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# Solution- Vectors Test-3 Class 12 Notes | EduRev

## Class 12 : Solution- Vectors Test-3 Class 12 Notes | EduRev

``` Page 1

CBSE TEST PAPER-03
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry)
Ans1. Vector
Ans2. ( ) ( )
2 2
1 2 5 a = + =

( ) ( )
2 2
2 1 5 b = + =

Ans3. D.R of   1,2,3 r are

1 4 9 4 r = + + =

D.C of
1 2 3
, ,
14 14 14
r are

Ans4.
( ) ( )
3 5 . 2 7 a b a b - +
   
2 2
2 2
6 11 . 35
. . . .
a a b b
a a a b b b a b b a
= - - ?
= = =
?
?
   
         
?
Ans5.
? ?
2 1 3
3 5 2
i j k
a b × =
- ?
 
( )
?
( )
?
( )
? ?
2 15 4 9 10 3
17 13 7
i j k
i j k
= - - - - - + - = - + +
?
?
Ans6. A (1, -2, -8), B (5, 0, -2), C (11, 3, 7)
? ?
? ?
? ?
1 2 8
5 0 2
11 3 7
OA i j k
OB i j k
OC i j k
= - - = - - = + +
 
?
 
?

?
? ?
4 2 6
AB OB OA
i j k
= - = + +
     
?
Page 2

CBSE TEST PAPER-03
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry)
Ans1. Vector
Ans2. ( ) ( )
2 2
1 2 5 a = + =

( ) ( )
2 2
2 1 5 b = + =

Ans3. D.R of   1,2,3 r are

1 4 9 4 r = + + =

D.C of
1 2 3
, ,
14 14 14
r are

Ans4.
( ) ( )
3 5 . 2 7 a b a b - +
   
2 2
2 2
6 11 . 35
. . . .
a a b b
a a a b b b a b b a
= - - ?
= = =
?
?
   
         
?
Ans5.
? ?
2 1 3
3 5 2
i j k
a b × =
- ?
 
( )
?
( )
?
( )
? ?
2 15 4 9 10 3
17 13 7
i j k
i j k
= - - - - - + - = - + +
?
?
Ans6. A (1, -2, -8), B (5, 0, -2), C (11, 3, 7)
? ?
? ?
? ?
1 2 8
5 0 2
11 3 7
OA i j k
OB i j k
OC i j k
= - - = - - = + +
 
?
 
?

?
? ?
4 2 6
AB OB OA
i j k
= - = + +
     
?
? ?
( )
3 2 3
BC OC OB
i j k
= - = + +
   
?
? ?
( )
3
4 2 6
2
3
2
i j k
BC AB
= + +
=
?
  
Thus BC AB
  
 and one point B is common there fore A, B, C are collinear.
Ans7.
? ?
4 2 a i j k = + +

?
? ?
? ?
3 2 7
2 4
b i j k
c i j k
= - +
= - +

?

?
? ?

. 0, . 0  . 15
Let d xi y j zk
ATQ d a d b and c d
= + +
= = =
 
?
      
4 2 0 (1)
3 2 7 0 (2)
2 4 15 (3)
x y z
x y z
x y z
+ + = - - - - - - - - - + = - - - - - - - - + = - - - - - - - On solving equation (i) and (ii)
28 4 6 7 2 12
32 , , 14
x y z
K
x k y k z k
= = =
+ - - - = = - = - Put x, y, z in equation (iii)
( ) ( ) ( ) 2 32 4 14 15
64 56 15
9 15
k k k
k k k
k
- - + - =
+ - =
=
15
9
5
3
k
k
=
=
5 160
32
3 3
x = × =
Page 3

CBSE TEST PAPER-03
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry)
Ans1. Vector
Ans2. ( ) ( )
2 2
1 2 5 a = + =

( ) ( )
2 2
2 1 5 b = + =

Ans3. D.R of   1,2,3 r are

1 4 9 4 r = + + =

D.C of
1 2 3
, ,
14 14 14
r are

Ans4.
( ) ( )
3 5 . 2 7 a b a b - +
   
2 2
2 2
6 11 . 35
. . . .
a a b b
a a a b b b a b b a
= - - ?
= = =
?
?
   
         
?
Ans5.
? ?
2 1 3
3 5 2
i j k
a b × =
- ?
 
( )
?
( )
?
( )
? ?
2 15 4 9 10 3
17 13 7
i j k
i j k
= - - - - - + - = - + +
?
?
Ans6. A (1, -2, -8), B (5, 0, -2), C (11, 3, 7)
? ?
? ?
? ?
1 2 8
5 0 2
11 3 7
OA i j k
OB i j k
OC i j k
= - - = - - = + +
 
?
 
?

?
? ?
4 2 6
AB OB OA
i j k
= - = + +
     
?
? ?
( )
3 2 3
BC OC OB
i j k
= - = + +
   
?
? ?
( )
3
4 2 6
2
3
2
i j k
BC AB
= + +
=
?
  
Thus BC AB
  
 and one point B is common there fore A, B, C are collinear.
Ans7.
? ?
4 2 a i j k = + +

?
? ?
? ?
3 2 7
2 4
b i j k
c i j k
= - +
= - +

?

?
? ?

. 0, . 0  . 15
Let d xi y j zk
ATQ d a d b and c d
= + +
= = =
 
?
      
4 2 0 (1)
3 2 7 0 (2)
2 4 15 (3)
x y z
x y z
x y z
+ + = - - - - - - - - - + = - - - - - - - - + = - - - - - - - On solving equation (i) and (ii)
28 4 6 7 2 12
32 , , 14
x y z
K
x k y k z k
= = =
+ - - - = = - = - Put x, y, z in equation (iii)
( ) ( ) ( ) 2 32 4 14 15
64 56 15
9 15
k k k
k k k
k
- - + - =
+ - =
=
15
9
5
3
k
k
=
=
5 160
32
3 3
x = × =
? ?
5 5
3 3
5 70
14
3 3
160 5 70
3 3 3
y
z
d i j k
= - = - = - × = - = - -  
?
Ans8.
( ) ( ) ( )
( ) . 0, . 0 . 0, a b c b c a c a b Given + = + = + =
        
( ) ( )
( ) ( ) ( )
2
.
. . . . .
a b c a b c a b c
a a a b c b b b a c c c a b
+ + = + + + +
= + + + + + + + +
        
            
2 2 2
9 16 25
a b c = + +
= + +
  
50
50
5 2
a b c
=
+ + =
=
  
Ans9.
? ?
9 4 4 a b i j k + = + -  
?
?
( ) ( )
ˆ
2
. 17
a b i k
a b a b
- = - +
+ - = -  
   
113
5
a b
a b
+ =
- =
 
 
( ) ( )
.
cos
17
113. 5
a b a b
a b a b
?
+ - =
+ - - =
   
   
1
17
cos
565
17
cos
565
?
?
- - =
- ? ?
=
? ?
? ?
Ans10.
? ?
2 1 3
1 3 2
i j k
a b × = - ?
 
? ?
( ) ( ) ( )
2 2 2
11 7
11 1 7
i j k
a b
= - - +
× = - + - +
?
 
171 3 19
3 19 3
sin 19
14 14. 14
a b
a b
?
= =
×
= = =
 
 
```
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