3 T 1 2 5 6 U 4 1 7 2 3 4 V W

p1

p2

S = { 2 , 3 , 4, 5 } T ={1,2 5 6 S T = { 1 , 2 3, 4 , 5 , 6 U ={1,2,3 S T U ={ 1 , 2 , 3 S T U = {2345} {1256} { 123 } = { 123456 } {123} = { 123 } S 3 T 1 2 5 6

} } } }

V = { 1 W = { V W = { V W = { 14 } { 34} = {4}

4 } 3, 4 } 4 }

p3

U 4 7 1 V W 2 3 4

10

(b)

Shade ( S T U U T S

(a)

Shade (V W )

V

W

U T S 1 2 3 4 5 2 3 4 6 1 V W

p1

p2

S={ 3 , 4, 5 T ={1,2, 3 S T = { 1 , 2 3, 4 , 5 U ={1, 5 5 S T U ={ 1 S T U = {345} {123 } {15} = { 1234 5} {15 } = {15 }

} } } } }

V = { 2, 3 W = { 3,4 V W = { 3 (V W ) = { 1 , 2 4 ( V W ) = ( {23} {34} ) = { 3 } = { 124}

} } } }

p3

U T S 1 2 3 4 5 6

1

V W 2 3 4

11

2. Angle between Line and Plane 3-D 2.1 (a) Line AR with plane ABCD S P D A Step Write line AR and plane ABCD Write the common point in the middle A 3 Write the other remaining point/s in the front 4 Choose point on the plane that is nearest to the remaining point. (Shade the plane, use arrow to denote the relative position) 2.1 (b) Line BS with plane CDSR S P D A B Q C R R A B Q C R

1 2

AR A BCD

R

A

C

1

Step Write the line BS and plane CDSR

BS CD S R

2

Write the common one in the middle

S

3

Write the other remaining point/s in the front

B

S

4

Choose point on the plane that is nearest to the remaining point. (Shade the plane, use arrow to denote the relative position)

B

S

C

12

2.2 Angle between plane and plane 2.2(a) Plane ARS with plane ABCD P D A 1 2 Step Write plane ARS with plane ABCD Write the common one in the middle A 3 Write the other remaining point/s at the front S Q C B ARS A BCD R

R/S

A

4

Choose a point that is nearer to the middle point Choose point on the plane that is nearest to the remaining point. (Shade the plane, use arrow to denote the relative position) 2.2 (b) Plane BRS with plane CDSR P

R/S

A

5

R/S

A

D

S Q D A B

R

C

1

Step Write plane BRS with plane CDSR

B RS CD R S

2

Write the common one in the middle R/S

3

Write the other remaining point/s at the front

B

R/S

4

Choose a point that is nearer to the middle point Choose point on the plane that is nearest to the remaining point. (Shade the plane, use arrow to denote the relative position)

B B

R/S R/S C

C

5

13

2.3. Determine the length of the sides or the angles. (a) (b) a=5 4 3 b=6 (d) (e) 8 10

(c)

a2+b2= c2

5

13

c=12 SOH CAH TOA

12=H 8=O 13=H do 5=A (f) (g) eo