A particle starting from the point (1, 2) moves in a straight line in the XY plane. Its coordinates at a later time are (2, 3). The path of the particle makes, with the x-axis, an angle of

1. 30° 

2. 45°

3. 60° 

4. data insufficient

A particle is projected from a horizontal plane (x-z plane) such that its velocity vector at time t is given by $\overrightarrow{\mathrm{v}}=a\hat{i}+\left(b-ct\right)\hat{j}$. Its range on this horizontal plane is given by 

1. $\frac{\mathrm{ba}}{\mathrm{c}}$

2. $\frac{2\mathrm{ba}}{\mathrm{c}}$

3. $\frac{3\mathrm{ba}}{\mathrm{c}}$

4. None 

A man is walking on the road with a speed 3 m/s. Rain is falling vertically at speed 3 m/s. At what angle from
the vertical, man has to hold his umbrella to avoid the rain drops?
1. ${45}^{\mathrm{o}}$ 

2. 30^o

3. 60^o 

4. 90^o

A river is 1 km wide. The banks are straight and parallel. The current is 5 km/h and is parallel to the banks. A boat has a maximum speed of 3 km/h in still water. In what direction should the boat head so as to arrive at point B directly opposite to its starting point A?

1. directly across the river

2. head $53°$ upstream from the line AB

3. head $37° \mathrm{upstream} \mathrm{from} \mathrm{the} \mathrm{line} \mathrm{AB}$

4. the trip from A to B is not possible with this speed

Two particles A and B, move with constant velocities \(\vec{v_1}\)   and  \(\vec{v_2}\) . At the initial moment their position vector are  \(\vec{r_1}\) and  \(\vec{r_2}\)  respectively. The condition for particles A and B for their collision to happen will be

1.  $\overrightarrow{{\mathrm{r}}_{1 }}.\overrightarrow{{\mathrm{v}}_1} =\overrightarrow{{\mathrm{r}}_{2 }}.\overrightarrow{{\mathrm{v}}_2}$

2.  $\overrightarrow{{\mathrm{r}}_1} \mathrm{x} \overrightarrow{{\mathrm{v}}_1} =\overrightarrow{{\mathrm{r}}_2} \mathrm{x} \overrightarrow{{\mathrm{v}}_2}$

3. $\overrightarrow{{\mathrm{r}}_1}- \stackrel{\rightharpoonup }{{\mathrm{r}}_2} =\overrightarrow{{\mathrm{v}}_1}- \overrightarrow{{\mathrm{v}}_2}$

4. $\frac{\overrightarrow{{\mathrm{r}}_1}- \overrightarrow{{\mathrm{r}}_2}}{\left|\overrightarrow{{\mathrm{r}}_1}- \overrightarrow{{\mathrm{r}}_2}\right|}= \frac{\overrightarrow{{\mathrm{v}}_2}- \overrightarrow{{\mathrm{v}}_1}}{\left|\overrightarrow{{\mathrm{v}}_2}- \overrightarrow{{\mathrm{v}}_1}\right|}$

A particle moves along a parabolic path y = 9x² in such a way that the x component of the velocity remains constant and has a value of $\frac{1}{3} {\mathrm{ms}}^{-1}$. It can be deduced that the acceleration of the particle will be

1. $\frac{1}{3} \hat{\mathrm{j}} {\mathrm{ms}}^{-2}$

2. $3 \hat{\mathrm{j}} {\mathrm{ms}}^{-2}$

3. $\frac{2}{3} \hat{\mathrm{j}} {\mathrm{ms}}^{-2}$

4. $2 \hat{\mathrm{j}} {\mathrm{ms}}^{-2}$

Three balls are thrown from the top of a building with equal speeds at different angles. When the balls strike the ground, their speeds are ${\mathrm{v}}_1,{\mathrm{v}}_2 \mathrm{and} {\mathrm{v}}_3$ respectively, then:
           

1. ${\mathrm{v}}_1>{\mathrm{v}}_2>{\mathrm{v}}_3$

2. ${\mathrm{v}}_3>{\mathrm{v}}_2={\mathrm{v}}_1$

3. ${\mathrm{v}}_1={\mathrm{v}}_2={\mathrm{v}}_3$

4. ${\mathrm{v}}_1<{\mathrm{v}}_2<{\mathrm{v}}_3$

Figure below shows a body of mass M moving with a uniform speed v on a circular path of radius, R. What is the change in acceleration in going from P₁ to P₂? 
              

1. zero

2. ${\mathrm{v}}^2/2\mathrm{R}$

3. $2{\mathrm{v}}^2/\mathrm{R}$

4. $\frac{{\mathrm{v}}^2}{\mathrm{R}}\times \sqrt{2}$

A particle is projected with a speed u at an angle $\mathrm{\theta }$ to the horizontal. Radius of curvature at highest point of its trajectory is

1. $\frac{{\mathrm{u}}^2{\cos }^2\mathrm{\theta }}{2\mathrm{g}}$

2. $\frac{\sqrt{3}{\mathrm{u}}^2{\cos }^2\mathrm{\theta }}{2\mathrm{g}}$

3. $\frac{{\mathrm{u}}^2{\cos }^2\mathrm{\theta }}{\mathrm{g}}$

4. $\frac{\sqrt{3}{\mathrm{u}}^2{\cos }^2\mathrm{\theta }}{\mathrm{g}}$

Two particles A and B are moving in a uniform circular motion in concentric circles of radii $r_A and r_B$ with speeds $v_A and v_B$ respectively. Their time periods of rotation are the same. The ratio of angular speed of A  to that of B will be:

1. 1: 1

2. $r_A : r_B$

3. $v_A : v_B$

4. $r_B : r_A$