If the velocity of a particle is \(\mathrm{v}=\mathrm{At}+\mathrm{Bt^{2}},\) where \(\mathrm{A}\) and \(\mathrm{B}\) are constants, then the distance travelled by it between \(1\) s and \(2\) s is:

1. $3\mathrm{A}+7\mathrm{B}$

2. $\frac{3}{2}\mathrm{A}+\frac{7}{3}\mathrm{B}$

3. $\frac{\mathrm{A}}{2}+\frac{\mathrm{B}}{3}$

4. $\frac{3}{2}\mathrm{A}+4\mathrm{B}$

A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to $\mathrm{v}\left(\mathrm{x}\right)$ $={\mathrm{\beta x}}^{-2\mathrm{n}}$ where $\mathrm{\beta }$ and \(\mathrm{n}\) are constants and \(\mathrm{x}\) is the position of the particle. The acceleration of the particle as a function of \(\mathrm{x}\) is given by:
1. $-2{\mathrm{n\beta }}^2{\mathrm{x}}^{-2\mathrm{n}-1}$
2. $-2{\mathrm{n\beta }}^2{\mathrm{x}}^{-4\mathrm{n}-1}$
3. $-2{\mathrm{\beta }}^2{\mathrm{x}}^{-2\mathrm{n}+1}$
4. $-2{\mathrm{n\beta }}^2{\mathrm{x}}^{-4\mathrm{n}+1}$

A car moves from \(\mathrm{X}\) to \(\mathrm{Y}\) with a uniform speed \(\mathrm{v_u}\) and returns to \(\mathrm{X}\) with a uniform speed \(\mathrm{v_d}.\) The average speed for this round trip is:

1. $\frac{2{\mathrm{v}}_{\mathrm{d}}{\mathrm{v}}_{\mathrm{u}}}{{\mathrm{v}}_{\mathrm{d}}+{\mathrm{v}}_{\mathrm{u}}}$

2. $\sqrt{{\mathrm{v}}_{\mathrm{u}}{\mathrm{v}}_{\mathrm{d}}}$

3. $\frac{{\mathrm{v}}_{\mathrm{d}}{\mathrm{v}}_{\mathrm{u}}}{{\mathrm{v}}_{\mathrm{d}}+{\mathrm{v}}_{\mathrm{u}}}$

4. $\frac{{\mathrm{v}}_{\mathrm{u}}+{\mathrm{v}}_{\mathrm{d}}}{2}$

A particle moves along a straight line and its position as a function of time is given by $\mathrm{x}={\mathrm{t}}^3-3{\mathrm{t}}^2+3\mathrm{t}+3$, then the particle: 

The coordinate of an object is given as a function of time by $\mathrm{x}=7\mathrm{t}-3{\mathrm{t}}^2$, where x is in metres and t is in seconds. Its average velocity over the interval t=0 to t=4 is will be:

1.  5 m/s

2.  -5 m/s

3.  11 m/s

4.  -11 m/s

The displacement time graph of a moving particle is shown in the figure below. The instantaneous velocity of the particle is negative at the point:

A boy standing at the top of a tower of 20 m height drops a stone. Assuming g=$10 {\mathrm{ms}}^{-2}$, the velocity with which it hits the ground will be:
1. 20 m/s                                         
2. 40 m/s
3. 5 m/s                                           
4. 10 m/s

A particle moves along a path \(ABCD\) as shown in the figure. The magnitude of the displacement of the particle from \(A\) to \(D\) is:

1. $(5+10\sqrt{2})$m
2. \(10\) m
3. $15\sqrt{2}$ m
4. \(15\) m

Which of the following four statements is false?

Two cars A and B are travelling in the same direction with velocities v₁ and v₂ $(v_1>v_2)$. When the car A is at a distance d behind car B, the driver of the car A applied the brake producing uniform retardation a. There will be no collision when-

1. $d<\frac{{(v_1-v_2)}^2}{2a}$

2. $d<\frac{v_1^2-v_2^2}{2a}$

3. $d>\frac{{(v_1-v_2)}^2}{2a}$

4. $d>\frac{v_1^2-v_2^2}{2a}$