33
LINES AND PLANES IN 3-DIMENSIONS Compulsary question in paper 2, section A, question 4 , and full marks given is 3 marks. ( paper 1, question 14.)
LINES AND PLANES IN
3-DIMENSIONS
Compulsary question in paper 2, section A, question 4 , and full marks given is 3 marks.
( paper 1, question 14.)
ANALYSIS OF SPM MATHEMATICS TOPIC : LINES AND PLANES IN 3-DPAPER 2 : SECTION A (3 MARKS)
YEAR :
2004 (N)Type
PYRAMID (2)CUBOID (1)
2005 (N)Type : RIGHT
PRISM (2)RIGHT
PYRAMID(1)
2006 (N)Type :
2007 (N)Type :RIGHT
PRISM(2)
2008 (N)Type :
CUBOID(2)
2009 (N)Type :
CUBOID(2)PYRAMID (1)
BILTopic / Content
K1 K2 J K1 K2 J K1 K2 J K1 K2 J K1 K2 J K1 K2 T
1
LINES AND PLANES IN 3-D
1 4 5 1 3 4 1 3 4 1 3 4 1 3 4 1 3 4
CUBOID
RIGHT PRISMPYRAMID WITH RECTANGULAR BASE
PYRAMID CUBOID
CUBOID
S
No Steps Solutions
1 Draw the line UP and shade the plane RPQ
U
S
P
R
Q
T
Question 1The Diagram shows a triangular prism. Identify the angle between the line UP and the plane RPQ
No Steps Solutions
2. 1. Arrange the line and plane in two rows
2. Find out the same alphabet – point P (identify the point of line UP that touches the plane RPQ, )
3. Draw 3 boxes,
4. Write point P in the middle and point U in the first box (point does not touch the plane)
5. Look at U, choose Which One is the Nearest
to U (WON) (Non-slashed alphabets) Between point R and Q, point R will be chosen and write it in the third box.6. Angle between line UP to the plane RPQ is
U P R P Q U P R P Q
U
S
P
R
Q
T
PU R
UPR
No Steps Solutions
1 Draw the line QT and shade the plane PQRS
Question 2The Diagram shows a cuboid with a horizontal rectangular base PQRS. Calculate the angle between the line QT and the base PQRS
P
V
SW
Q
R
UT
16 cm
6 cm
8 cm
No Steps Solutions
2. 1. Arrange the line and plane in two rows
2. Find out the same alphabet – point Q (identify the point of line QT that touches the plane PQRS, )
3. Draw 3 boxes,
4. Write point Q in the middle and point T in the first box. (point does not touch the plane)5. Look at T, choose Which One is the Nearest to T (WON) (Non-slashed alphabets- Between point P, R or S, point S will be chosen and write it in the third box.6. Angle between line UP to the plane RPQ is
Q T P Q R S Q T P Q R S
QT S
TQS P
VS
W
Q
R
UT
16 cm
6 cm
8 cm
No Steps Solutions
3 Refer to the points you have obtained in step 2. (point T,Q,S) Draw ΔTQS. Mark the right angle.
P
V
SW
Q
R
U
T
16 cm
6 cm
8 cm
T
S Q
No Steps Solutions
4 With the information given in the question, label the length of the sides of ΔTQS. At least two sides of the length must be known. Use Pythagoras Theorem if necessary
P
V
SW
Q
R
U
T
16 cm
6 cm
8 cm
T
S Q
8 cm
09.17616 22
No Steps Solutions
5 Mark the opposite side,TSthe adjacent side,SQ
P
V
SW
Q
R
U
T
16 cm
6 cm
8 cm
T
S Q
8 cm
09.17616 22
TQS
P
V
SW
Q
R
U
16 cm
6 cm
8 cm
T
09.17
8tan TQS
'52508.25 orTQS
T
S Q
8 cm
09.17616 22 are
No Steps Solutions
1 Shade the plane PBC the plane BCRQ
Question 4The diagram shows a prism with cross section BCRQ. Given T and U are the midpoint of AD and BC respectively, P and Q are right above T and U respectively and PQRS is a square.
UT
C
9 cm
BA
D
R
QP
S
15 cm
12 cm
No Steps Solutions
2. 1. Arrange the plane PCB and plane BCRQ in two rows
2. Find out the same alphabet (identify the points of PCB that touch the plane BCRQ, )
3. Draw 3 boxes, 4. Write point CB in the middle and point P in the first box.
P C B B C R Q P C B B C R Q
CBP
UT
C
9 cm
BA
D
R
QP
S
15 cm
12 cm
No Steps Solutions
2. 5. Look at P, choose WON – Slashed alphabet BU = CU, SO CHOOSE THE MIDPOINT OF BC (U) 6. Look at P choose Which One is the Nearest to P (WON) (Non-slashed alphabets - point at the plane BCRQ, point Q is the nearest to P, Write point Q in the third box.7. Angle between plane PCB to the plane BCRQ is
P C B B C R Q
BCP
UT
C
9 cm
BA
D
R
QP
S
15 cm
12 cm
P U Q
PUQ
No Steps Solutions
3 Refer to the points you have obtained in step 2. (point P,U,Q) Draw ΔPUQ. Mark the right angle.
P
U
Q
UT
C
9 cm
BA
D
R
QP
S
15 cm
12 cm
No Steps Solutions
4 With the information given in the question, label the length of the sides of ΔPUQ. At least two sides of the length must be known. Use Pythagoras Theorem if necessary
P
U
Q
UT
C
9 cm
BA
D
R
QP
S
15 cm
12 cm
9 cm
12 cm
No Steps Solutions
5 Mark the opposite side PQthe adjasent side UQ
P
U
Q
UT
C
9 cm
BA
D
R
QP
S
15 cm
12 cm
9 cm
12 cm
PUQ
12
9tan PUQ
'523687.36 orPUQ
P
U
Q9 cm
12 cm
UT
C
9 cm
BA
D
R
QP
S
15 cm
12 cm
are
No Steps Solutions
1 Shade the plane PRV and the plane QRVU
Question 5The diagram shows a cuboid with base TUVW, Calculate the angle between PRV and the plane QRVU
R
Q
S
P
T U
VW
5 cm
12 cm
4 cm
R
No Steps Solutions
2. 1. Arrange the plane PRV and plane QRVU in two rows
2. Find out the same alphabet (identify the points of PRV that touch the plane QRVU, )
3. Draw 3 boxes, 4. Write point RV at the middle and point P in the first box.
P R V Q R V U P R V Q R V U
RVP
R
Q
S
P
T U
VW
5 cm
12 cm
4 cm
R
No Steps Solutions
2. 5. Look at P, choose WON – Slashed alphabet (Between point R and V, point which is nearer to point P. Point R is chosen and write it in the middle box.
6. Look at P choose Which One is the Nearest to P (WON) (Non-slashed alphabets – Between point Q and and U, point Q is chosen and write it in the third box, 7. Angle between plane PRV to the plane QRUV is
P R V Q R V U
RVP
P R Q
PRQR
Q
S
P
T U
VW
5 cm
12 cm
4 cm
R
No Steps Solutions
3 Refer to the points you have obtained in step 2. (point P,R,Q) Draw ΔPRQ. Mark the right angle.
P
R
Q
R
Q
S
P
T U
VW
5 cm
12 cm
4 cm
R
No Steps Solutions
4 With the information given in the question, label the length of the sides of ΔPRQ. At least two sides of the length must be known. Use Pythagoras Theorem if necessary
P
R
Q
12 cm
5 cm
R
Q
S
P
T U
VW
5 cm
12 cm
4 cm
Diagram 5
R
No Steps Solutions
5 Mark the opposite side,PQthe adjacent side, RQ
P
R
Q
12 cm
5 cm
R
Q
S
P
T U
VW
5 cm
12 cm
4 cm
R
PRQ
5
12tan PRQ
'226738.67 orPRQ
P
U
Q12 cm
5 cm
R
Q
S
P
T U
VW
5 cm
12 cm
4 cm
R
are
Try this..Try this..
HH
EE
UUGG
TT
LL MM
NN
RR
FF
PP
Name the angle between the plane LUM with the plane LPNM
Name the angle between the plane LUM with the plane LPNM
Answer : URTAnswer : URT
Find the angle between the plane JFE and the plane DEF.
L
E
DF
J
M5
513
5
M
L
E
D
135
5
F
J
5
Identify the angle JMD
22 513
5
tan JMD
JMD'3722
62.22o
o
OR
