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# Solution- Vectors Test-2 Class 12 Notes | EduRev

## Class 12 : Solution- Vectors Test-2 Class 12 Notes | EduRev

``` Page 1

CBSE TEST PAPER-02
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry)
Ans1. Vector
Ans2.
? ?
2 3 a i j k = + +

?
? ?
1 4 9 14
3 2 1
9 4 1 14
a
b i j k
b
= + + =
= + +
= + + =


?

Ans3. D.R of   1,1,1 r are


1 1 1 3
1 1 1
.  , ,
3 3 3
r
D C of are
= + + =

Ans4.
( ) ( )
2
. a b a b a b - = - -      
2
. . . .
2 .
a a a b b a b b
a a b b
= - - +
= - +
       
   
4 2 4 9
5
5 a b
= - × +
=
- =
 
Ans5.
?
3 2 b i k = +

?
? ?
2 8 6 4 a i j k = + +

?
Page 2

CBSE TEST PAPER-02
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry)
Ans1. Vector
Ans2.
? ?
2 3 a i j k = + +

?
? ?
1 4 9 14
3 2 1
9 4 1 14
a
b i j k
b
= + + =
= + +
= + + =


?

Ans3. D.R of   1,1,1 r are


1 1 1 3
1 1 1
.  , ,
3 3 3
r
D C of are
= + + =

Ans4.
( ) ( )
2
. a b a b a b - = - -      
2
. . . .
2 .
a a a b b a b b
a a b b
= - - +
= - +
       
   
4 2 4 9
5
5 a b
= - × +
=
- =
 
Ans5.
?
3 2 b i k = +

?
? ?
2 8 6 4 a i j k = + +

?
? ?
2 3 0 2
8 6 4
i j k
b a × =
?
 
( )
?
( )
?
( ) 0 12 12 16 18 0 i j k = - - - + - ?
? ?
( ) ( ) ( )
2 2 2
12 4 18
2 12 4 18
i j k
b a
- + +
× = - + +
?
 
484
22
=
=
Ans6. (i)
( ) ( )
2 1 3 2
2 1
a b a b
OR
+ + - =
+
   
 
5
3
a
=

(ii)
( ) ( )
2 3 2
2 1
a b a b
OR
+ - - =
-    
 
2 2 3 2
1
4
a b a b
b a
+ - +
=
= -    
 
Ans7. 2 3 5 OA a b c = - + +
    
2 3
7
3 2
OB a b c
OC a c
OA OB OA a b c
= + +
= - = - = - -     
  
        
( )
6 2 4
2 3 2
2
BC OC OB a b c
a b c
BC AB
= - = - - = - - =
       
  
   
Thus || AB BC
   
but one point B is common to both vectors hence A, B, C are collinear.
Page 3

CBSE TEST PAPER-02
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry)
Ans1. Vector
Ans2.
? ?
2 3 a i j k = + +

?
? ?
1 4 9 14
3 2 1
9 4 1 14
a
b i j k
b
= + + =
= + +
= + + =


?

Ans3. D.R of   1,1,1 r are


1 1 1 3
1 1 1
.  , ,
3 3 3
r
D C of are
= + + =

Ans4.
( ) ( )
2
. a b a b a b - = - -      
2
. . . .
2 .
a a a b b a b b
a a b b
= - - +
= - +
       
   
4 2 4 9
5
5 a b
= - × +
=
- =
 
Ans5.
?
3 2 b i k = +

?
? ?
2 8 6 4 a i j k = + +

?
? ?
2 3 0 2
8 6 4
i j k
b a × =
?
 
( )
?
( )
?
( ) 0 12 12 16 18 0 i j k = - - - + - ?
? ?
( ) ( ) ( )
2 2 2
12 4 18
2 12 4 18
i j k
b a
- + +
× = - + +
?
 
484
22
=
=
Ans6. (i)
( ) ( )
2 1 3 2
2 1
a b a b
OR
+ + - =
+
   
 
5
3
a
=

(ii)
( ) ( )
2 3 2
2 1
a b a b
OR
+ - - =
-    
 
2 2 3 2
1
4
a b a b
b a
+ - +
=
= -    
 
Ans7. 2 3 5 OA a b c = - + +
    
2 3
7
3 2
OB a b c
OC a c
OA OB OA a b c
= + +
= - = - = - -     
  
        
( )
6 2 4
2 3 2
2
BC OC OB a b c
a b c
BC AB
= - = - - = - - =
       
  
   
Thus || AB BC
   
but one point B is common to both vectors hence A, B, C are collinear.
Ans8. A vector which is ? to both
( ) ( )
a b are a b + -    
is giving by
( ) ( )
? ?
2 3 4
0 1 2
i j k
a b a b + × - =
- - ?
   
? ?
? ?
2 4 2
2 4 2
i j k
Let c i j k
= - + - = - + - ?

?
4 16 4
24
2 6
c = + +
=
=

Req. unit vector is
? ?
1 2 1
6 6 6
c
i j k
c
= - + - 
?

Ans9.
? ?
2 4 5 a i j k = + - 
?
? ?
( )
? ?
3
2 6 2
b i j k
a b i j k
?
?
= + +
+ = + + - 
?
 
?
Unit vector along
( )
? ?
( ) ( )
( )
? ?
( )
2 2
2
2
2 6 2
2 6 + (-2)
2 6 2
2 40
a b
a b
a b
i j k
i j k
?
?
?
?
+
+ =
+
+ + - =
+ +
+ + - =
+ +
 
 
 
?
?
( )
? ?
( )
( )
? ?
( )
2
. 1
2 6 2
. 1
2 40
ATQ c a b
i j k
i j k
?
?
+ =
? ?
+ + - + + = ? ?
? ?
+ +
? ?
  
?
?
Page 4

CBSE TEST PAPER-02
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry)
Ans1. Vector
Ans2.
? ?
2 3 a i j k = + +

?
? ?
1 4 9 14
3 2 1
9 4 1 14
a
b i j k
b
= + + =
= + +
= + + =


?

Ans3. D.R of   1,1,1 r are


1 1 1 3
1 1 1
.  , ,
3 3 3
r
D C of are
= + + =

Ans4.
( ) ( )
2
. a b a b a b - = - -      
2
. . . .
2 .
a a a b b a b b
a a b b
= - - +
= - +
       
   
4 2 4 9
5
5 a b
= - × +
=
- =
 
Ans5.
?
3 2 b i k = +

?
? ?
2 8 6 4 a i j k = + +

?
? ?
2 3 0 2
8 6 4
i j k
b a × =
?
 
( )
?
( )
?
( ) 0 12 12 16 18 0 i j k = - - - + - ?
? ?
( ) ( ) ( )
2 2 2
12 4 18
2 12 4 18
i j k
b a
- + +
× = - + +
?
 
484
22
=
=
Ans6. (i)
( ) ( )
2 1 3 2
2 1
a b a b
OR
+ + - =
+
   
 
5
3
a
=

(ii)
( ) ( )
2 3 2
2 1
a b a b
OR
+ - - =
-    
 
2 2 3 2
1
4
a b a b
b a
+ - +
=
= -    
 
Ans7. 2 3 5 OA a b c = - + +
    
2 3
7
3 2
OB a b c
OC a c
OA OB OA a b c
= + +
= - = - = - -     
  
        
( )
6 2 4
2 3 2
2
BC OC OB a b c
a b c
BC AB
= - = - - = - - =
       
  
   
Thus || AB BC
   
but one point B is common to both vectors hence A, B, C are collinear.
Ans8. A vector which is ? to both
( ) ( )
a b are a b + -    
is giving by
( ) ( )
? ?
2 3 4
0 1 2
i j k
a b a b + × - =
- - ?
   
? ?
? ?
2 4 2
2 4 2
i j k
Let c i j k
= - + - = - + - ?

?
4 16 4
24
2 6
c = + +
=
=

Req. unit vector is
? ?
1 2 1
6 6 6
c
i j k
c
= - + - 
?

Ans9.
? ?
2 4 5 a i j k = + - 
?
? ?
( )
? ?
3
2 6 2
b i j k
a b i j k
?
?
= + +
+ = + + - 
?
 
?
Unit vector along
( )
? ?
( ) ( )
( )
? ?
( )
2 2
2
2
2 6 2
2 6 + (-2)
2 6 2
2 40
a b
a b
a b
i j k
i j k
?
?
?
?
+
+ =
+
+ + - =
+ +
+ + - =
+ +
 
 
 
?
?
( )
? ?
( )
( )
? ?
( )
2
. 1
2 6 2
. 1
2 40
ATQ c a b
i j k
i j k
?
?
+ =
? ?
+ + - + + = ? ?
? ?
+ +
? ?
  
?
?
( )
( )
( )
2
2
2 6 2
1
2 40
2 4 2 40
?
?
? ?
+ + - =
+ +
+ + = + +
( )
2
2
.
36 12 2 40
1
sq both site
? ? ?
?
+ + = + +
=

Ans10. A (1, 1, 2) B(2, 3, 5) C (1, 5, 5)
? ?
2 OA i j k = + +
 
?
? ?
? ?
2 3 5
5 5
OB i j k
OC i j k
= + +
= + +
 
?

?
? ?
? ?
2 2
4 3
OA OB OA i j k
AC OC OA j k
= - = + +
= - = +
     
?
   
Ë† Ë† Ë†
1 2 2
0 4 3
Ë† Ë† Ë†
2 3 4
i j k
AB AC
i j k
× =
= - - +
  
( ) ( ) ( )
2 2 2
1

2
1
2 3 4
2
1
29 .
2
Arae of ABC AB AC
sq unit
? = ×
= - + - +
=
  
```
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