11. The position vector of a particle is r^→ = (a cos ωt)î + (a sin ωt)ĵ , The velocity of the particle is
    

1. 1. directed towards the origin

   
   
   

   2. ​directed away from the origin ​

   
   
   

   3. parallel to the position vector

   
   
   

   4. perpendicular to the position vector

   
   
   

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1. View Hint View Answer Discuss in Forum

   r^→ = (a cos ωt)î + (a sin ωt)ĵ
   

   v→ =	dr→	=	d	{ (a cos ωt)î + (a sin ωt)ĵ }
   	dt		dt

   
   = (-aω sin ωt)î + (aω cos ωt)ĵ
   = ω [ (-a sin ωt)î + (a cos ωt)ĵ ]
   

   Slope of position vector =	a sin ωt	= tan ωt
   	a cos ωt

   
   

   & slope of velocity vector =	-a cos ωt	=	-1
   	a sin ωt		tan ωt

   
   ∴ velocity is perpendicular to the displacement.
   

   Correct Option: D

   r^→ = (a cos ωt)î + (a sin ωt)ĵ
   

   v→ =	dr→	=	d	{ (a cos ωt)î + (a sin ωt)ĵ }
   	dt		dt

   
   = (-aω sin ωt)î + (aω cos ωt)ĵ
   = ω [ (-a sin ωt)î + (a cos ωt)ĵ ]
   

   Slope of position vector =	a sin ωt	= tan ωt
   	a cos ωt

   
   

   & slope of velocity vector =	-a cos ωt	=	-1
   	a sin ωt		tan ωt

   
   ∴ velocity is perpendicular to the displacement.
   

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12. Two particles A and B are connected by a rigid rod AB. The rod slides along perpendicular rails as shown here. The velocity of A to the left is 10 m/s. What is the velocity of B when angle α = 60° ?​​​
    
    

1. 1. ​5.8 m/s

   
   
   

   2. 9.8 m/s

   
   
   

   3. 10 m/s

   
   
   

   4. 17.3 m/s

   
   
   

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1. View Hint View Answer Discuss in Forum

   Let after 1 sec angle become 60°. When the end A moves by 10 m left, the end B moves upward by BB′ = 10 × √3 = 10 × 1.73 = 17.3 m / s
   
   

   Correct Option: D

   Let after 1 sec angle become 60°. When the end A moves by 10 m left, the end B moves upward by BB′ = 10 × √3 = 10 × 1.73 = 17.3 m / s
   
   

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13. If a unit vector is represented by 0.5î + 0.8ĵ + ck̂, the value of c is​
    

1. 1. 1

   
   
   

   2. √0.11

   
   
   

   3. √0.01

   
   
   

   4. 0.39

   
   
   

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1. View Hint View Answer Discuss in Forum

   r̂ = 0.5î + 0.8ĵ + ck̂
   | r̂ | = 1 = √(0.5)² + (0.8)² + c²
   (0.5)² + (0.8)² + c² = 1
   c² = 0.11 ⇒ c = √0.11
   

   Correct Option: B

   r̂ = 0.5î + 0.8ĵ + ck̂
   | r̂ | = 1 = √(0.5)² + (0.8)² + c²
   (0.5)² + (0.8)² + c² = 1
   c² = 0.11 ⇒ c = √0.11
   

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14. A particle moves in a plane with constant acceleration in a direction different from the initial velocity. The path of the particle is​​
    

1. 1. an ellipse

   
   
   

   2. ​a parabola

   
   
   

   3. an arc of a circle

   
   
   

   4. a straight line

   
   
   

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1. View Hint View Answer Discuss in Forum

   NA

   Correct Option: B

   NA

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15. A particle starting from the origin (0, 0) moves in the (x, y) plane. Its coordinates at a later time are (√3 , 3). The path of the particle makes with the x-axis an angle of
    

1. 1. 45°

   
   
   

   2. 60°

   
   
   

   3. 0°

   
   
   

   4. 30°

   
   
   

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1. View Hint View Answer Discuss in Forum

   
   Let θ be the angle which the particle makes with x axis. ​
   

   From figure, tan θ =	3	= √3
   	√3

   
   ⇒ θ = tan ^-1 (√3) = 60°

   Correct Option: B

   
   Let θ be the angle which the particle makes with x axis. ​
   

   From figure, tan θ =	3	= √3
   	√3

   
   ⇒ θ = tan ^-1 (√3) = 60°

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