20[A Math CD]

1 © Penerbitan Pelangi Sdn. Bhd.

Paper 1

1.

FE

H

J

G

M

LK

FGJK is an inclined plane.

Answer: C

2.

P Q

RS

T

O

The orthogonal projection of line TQ onto plane PQRS is OQ.

Answer: B

3.

P

M

QR

T

The angle between line TM and the plane PQR is ∠TMR.

Answer: A

4.

E F

GH

J K

LM

The angle between line JG and the base is ∠JGE.

Answer: C

5.

E

FG

H

JK

LM

The angle between line LE and plane EHMJ is ∠MEL.

Answer: D

6.

P Q

RS

N

T

The angle between line TQ and the base PQRS is ∠TQN.

Answer: A

CHAPTER

20 Lines and Planes in 3-Dimensions

CHAPTER

2

Mathematics SPM Chapter 20

© Penerbitan Pelangi Sdn. Bhd.

7.

E

F

G

H

M

The angle between plane GEH and plane GEF is ∠HMF.

Answer: A

8.

E F

GH

K

J M

L

The angle between plane FGJK and plane EFGH is ∠EFK or ∠HGJ.

Answer: C

9.

P Q

RS

T U

VW

The angle between plane PQV and the base PQRS is ∠RQV.

Answer: D

Paper 2

1.

E F

G

9 cm12 cm

8 cm

H x

J

tan x = 8–––12

x = 33.7°

The angle between line JG and plane EFGH is 33.7°.

2.

A B

C10 cm

9 cm D

E

x

tan x = 9–––10

x = 41°59′

The angle between line EC and plane ABCD is 41°59′.

3.

E F12 cm5 cm

10 cm

GHx

K

J M

L

HF 2 = HE 2 + EF 2

HF = 52 + 122

= 13 cm

tan x = JH––––HF

= 10–––13

x = 37°34′

The angle between line JF and the base EFGH is 37°34′.

4.

S R

Qx

7 cm7 cm

24 cm

P

T

QS2 = SR2 + RQ2

QS = 72 + 72

= 98 cm

3



tan x = TQ––––QS

= 24––––98

x = 67°35′

The angle between line ST and the base PQRS is 67°35′.

5.

12 cm

E F

H

7 cm

G

sin ∠HFG = 712

∠HFG = 35°419

The angle between line HF and the base EFG is 35°419.

6.

A Bx

C

8 cm

7 cm6 cm

D

E H

FG

tan x = 6—8

x = 36°52′

The angle between plane BCEF and the base ABCD is 36°52′.

7.

7 cm3 cm

4 cmA

BCx

F

E

D

tan x = 4—3

x = 53°8′

The angle between plane ABFE and the base ABCD is 53°8′.

8.

5 cm

10 cm6 cm

P

Q

Mx

R

T

PM 2 = PQ2 – QM2

PM = 102 – 62

= 8 cm

tan x = TP––––PM

= 5—8

x = 32°

The angle between plane TQR and the base PQR is 32°.

Paper 1

1. J M

H

GN

F

EK

The angle between plane FHJ and plane KHJ is ∠KMF.

Answer: D

4



2.

E F

GH

K

J M

L

The angle between plane KGJ and the base EFGH is ∠JGH.

Answer: B

3.

R

L

M

N

P

TQ

The angle between line TN and the base LMNP is ∠TNL.

Answer: A

4.

M N

P

TS

R U

Q

The angle between the plane SNP and the plane PNTU is ∠SNT.

Answer: D

5.

P

Q

R

S

T

The angle between line TS and the base PQRS is ∠TSQ.

Answer: B

6.

E F

V

GHU

K

T

J M

L

The angle between the plane FTL and the plane FGML is ∠TUV.

Answer: D

7.

S T

QRM

L P

N

The angle between plane PRSN and plane LMNP is ∠SNM.

Answer: A

Paper 2

1.

A B10 cm

3 cm

3 cm

7 cm CD

E H

F

M

N L

G

(a) The angle between the plane MBC and the base ABCD is ∠MLN.

(b) tan ∠MLN = MN––––NL

= 7––10

∠MLN = 35°

5



2.

W 12 cm

10 cm

5 cm

15 cm

X

YZ

P

M

S R

Q

(a) The angle between line XM and plane WZSP is ∠XMW.

(b) MW 2 = PM2 + PW2

MW = 52 + 152

= 250 cm

tan ∠XMW = WX––––MW

= 12–––––250

∠XMW = 37.2°

3. E H

C

BA

F DG

10 cm

15 cm

6 cm

7 cm

(a) The angle between the line CE and the plane ADEF is ∠CED.

(b) tan ∠CED = CDED

= 107

∠CED = 55°

4. (a)

A

FD

B

CG

HE

10 cm

7 cm

4 cm

The angle between the plane DFG and the plane ABGF is ∠AFD.

(b) tan ∠AFD = 104

∠AFD = 68°12′

5. F

A x

E

CB 16 cm

8 cm

7 cm

D

tan x = ED––––AD

= 7–––16

x = 23°38′

The angle between plane EAB and the base ABCD is 23°38′.

6. (a)

F

E

C

BA

D

7 cm

8 cm

6 cm

The angle between the plane BCEF and the base ABCD is ∠ABF or ∠DCE.

(b) tan ∠ABF = 68

∠ABF = 36°52′

7. J K

Gx

FE 15 cm8 cm

9 cm

H

EG2 = EF 2 + FG2

EG = 152 + 82

= 17 cm

tan x = KG––––EG

= 9–––17

x = 27°54′

The angle between line EK and the base EFGH is ∠KEG and ∠KEG = 27°54′.

6



Paper 1

1.

E

FG

H

KJ

The angle between line FK and plane GHK is ∠GKF.

Answer: D

2.

E F

GH

NJ K

LM

The angle between line FN and plane EHMJ is ∠ENF.

Answer: A

3.

E

F

M

G

H

The angle between line HG and plane EFH is ∠GHM.

Answer: C

4.

P Q

RS

T

The angle between line QT and plane PST is ∠PTQ.

Answer: D

5.

P Q

RS

T U

VW X

The angle between plane PWV and plane RSWV is ∠PWS.

Answer: A

6.

PX Q

R

V

The angle between the base PQR and plane VPQ is ∠RXV.

Answer: B

7.

P Q

RS

T U

The angle between plane QST and plane RSTU is ∠RSQ.

Answer: B

8.

E F

GH

JM

V

The angle between plane VFG and the base EFGH is ∠VJM.

Answer: C

7



9.

E F

GHK

YJ

X

M L

The angle between plane XFGM and plane JKLM is ∠GML.

Answer: D

10. T U

S

YP Q

X

R

The angle between plane UPS and the base PQRS is ∠UYX.

Answer: A

11.

E F

GHP

R Q

LM

KJ

The angle between plane PGL and plane HGLM is ∠RQP.

Answer: B

12.

J K

PQ

LM

N

The angle between plane NKL and plane NJM is ∠PNQ.

Answer: C

Paper 2

1.

A 6 cm

12 cm

3 cm

4 cmB

CD

M

V

x

BM 2 = BC2 + CM 2

BM = 42 + 32

= 5 cm

tan x = VM––––BM

= 12–––5

x = 67°23′

The angle between line VB and the base ABCD is ∠VBM and ∠VBM = 67°23′.

2.

DC

V

AB

M

12 cm

12 cm

13 cm

x

BD2 = BC2 + CD2

BD = 122 + 122

= 288 cm

BM = 288–––––2

= 8.485 cm

tan x = VM––––BM

= 13–––––8.485

x = 56°52′

The angle between line VB and the base ABCD is ∠VBM and ∠VBM = 56°52′.

8



3.

A B

CD

V

12 cm

9 cm

8 cm

x

BD2 = BC2 + CD2

BD = 92 + 122

= 15 cm

tan x = VD––––BD

= 8–––15

x = 28°4′

The angle between line VB and the base ABCD is ∠VBD and ∠VBD = 28°4′.

4.

A

D

C B

V

10 cm

5 cm

5 cm

9 cm

x

BD2 = AB2 – AD2

BD = 102 – 52

= 75 cm

tan x = VB––––DB

= 9––––75

x = 46°6′

The angle between line VD and the base ABC is ∠VDB and ∠VDB = 46°6′.

5.

E F12 cm

5 cm6 cm GH

K

J M

L

x

FH 2 = HE2 + EF2

FH = 52 + 122

= 13 cm

tan x = JH––––FH

= 6–––13

x = 24°479

The angle between line JF and the base EFGH is 24°479.

6.

A

B10 cm

7 cm

8 cm

6 cm 7 cm

4 cm

CJ

D

EH

GF

x

DJ = 6 cm AJ2 = AD2 + DJ 2

AJ = 82 + 62

= 10 cm

tan x = EJ––––AJ

= 7–––10

x = 35°

The angle between line EA and the base ABCD is 35°.

9



7.

L MT

9 cm 6 cm

3 cm

5 cm

N

RQ

P x

PT2 = PN2 + NT 2

PT = 92 + 32

= 90 cm

RT 2 = RN2 – TN2

RT = 52 – 32

= 4 cm

tan x = RT––––PT

= 4––––90

x = 22°52′

The angle between line RP and the base LMNP is 22°52′.

8.

A B

C

24 cm

7 cm

D

E H

F G

x

tan x = EH––––HC

= 24–––7

x = 73°44′

The angle between line HC and plane BCEF is ∠ECH and ∠ECH = 73°44′.

9.

4 cm

6 cm TMU

S

P Q

RNx

tan x = PS––––NS

= 4––6

x = 33°41′

The angle between plane PMN and plane RSUT is ∠PNS and ∠PNS = 33°41′.

10.

BA

8 cm6 cm

7 cm

D C

V

x

tan x = VC––––DC

= 7––8

x = 41°11′

The angle between plane VAD and the base ABCD is ∠VDC and ∠VDC = 41°11′.

11.

BMA

8 cm

9 cm

6 cm

CN

D

E F

GH

x

tan x = GN––––MN

= 9—8

x = 48°22′

The angle between plane ABG and the base ABCD is 48°22′.

10



12.

8 cm6 cm

15 cm

E FN

GH

J M

KL

x

HF 2 = HE2 + EF2

HF = 62 + 82

= 10 cm

HN = 10–––2

= 5 cm

tan x = JH––––HN

= 15–––5

x = 71°34′

The angle between plane JEG and the base EFGH is 71°34′.

13.

A E

M

B

D

9 cm

9 cm15 cm

x

AM 2 = AB2 – BM2

AM = 152 – 92

= 12 cm

tan x = BM––––AM

= 9–––12

x = 36°52′

∠BAE = 2x = 73°44′

The angle between plane ABD and plane AED is ∠BAE and ∠BAE = 73°44′.

14.

A B

CD

MON

12 cm

5 cm

V

x

tan x = OM––––VO

= 5–––12

x = 22°37′

∠MVN = 2x = 45°14′

The angle between plane VAD and plane VBC is ∠MVN and ∠MVN = 45°14′.

15.

E F13 cm

13 cm

13 cm

G

O

H

J

K L

M

Nx

HF 2 = HE 2 + EF 2

HF = 132 + 132

= 338 cm

OH = 338–––––2

= 9.192 cm

NH = 13–––2

= 6.5 cm

tan x = NH––––OH

= 6.5––––––9.192

x = 35°16′

The angle between plane NEG and the base EFGH is 35°16′.

11



16.

L M

T

U

O

PN

QR

12 cm

4 cm

5 cm

x

QT 2 = QN 2 – TN 2

QT = 52 – 42

= 3 cm UO = QT = 3 cm

OT = 12–––2

= 6 cm

tan x = UO––––OT

= 3—6

x = 26°34′

The angle between plane UMN and the base LMNP is 26°34′.

Documents

20[A Math CD]