A particle is moving along positive x-axis. Its position varies as $\mathrm{x}={\mathrm{t}}^{3}-3{\mathrm{t}}^{2}+12\mathrm{t}+20$, where x is in meters and t is in seconds.

Initial acceleration of the particle is

(A)  Zero

(B)

(C)

(D)

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Differentiation
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A particle is moving along positive x-axis. Its position varies as $\mathrm{x}={\mathrm{t}}^{3}-3{\mathrm{t}}^{2}+12\mathrm{t}+20$, where x is in meters and t is in seconds.

Velocity of the particle when its acceleration zero is

(A)  1 m/s

(B)  3 m/s

(C)  6 m/s

(D)  9 m/s

Concept Questions :-

Differentiation
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Two forces  and  are acting on a particle.

The resultant force acting on particle is:

(A)  $2\stackrel{^}{\mathrm{i}}+5\stackrel{^}{\mathrm{j}}+4\stackrel{^}{\mathrm{k}}$

(B)  $2\stackrel{^}{\mathrm{i}}-5\stackrel{^}{\mathrm{j}}-4\stackrel{^}{\mathrm{k}}$

(C)  $\stackrel{^}{\mathrm{i}}-3\stackrel{^}{\mathrm{j}}-2\stackrel{^}{\mathrm{k}}$

(D)  $\stackrel{^}{\mathrm{i}}-\stackrel{^}{\mathrm{j}}-\stackrel{^}{\mathrm{k}}$

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Two forces  and  are acting on a particle.

The angle between  is:

(A)  $\mathrm{\theta }={\mathrm{cos}}^{-1}\left(\frac{3}{2\sqrt{5}}\right)$

(B)  $\mathrm{\theta }={\mathrm{cos}}^{-1}\left(\frac{3}{5\sqrt{2}}\right)$

(C)  $\mathrm{\theta }={\mathrm{cos}}^{-1}\left(\frac{2}{3\sqrt{5}}\right)$

(D)  $\mathrm{\theta }={\mathrm{cos}}^{-1}\left(\frac{\sqrt{3}}{5}\right)$

Concept Questions :-

Resultant of Vectors
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Two forces  and  are acting on a particle.

The magnitude of the component of force ${\stackrel{\to }{\mathrm{F}}}_{1}$ along force ${\stackrel{\to }{\mathrm{F}}}_{2}$ is:

(A)  $\frac{5}{6}$

(B)  $\frac{5}{3}$

(C)  $\frac{6}{5}$

(D)  $\frac{5}{2}$

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Resultant of Vectors
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$\mathrm{A}=4\mathrm{i}+4\mathrm{j}-4\mathrm{k}$ and $\mathrm{B}=3\mathrm{i}+\mathrm{j}+4\mathrm{k}$, then angle between vectors A and B is:

(A)  $180°$

(B)  $90°$

(C)  $45°$

(D)  $0°$

Concept Questions :-

Resultant of Vectors
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A curve is governed by the equation y=sinx, then what is the area enclosed by the curve and x-axis between x =0 and x =$\mathrm{\pi }$ is (shaded region)

1. 1 units

2. 2 units

3. 3 units

4. 4 units

Concept Questions :-

Integration
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Calculate the area of disk of radius 'a' using integration

1. $\frac{{\mathrm{\pi a}}^{2}}{2}$

2. ${\mathrm{\pi a}}^{2}$

3. $\frac{3}{2{\mathrm{\pi a}}^{2}}$

4. $2{\mathrm{\pi a}}^{2}$

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Integration
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The acceleration of a particle starting from rest varies with time according to relation, . Find the velocity of the particle at time instant t.

1. $\alpha {t}^{2}+\beta t$

2. $\alpha {t}^{2}+\frac{\beta t}{2}$

3. $\frac{\alpha {t}^{2}}{2}+\beta t$

4. $2\alpha {t}^{2}+\beta t$

Concept Questions :-

Integration
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The displacement of particle is zero at t=0 and at t=t it is x. It starts moving in the x direction with velocity, which varies as $v=k\sqrt{x}$, where k is constant. The velocity-

1. varies with time

2. is independent to time

3. inversely proportional to time

4. inversely proportional to acceleration

Concept Questions :-

Integration
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