A particle is moving along the path y = ${\mathrm{x}}^2$ from x = 0 m to x = 2 m. Then the distance traveled by the particle is:

1.  4 m

2.  $\sqrt{20} \mathrm{m}$

3.  $>\sqrt{20} \mathrm{m}$

4.  $<\sqrt{20} \mathrm{m}$

A vector \(\vec{a}\) is turned without a change in its length through a small angle \(d\theta\). The values of \(\left | \Delta \vec{a} \right |\) and \(\Delta a\) are respectively:
1. $\mathrm{}$\(0,\)  \(ad\theta\)
2. \(ad\theta,\) \(0\)
3. \(0,\) \(0\)
4. none of these

A particle starting from the origin \((0,0)\) moves in a straight line in the \((x,y)\) plane. Its coordinates at a later time are ($\sqrt{3}$, \(3\)). The path of the particle makes an angle of __________ with the x-axis:
1. \(30^\circ\)
2. \(45^\circ\)
3. \(60^\circ\)
4. \(0\)

A man starts from point 'P' and goes 20 m due East, then 5 m due North, then 35 m due West, and finally 25 m due South. His displacement from point 'P' is:

A body moved along x-axis and y-axis according to the equation,

$x=2\sin \theta$
$y=3\cos \theta$

What is the trajectory followed by the body?

1.  

2.  

3.  

4.  

A boy runs on a circular track of radius \(\mathrm{R}\) (in km) with a speed of $\frac{\mathrm{\pi R}}{2}$ km/h in the clockwise direction for \(3\) h and then with $\mathrm{\pi R}$ km/h in the anticlockwise direction for \(1\) h. The magnitude of his displacement will be: 
1.  $\frac{\mathrm{\pi R}}{2}$

2.  $\frac{\mathrm{R}}{\sqrt{2}}$

3.  $\frac{3\mathrm{\pi R}}{2}$

4.  $\sqrt{2}\mathrm{R}$

A body starts moving from rest on a horizontal ground such that the position vector of the body with respect to its starting point is given by $\mathrm{r}$ $=$ $2\mathrm{t}\hat{\mathrm{i}}$ $+$ $3{\mathrm{t}}^2\hat{\mathrm{j}}$. The equation of the trajectory of the body is:

1. y = 1.5 x

2. y = 0.75 ${\mathrm{x}}^2$

3. y = 1.5 ${\mathrm{x}}^2$

4. y = 0.45 ${\mathrm{x}}^2$

Coordinates of a particle as a function of time t are x = 2t, y = 4t. It can be inferred that the path of the particle will be:

1. Straight line

2. Ellipse

3. Parabola

4. Hyperbola

A particle is moving on a circular path of radius \(R.\) When the particle moves from point \(A\) to \(B\) (angle \( \theta\)), the ratio of the distance to that of the magnitude of the displacement will be:

1. $\frac{\mathrm{\theta }}{\sin {\displaystyle \frac{\mathrm{\theta }}{2}}}$

2. $\frac{\mathrm{\theta }}{2\sin {\displaystyle \frac{\mathrm{\theta }}{2}}}$

3. $\frac{\mathrm{\theta }}{2\cos {\displaystyle \frac{\mathrm{\theta }}{2}}}$

4. $\frac{\mathrm{\theta }}{{\displaystyle \cos \frac{\mathrm{\theta }}{2}}}$

A particle starts from (-1, 0) in the x-y plane along a straight path making an angle of 60° with the X-axis. Which of the following will not be the coordinate through which the particle will pass?

(1) (0, $\sqrt{3}$)

(2) (1, 2$\sqrt{3}$)

(3) (2, 3$\sqrt{3}$)

(4) (4, 4$\sqrt{3}$)