Solution- Three Dimensional Geometry Test-3 Class 12 Notes | EduRev

Class 12 : Solution- Three Dimensional Geometry Test-3 Class 12 Notes | EduRev

 Page 1


CBSE TEST PAPER-08 
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) 
Topic: - Three Dimensional Geometry [ANSWERS] 
Ans1. a = -18, b=12, c= -4 
a
2
+b
2
+c
2
 = (-18)
2
 + (12)
2
 + (-4)
2
 
= 484 
18 18 9
22 11 484
12 6
22 11
4 2
22 11
l
m
n
- - - = = =
= =
- - = =
Ans2. 
? ?
? ?
1
2
1 2
1 2
2 2
3 2 6
. 19
cos
21
b i j k
b i j k
b b
b b
?
= + +
= + +
= =

?

?
 
 
Ans3. The equations of the line AB are 
1 1 1
2 1 2 1 2 1
2 1 3
5 2 0 1 5 3
2 1 3
 (1)
3 1 2
x x y y z z
x x y y z z
x y z
x y z
- - - = =
- - - - - - = =
- - - - - - = =
- If A, B, C are collinear, C lies in equation (1) 
4 2 3 1 1 3
3 1 2
2 2 2
- - - - - = =
- - = - = - Hence A,B,C are collinear 
Ans4. 
? ?
1 2 4 a i j k = + - 
?
? ?
2
1 2
3 3 5
since  
a i j k
b b
= + - =

?
 
Hence line are parallel 
Page 2


CBSE TEST PAPER-08 
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) 
Topic: - Three Dimensional Geometry [ANSWERS] 
Ans1. a = -18, b=12, c= -4 
a
2
+b
2
+c
2
 = (-18)
2
 + (12)
2
 + (-4)
2
 
= 484 
18 18 9
22 11 484
12 6
22 11
4 2
22 11
l
m
n
- - - = = =
= =
- - = =
Ans2. 
? ?
? ?
1
2
1 2
1 2
2 2
3 2 6
. 19
cos
21
b i j k
b i j k
b b
b b
?
= + +
= + +
= =

?

?
 
 
Ans3. The equations of the line AB are 
1 1 1
2 1 2 1 2 1
2 1 3
5 2 0 1 5 3
2 1 3
 (1)
3 1 2
x x y y z z
x x y y z z
x y z
x y z
- - - = =
- - - - - - = =
- - - - - - = =
- If A, B, C are collinear, C lies in equation (1) 
4 2 3 1 1 3
3 1 2
2 2 2
- - - - - = =
- - = - = - Hence A,B,C are collinear 
Ans4. 
? ?
1 2 4 a i j k = + - 
?
? ?
2
1 2
3 3 5
since  
a i j k
b b
= + - =

?
 
Hence line are parallel 
? ?
? ?
2
2
2 1
2
( ) 2 3 6
2 1 1
( )
9 14 4
49
293 293
7 49
a a i j k
i j k
b a a
b a a
d
b
i j k
- = - - × - =
- × - =
- + - =
 
?
  
  

?
Ans5. 
0 0 0
2 2 1
x y z - - - = =
1 1 1
2 2 2
1 2
1 2
2 2 2 2 2
1
5 2 3
4 1 8
2, 2, 1
4, 1, 8
.
cos
2(4) 2(1) 1(8)
2 2 1 4 1 8
8 2 8
9 81
18
27
2
3
2
cos
3
x y z
a b c
a b c
b b
b b
?
?
- - - - = =
= = =
= = =
=
+ +
=
+ + + +
+ +
=
=
=
? ?
=
? ?
? ?
 
 
Ans6. 
? ?
1 a i j k = - - - 
?
? ?
? ?
? ?
2
1
2
3 5 7
7 6 1
2
a i j k
b i j k
b i j k
= + +
= - +
= - +

?

?

?
Page 3


CBSE TEST PAPER-08 
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) 
Topic: - Three Dimensional Geometry [ANSWERS] 
Ans1. a = -18, b=12, c= -4 
a
2
+b
2
+c
2
 = (-18)
2
 + (12)
2
 + (-4)
2
 
= 484 
18 18 9
22 11 484
12 6
22 11
4 2
22 11
l
m
n
- - - = = =
= =
- - = =
Ans2. 
? ?
? ?
1
2
1 2
1 2
2 2
3 2 6
. 19
cos
21
b i j k
b i j k
b b
b b
?
= + +
= + +
= =

?

?
 
 
Ans3. The equations of the line AB are 
1 1 1
2 1 2 1 2 1
2 1 3
5 2 0 1 5 3
2 1 3
 (1)
3 1 2
x x y y z z
x x y y z z
x y z
x y z
- - - = =
- - - - - - = =
- - - - - - = =
- If A, B, C are collinear, C lies in equation (1) 
4 2 3 1 1 3
3 1 2
2 2 2
- - - - - = =
- - = - = - Hence A,B,C are collinear 
Ans4. 
? ?
1 2 4 a i j k = + - 
?
? ?
2
1 2
3 3 5
since  
a i j k
b b
= + - =

?
 
Hence line are parallel 
? ?
? ?
2
2
2 1
2
( ) 2 3 6
2 1 1
( )
9 14 4
49
293 293
7 49
a a i j k
i j k
b a a
b a a
d
b
i j k
- = - - × - =
- × - =
- + - =
 
?
  
  

?
Ans5. 
0 0 0
2 2 1
x y z - - - = =
1 1 1
2 2 2
1 2
1 2
2 2 2 2 2
1
5 2 3
4 1 8
2, 2, 1
4, 1, 8
.
cos
2(4) 2(1) 1(8)
2 2 1 4 1 8
8 2 8
9 81
18
27
2
3
2
cos
3
x y z
a b c
a b c
b b
b b
?
?
- - - - = =
= = =
= = =
=
+ +
=
+ + + +
+ +
=
=
=
? ?
=
? ?
? ?
 
 
Ans6. 
? ?
1 a i j k = - - - 
?
? ?
? ?
? ?
2
1
2
3 5 7
7 6 1
2
a i j k
b i j k
b i j k
= + +
= - +
= - +

?

?

?
 
? ?
? ?
2
1
1 2
2 2 2
1 2
2 1
1 2
1 2
4 6 8
7 6 1
1 2 1
4 6 8
( 4) ( 6) ( 8)
116
2 29
( )( )
116 11658
2 29 2 29
2 29
a a i j k
i j k
b b
i j k
b b
a a b b
d
b b
- = + +
× = - - = - - - × = - + - + - =
=
- ×
=
×
- = =
=

?
?
 
 
 
 
Ans7. 
? ?
5 2 4 a i j k = + - 
?
? ?
2 3 N i j k = + -  
?
Vector equation is 
( )
? ? ? ?
. 0
[ (5 2 4 )].(2 3 ) 0
r a N
r i j k i j k
- =
- + - + - =
   

? ?
 
Cartesian equation is 
? ?
? ? ? ? ? ? ?
? ? ? ?
[ 5 2 4 ].[2 3 ] 0
(( 5) ( 2) ( 4) ).(2 3 ) 0
2( 5) 3( 2) ( 4) 0
2 10 3 6 4 0
2 3 20
r xi y j zk
xi y j zk j j k i j k
x i y j z k i j k
x y z
x y z
x y z
= + +
+ + - - + + - =
- + - + + + - =
- + - - + =
- + - - - =
+ - =

?
? ?
? ?
 
Ans8. 
? ?
[(5 2 ) (3 ) (2 ) ] 15 r t i t j s t k - + - + + =

?
? ? ? ?
( ).[( 2 ) (3 ) (2 ) ] 15
( 2 ) (3 ) (2 ) 15
xi y j zk s t i t j s t k
s t n t y s t z
+ + - + - + + =
- + - + + - =
? ?
 
Page 4


CBSE TEST PAPER-08 
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) 
Topic: - Three Dimensional Geometry [ANSWERS] 
Ans1. a = -18, b=12, c= -4 
a
2
+b
2
+c
2
 = (-18)
2
 + (12)
2
 + (-4)
2
 
= 484 
18 18 9
22 11 484
12 6
22 11
4 2
22 11
l
m
n
- - - = = =
= =
- - = =
Ans2. 
? ?
? ?
1
2
1 2
1 2
2 2
3 2 6
. 19
cos
21
b i j k
b i j k
b b
b b
?
= + +
= + +
= =

?

?
 
 
Ans3. The equations of the line AB are 
1 1 1
2 1 2 1 2 1
2 1 3
5 2 0 1 5 3
2 1 3
 (1)
3 1 2
x x y y z z
x x y y z z
x y z
x y z
- - - = =
- - - - - - = =
- - - - - - = =
- If A, B, C are collinear, C lies in equation (1) 
4 2 3 1 1 3
3 1 2
2 2 2
- - - - - = =
- - = - = - Hence A,B,C are collinear 
Ans4. 
? ?
1 2 4 a i j k = + - 
?
? ?
2
1 2
3 3 5
since  
a i j k
b b
= + - =

?
 
Hence line are parallel 
? ?
? ?
2
2
2 1
2
( ) 2 3 6
2 1 1
( )
9 14 4
49
293 293
7 49
a a i j k
i j k
b a a
b a a
d
b
i j k
- = - - × - =
- × - =
- + - =
 
?
  
  

?
Ans5. 
0 0 0
2 2 1
x y z - - - = =
1 1 1
2 2 2
1 2
1 2
2 2 2 2 2
1
5 2 3
4 1 8
2, 2, 1
4, 1, 8
.
cos
2(4) 2(1) 1(8)
2 2 1 4 1 8
8 2 8
9 81
18
27
2
3
2
cos
3
x y z
a b c
a b c
b b
b b
?
?
- - - - = =
= = =
= = =
=
+ +
=
+ + + +
+ +
=
=
=
? ?
=
? ?
? ?
 
 
Ans6. 
? ?
1 a i j k = - - - 
?
? ?
? ?
? ?
2
1
2
3 5 7
7 6 1
2
a i j k
b i j k
b i j k
= + +
= - +
= - +

?

?

?
 
? ?
? ?
2
1
1 2
2 2 2
1 2
2 1
1 2
1 2
4 6 8
7 6 1
1 2 1
4 6 8
( 4) ( 6) ( 8)
116
2 29
( )( )
116 11658
2 29 2 29
2 29
a a i j k
i j k
b b
i j k
b b
a a b b
d
b b
- = + +
× = - - = - - - × = - + - + - =
=
- ×
=
×
- = =
=

?
?
 
 
 
 
Ans7. 
? ?
5 2 4 a i j k = + - 
?
? ?
2 3 N i j k = + -  
?
Vector equation is 
( )
? ? ? ?
. 0
[ (5 2 4 )].(2 3 ) 0
r a N
r i j k i j k
- =
- + - + - =
   

? ?
 
Cartesian equation is 
? ?
? ? ? ? ? ? ?
? ? ? ?
[ 5 2 4 ].[2 3 ] 0
(( 5) ( 2) ( 4) ).(2 3 ) 0
2( 5) 3( 2) ( 4) 0
2 10 3 6 4 0
2 3 20
r xi y j zk
xi y j zk j j k i j k
x i y j z k i j k
x y z
x y z
x y z
= + +
+ + - - + + - =
- + - + + + - =
- + - - + =
- + - - - =
+ - =

?
? ?
? ?
 
Ans8. 
? ?
[(5 2 ) (3 ) (2 ) ] 15 r t i t j s t k - + - + + =

?
? ? ? ?
( ).[( 2 ) (3 ) (2 ) ] 15
( 2 ) (3 ) (2 ) 15
xi y j zk s t i t j s t k
s t n t y s t z
+ + - + - + + =
- + - + + - =
? ?
 
Ans9. Equations of any plane through the intersection of given planes are be written is 
( 1) (2 3 4 5) 0
(1 2 ) (1 3 ) (1 4 ) 1 5 0...(1)
x y z x y z
x y z
?
? ? ? ?
+ + - + + + - =
+ + + + + - - = 
This plane is it right angle to the plane x-y+z=0 
(1 2 )(1) (1 3 )( 1) (1 4 )(1) 0
1/ 3
 equation (1)
2 3 4 5
1 1 1 1 0
3 3 3 3
2 0
put in
x y Z
x z
? ? ?
?
?
+ + + - + + =
= - ? ? ? ? ? ?
- + - + - - + =
? ? ? ? ? ?
? ? ? ? ? ?
- + =
Ans10. (2 2 ) (3 4 2 ) r i j k i j k ? = - + + + +
? ?
? ? ? ?
2 1 2
.......( )
3 4 2
3 2,4 1, 2 2
.( ) 5
( ).( ) 5
5.....( )
. ( )
0
(2, 1,2)
x y z
i
coordinets are
and r i j k
xi y j zk i j k
x y z ii
coordinatelies in eq ii
we get
?
? ? ?
?
- + - = = =
+ - +
- + =
+ + - + =
- + =
=
- 
?
? ?
Are the coordinate of the point of intersection of the given line and the plane 
( 1, 5, 10) (2, 1, 2) and - - - - 2 2 2
req. distance (2 1) ( 1 5) (2 10)
13
= + + - + + +
=
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