A motor boat of mass m moving along a lake with velocity V0. At t=0, the engine of the boat is shut down. Magnitude of resistance force offered to the boat is equal to rV. (V is instantaneous speed). What is the total distance covered till it stops completely? Hint: Fx=mVdVdx=-rV

(1)  mV0/r

(2)  3 mV0/2r

(3)  mV0/2r

(4)  2mV0/r

Subtopic:  Integration |
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A particle is moving along positive \(x\text-\)axis. Its position varies as \(x = t^3-3t^2+12t+20,\) where \(x\) is in meters and \(t\) is in seconds. The velocity of the particle when its acceleration zero is:
1. \(1~\text{m/s}\)
2. \(3~\text{m/s}\)
3. \(6~\text{m/s}\)
4. \(9~\text{m/s}\)

Subtopic:  Differentiation |
 71%
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Two forces F1=2i^+2j^ N and F2=3j^+4k^ N are acting on a particle.

The resultant force acting on particle is:

(A)  2i^+5j^+4k^

(B)  2i^-5j^-4k^

(C)  i^-3j^-2k^

(D)  i^-j^-k^

Subtopic:  Resultant of Vectors |
 84%
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A=4i+4j-4k and B=3i+j+4k, then angle between vectors A and B is:

(1)  180°

(2)  90°

(3)  45°

(4)  0°

Subtopic:  Resultant of Vectors |
 77%
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If vectors \(\overrightarrow{{A}}=\cos \omega t \hat{{i}}+\sin \omega t \hat{j}\) and \(\overrightarrow{{B}}=\cos \left(\frac{\omega t}{2}\right)\hat{{i}}+\sin \left(\frac{\omega t}{2}\right) \hat{j}\) are functions of time. Then, at what value of \(t\) are they orthogonal to one another?
1. \(t = \frac{\pi}{4\omega}\)
2. \(t = \frac{\pi}{2\omega}\)
3. \(t = \frac{\pi}{\omega}\)
4. \(t = 0\)

Subtopic:  Scalar Product |
 61%
From NCERT
NEET - 2015
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Six vectors a through f have the magnitudes and directions indicated in the figure. Which of the following statements is true? 

1. b+c=f

2. d+c=f

3. d+e=f

4. b+e=f

Subtopic:  Resultant of Vectors |
 75%
From NCERT
AIPMT - 2010
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\(\overrightarrow{A}\) and \(\overrightarrow B\) are two vectors and \(\theta\) is the angle between them. If \(\left|\overrightarrow A\times \overrightarrow B\right|= \sqrt{3}\left(\overrightarrow A\cdot \overrightarrow B\right),\) then the value of \(\theta\) will be:

1. \(60^{\circ}\) 2. \(45^{\circ}\)
3. \(30^{\circ}\) 4. \(90^{\circ}\)
Subtopic:  Scalar Product | Vector Product |
 80%
From NCERT
AIPMT - 2007
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If a curve is governed by the equation y = sinx, then the area enclosed by the curve and x-axis between x = 0 and x = π is (shaded region):

              
1. \(1\) unit
2. \(2\) units
3. \(3\) units
4. \(4\) units

Subtopic:  Integration |
 59%
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The acceleration of a particle starting from rest varies with time according to relation, a=α t+β. The velocity of the particle at time instant \(t\) is: \(\left(\text{Here,}~ a=\frac{dv}{dt}\right)\)

1. αt2+βt

2. αt2+βt2

3. αt22+βt

4. 2αt2+βt

Subtopic:  Integration |
 85%
From NCERT
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The displacement of the particle is zero at \(t=0\) and at \(t=t\) it is \(x\). It starts moving in the \(x\)-direction with a velocity that varies as \(v = k \sqrt{x}\), where \(k\) is constant. The velocity will: (Here, \(v=\frac{dx}{dt}\))

1. vary with time.
2. be independent of time.
3. be inversely proportional to time.
4. be inversely proportional to acceleration.
Subtopic:  Integration |
 52%
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