1 ) Which pair of following forces will never give resultant of 2N ? A ) 2 N and 2 N , B ) 1 N and 1 N , C ) 1 N and 3 N , D ) 1 N and 4 N ,

1 )  Which pair of following forces will never give resultant of 2N ?

A )  2 N  and  2 N ,                                       B )  1 N  and  1 N ,

C )  1 N  and  3 N ,                                       D )  1 N  and  4 N ,

Correct Answer : -  D )  1 N  and  4 N ,

Solution : -

We have a simple trick to check whether a number is resultant or not : -

| A – B |  <=  R  <=  | A + B |  ,

So , first we put the values of option A ,

| 2 – 2 |  <=  R  <=  | 2 + 2 |  ,

0  <=  R  <=  4 ,

So all the values in between 0 and 4 , also including 0 and 4 are resultants .  So 2 is resultant . This option is incorrect .

Now , we put the values of option D ,

| 1 – 4 |  <=  R  <=  | 1 + 4 |  ,

| – 3 |  <=  R  <=  | 5 |  ,

Now values are under mode , so – ve sign will change into + ve .

3  <=  R  <=  5 ,

So all the values in between 3 and 5 , also including 3 and 5 are resultants .  So 2 is not resultant .  So , D option is correct .

 

2 )  Vector product : -

i x j = k  ,  j x k = i  ,  k x i = j .

 

3 )  Angular momentum is axial vector .

 

4 )  | A • B |  =  | A x B |  : -

A ) 0 degree ,                                                                        B ) 45 degree ,

C ) 90 degree ,                                                                     D ) None of these ,

Correct Answer : -   B ) 45 degree ,

 

5 )  | A + B |  =  | A - B |  : -

A ) 0 degree ,                                                                        B ) 45 degree ,

C ) 90 degree ,                                                                     D ) None of these ,

Correct Answer : -   C ) 90 degree ,

 

6 ) Find angle between these two vectors   - 2i + 3j + k  and  i + 2j – 4k  is : -

A ) 0 degree ,                                                                        B ) 90 degree ,

C ) 180 degree ,                                                                   D ) None of these ,

Correct Answer : -   B ) 90 degree ,

Solution : -

We have formula : -

Cos θ   =   A • B   /   ( | A |  | B | )   . . . . . . . . . . eq. 1

∴     A  =  - 2i + 3j + k   and   B = i + 2j – 4k   ,

∴      A •B   =   ( - 2 ) ( 1 ) + ( 2 ) ( 3 ) + ( - 4 ) ( 1 )   =  0 .

∴      | A |  =  √ {( - 2 ) ^ 2  +  ( 2 ) ^ 2  +  ( - 4 ) ^ 2}   =  √ 14 .

∴      | B |  =  √ {( 1 ) ^ 2  +  ( 2 ) ^ 2  +  ( - 4 ) ^ 2}   =  √ 21 .

So ,  put these all values in eq. 1 ;

Cos θ   =   0  /  ( √ 14 • √ 21 )  ,

Cos θ   =   0 ,

So ,   θ  =  90 °  ,  Answer .