Page 1
CBSE TEST PAPER-06
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry)
Topic: - Three Dimensional Geometry [ANSWERS]
Ans1. 1,0,0, 0,1,0 0,0,1
Ans2. Let a and b
be the p.v of the points A (-1,0,2) and B ( 3, 4 6)
? ? ?
( )
( 2 ) (4 4 4 )
r a b a
i j i j k
?
?
= + - = - + + + +
? ?
Ans3. Let a 1 = 3, b 1 = 4, c 1 = 5 and a 2 = 4, b 2 = -3, c 2 = 5
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
0
1
cos
2
60
a a b b c c
a b c a b c
?
?
? ?
+ +
? ? = =
? ?
+ + + +
? ?
=
Ans4.
? ? ? ?
5 2 4 , 3 2 8 a i j k b i j k = + - = + -
? ?
Vector equation of line is
? ? ? ?
5 2 4 (3 2 8 )
r a b
i j k i j k
?
?
= +
= + - + + -
? ?
Cartesian equation is
? ? ? ? ? ?
? ? ? ?
5 2 4 (3 2 8 )
(5 3 ) (2 2 ) ( 4 8 )
5 3 , 2 2 , 4 8
5 2 4
3 2 8
xi y j zk i j k i j k
xi y j zk i j k
x y z
x y z
?
? ? ?
? ? ?
+ + = + - + + - + + = + + + + - - = + = + = - - - - +
= =
- ? ? ?
? ?
Ans5. Let ? is the angle between the given lines
? ?
? ?
1
2
2
1
2
1
2
3 5 4
cos
b i j k
and
b i j k
b b
b b
?
= - - = - - ·
=
?
?
Page 2
CBSE TEST PAPER-06
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry)
Topic: - Three Dimensional Geometry [ANSWERS]
Ans1. 1,0,0, 0,1,0 0,0,1
Ans2. Let a and b
be the p.v of the points A (-1,0,2) and B ( 3, 4 6)
? ? ?
( )
( 2 ) (4 4 4 )
r a b a
i j i j k
?
?
= + - = - + + + +
? ?
Ans3. Let a 1 = 3, b 1 = 4, c 1 = 5 and a 2 = 4, b 2 = -3, c 2 = 5
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
0
1
cos
2
60
a a b b c c
a b c a b c
?
?
? ?
+ +
? ? = =
? ?
+ + + +
? ?
=
Ans4.
? ? ? ?
5 2 4 , 3 2 8 a i j k b i j k = + - = + -
? ?
Vector equation of line is
? ? ? ?
5 2 4 (3 2 8 )
r a b
i j k i j k
?
?
= +
= + - + + -
? ?
Cartesian equation is
? ? ? ? ? ?
? ? ? ?
5 2 4 (3 2 8 )
(5 3 ) (2 2 ) ( 4 8 )
5 3 , 2 2 , 4 8
5 2 4
3 2 8
xi y j zk i j k i j k
xi y j zk i j k
x y z
x y z
?
? ? ?
? ? ?
+ + = + - + + - + + = + + + + - - = + = + = - - - - +
= =
- ? ? ?
? ?
Ans5. Let ? is the angle between the given lines
? ?
? ?
1
2
2
1
2
1
2
3 5 4
cos
b i j k
and
b i j k
b b
b b
?
= - - = - - ·
=
?
?
? ? ? ?
? ? ? ?
( 2 ) (3 5 4 )
2 3 5 4
3 5 8 16
6 50 50
16
6 5 2
16 3
2 3 5 2 3
816
i j k i j k
i j k i j k
- - · - - - - - - + +
= =
×
× × ×
=
? ?
? ?
3
2
1
3 5
8 3
cos
15
8 3
cos
15
?
?
- × ×
=
? ?
=
? ?
? ?
? ?
Ans6.
? ? ? ?
1 1 2 , a i j k b i j k = + + = - +
? ?
? ? ? ?
( )
? ?
? ?
?
? ? ?
?
2 1
2 1 1 2
1 2
2 1
1 2
2 , 2 2
.( )
3 2
1 1 1
2 1 2
3 3
( 3 2 ).( 3 3
3 3
3 6 9 3
9 9 3 2 2
a i j k b i j k
a a b b
d
b b
a a i j k
i j k
b b
i k
i j k i k
d
i k
= - - = + +
- ×
=
×
- = - - × = - = - +
- - - +
=
- +
- - = = =
+
? ?
?
?
?
? ?
?
Page 3
CBSE TEST PAPER-06
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry)
Topic: - Three Dimensional Geometry [ANSWERS]
Ans1. 1,0,0, 0,1,0 0,0,1
Ans2. Let a and b
be the p.v of the points A (-1,0,2) and B ( 3, 4 6)
? ? ?
( )
( 2 ) (4 4 4 )
r a b a
i j i j k
?
?
= + - = - + + + +
? ?
Ans3. Let a 1 = 3, b 1 = 4, c 1 = 5 and a 2 = 4, b 2 = -3, c 2 = 5
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
0
1
cos
2
60
a a b b c c
a b c a b c
?
?
? ?
+ +
? ? = =
? ?
+ + + +
? ?
=
Ans4.
? ? ? ?
5 2 4 , 3 2 8 a i j k b i j k = + - = + -
? ?
Vector equation of line is
? ? ? ?
5 2 4 (3 2 8 )
r a b
i j k i j k
?
?
= +
= + - + + -
? ?
Cartesian equation is
? ? ? ? ? ?
? ? ? ?
5 2 4 (3 2 8 )
(5 3 ) (2 2 ) ( 4 8 )
5 3 , 2 2 , 4 8
5 2 4
3 2 8
xi y j zk i j k i j k
xi y j zk i j k
x y z
x y z
?
? ? ?
? ? ?
+ + = + - + + - + + = + + + + - - = + = + = - - - - +
= =
- ? ? ?
? ?
Ans5. Let ? is the angle between the given lines
? ?
? ?
1
2
2
1
2
1
2
3 5 4
cos
b i j k
and
b i j k
b b
b b
?
= - - = - - ·
=
?
?
? ? ? ?
? ? ? ?
( 2 ) (3 5 4 )
2 3 5 4
3 5 8 16
6 50 50
16
6 5 2
16 3
2 3 5 2 3
816
i j k i j k
i j k i j k
- - · - - - - - - + +
= =
×
× × ×
=
? ?
? ?
3
2
1
3 5
8 3
cos
15
8 3
cos
15
?
?
- × ×
=
? ?
=
? ?
? ?
? ?
Ans6.
? ? ? ?
1 1 2 , a i j k b i j k = + + = - +
? ?
? ? ? ?
( )
? ?
? ?
?
? ? ?
?
2 1
2 1 1 2
1 2
2 1
1 2
2 , 2 2
.( )
3 2
1 1 1
2 1 2
3 3
( 3 2 ).( 3 3
3 3
3 6 9 3
9 9 3 2 2
a i j k b i j k
a a b b
d
b b
a a i j k
i j k
b b
i k
i j k i k
d
i k
= - - = + +
- ×
=
×
- = - - × = - = - +
- - - +
=
- +
- - = = =
+
? ?
?
?
?
? ?
?
Ans7.
? ?
.(6 3 2 ) 1 r i j k - - = -
?
? ?
? ?
.( 6 3 2 ) 1....(1)
6 3 2 36 9 4 7
r i j k
i j k
- + + =
- + + = + + =
? ?
?
Dividing equation 1 by 7
? ?
? ? ?
6 3 2 1
.
7 7 7 7
6 3 2
[ .
7 7 7
r i j k
n i j k r n d
- ? ?
+ + =
? ?
? ?
- = + + =
?
?
?
Hence direction cosines of
?
n is
6 3 2
, ,
7 5 7
- Ans8. Comparing the giving eq of the planes with the equations
A 1 x +B 1y +C 1Z + D = 0 , A 2 x + B 2y + C 2 Z + D 2 = 0
A 1 = 3, B 1 = -6, C 1 = 2
A 2 = 2, B 2 = 2, C 2 = -2
1 2 1 2 1 2
2 2 2 2 2
1 12 1 2 2 2
1
cos
10
7 2 3
5 5 3
21 7 3
5 3
21
A A B B C C
A B C A B C
COS
?
?
- + +
=
+ + + +
- =
×
= =
? ?
=
? ?
? ?
? ?
Ans9.
? ? ? ?
1 2
1
2
, 2 3 4
5, 6
n i j k n i j k
d d
= + + = + +
= - =
? ?
Using the relation
? ?
? ?
? ? ? ?
1 2 1 2
.( )
.[(1 2 ) (1 3 ) (1 4 ) ] 6 5 ......(1)
( )[(1 2 ) (1 3 ) (1 4 ) ] 6 5
(1 2 ) (1 3 ) (1 4 ) 6 5
r n n d d
r i j k
taking r xi y j zk
xi y j zk i j k
x y z
? ?
? ? ? ?
? ? ? ?
? ? ? ?
+ = +
+ + + + + = - = + +
+ + + + + + + = - + + + + + = -
?
?
? ?
Page 4
CBSE TEST PAPER-06
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry)
Topic: - Three Dimensional Geometry [ANSWERS]
Ans1. 1,0,0, 0,1,0 0,0,1
Ans2. Let a and b
be the p.v of the points A (-1,0,2) and B ( 3, 4 6)
? ? ?
( )
( 2 ) (4 4 4 )
r a b a
i j i j k
?
?
= + - = - + + + +
? ?
Ans3. Let a 1 = 3, b 1 = 4, c 1 = 5 and a 2 = 4, b 2 = -3, c 2 = 5
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
0
1
cos
2
60
a a b b c c
a b c a b c
?
?
? ?
+ +
? ? = =
? ?
+ + + +
? ?
=
Ans4.
? ? ? ?
5 2 4 , 3 2 8 a i j k b i j k = + - = + -
? ?
Vector equation of line is
? ? ? ?
5 2 4 (3 2 8 )
r a b
i j k i j k
?
?
= +
= + - + + -
? ?
Cartesian equation is
? ? ? ? ? ?
? ? ? ?
5 2 4 (3 2 8 )
(5 3 ) (2 2 ) ( 4 8 )
5 3 , 2 2 , 4 8
5 2 4
3 2 8
xi y j zk i j k i j k
xi y j zk i j k
x y z
x y z
?
? ? ?
? ? ?
+ + = + - + + - + + = + + + + - - = + = + = - - - - +
= =
- ? ? ?
? ?
Ans5. Let ? is the angle between the given lines
? ?
? ?
1
2
2
1
2
1
2
3 5 4
cos
b i j k
and
b i j k
b b
b b
?
= - - = - - ·
=
?
?
? ? ? ?
? ? ? ?
( 2 ) (3 5 4 )
2 3 5 4
3 5 8 16
6 50 50
16
6 5 2
16 3
2 3 5 2 3
816
i j k i j k
i j k i j k
- - · - - - - - - + +
= =
×
× × ×
=
? ?
? ?
3
2
1
3 5
8 3
cos
15
8 3
cos
15
?
?
- × ×
=
? ?
=
? ?
? ?
? ?
Ans6.
? ? ? ?
1 1 2 , a i j k b i j k = + + = - +
? ?
? ? ? ?
( )
? ?
? ?
?
? ? ?
?
2 1
2 1 1 2
1 2
2 1
1 2
2 , 2 2
.( )
3 2
1 1 1
2 1 2
3 3
( 3 2 ).( 3 3
3 3
3 6 9 3
9 9 3 2 2
a i j k b i j k
a a b b
d
b b
a a i j k
i j k
b b
i k
i j k i k
d
i k
= - - = + +
- ×
=
×
- = - - × = - = - +
- - - +
=
- +
- - = = =
+
? ?
?
?
?
? ?
?
Ans7.
? ?
.(6 3 2 ) 1 r i j k - - = -
?
? ?
? ?
.( 6 3 2 ) 1....(1)
6 3 2 36 9 4 7
r i j k
i j k
- + + =
- + + = + + =
? ?
?
Dividing equation 1 by 7
? ?
? ? ?
6 3 2 1
.
7 7 7 7
6 3 2
[ .
7 7 7
r i j k
n i j k r n d
- ? ?
+ + =
? ?
? ?
- = + + =
?
?
?
Hence direction cosines of
?
n is
6 3 2
, ,
7 5 7
- Ans8. Comparing the giving eq of the planes with the equations
A 1 x +B 1y +C 1Z + D = 0 , A 2 x + B 2y + C 2 Z + D 2 = 0
A 1 = 3, B 1 = -6, C 1 = 2
A 2 = 2, B 2 = 2, C 2 = -2
1 2 1 2 1 2
2 2 2 2 2
1 12 1 2 2 2
1
cos
10
7 2 3
5 5 3
21 7 3
5 3
21
A A B B C C
A B C A B C
COS
?
?
- + +
=
+ + + +
- =
×
= =
? ?
=
? ?
? ?
? ?
Ans9.
? ? ? ?
1 2
1
2
, 2 3 4
5, 6
n i j k n i j k
d d
= + + = + +
= - =
? ?
Using the relation
? ?
? ?
? ? ? ?
1 2 1 2
.( )
.[(1 2 ) (1 3 ) (1 4 ) ] 6 5 ......(1)
( )[(1 2 ) (1 3 ) (1 4 ) ] 6 5
(1 2 ) (1 3 ) (1 4 ) 6 5
r n n d d
r i j k
taking r xi y j zk
xi y j zk i j k
x y z
? ?
? ? ? ?
? ? ? ?
? ? ? ?
+ = +
+ + + + + = - = + +
+ + + + + + + = - + + + + + = -
?
?
? ?
? ?
? ?
? ?
( 6) (2 3 4 4 5) 0....(2)
int (1,1,1)
3
14
(1)
3 9 6 15
. 1 1 1 6
7 14 7 14
10 23 13 69
.
7 14 7 14
.(20 23 26 ) 69
x y z x y y z
plane passes through the po
put in eq
r i j k
r i j k
r i j k
?
?
?
+ + - + + + + + =
=
? ? ? ? ? ? ? ?
+ + + + + = - ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ?
+ + =
? ?
? ?
+ + =
?
?
?
Ans10. Given points are A(3,-4,-5)
B(2,-3,1)
Direction ration of AB are 3-2, -4+3, -5-1
1,-1,-6
Eq. of line AB
3 4 5
( )
1 1 6
3, 4, 6 5
( 3, 4, 6 5)
2 7
2( 3) ( 4) ( 6 5) 7
2
(1, 2,7)
x y Z
say
x y Z
let
lies in
the plane x y Z
?
? ? ?
? ? ?
? ? ?
?
- + +
= = =
- - = + = - - = - - + - - - - + + =
+ + - - + - - =
= - - are the required point
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