If rain falls vertically with a velocity ${\mathrm{V}}_{\mathrm{r}}$ wrt wind and wind blows with a velocity ${\mathrm{V}}_{\mathrm{w}}$ from east to west, then a person standing on the roadside should hold the umbrella in the direction

1. $\tan <mspace\; width="1em"></mspace>\theta <mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{V_w}{V_r}$

2. $\tan <mspace\; width="1em"></mspace>\theta \mathit{\hspace{0.17em}}\mathit{=}\mathit{\hspace{0.17em}}\frac{{\mathit{V}}_{\mathit{r}}}{{\mathit{V}}_{\mathit{w}}}$

3. $\tan <mspace\; width="1em"></mspace>\theta \mathit{\hspace{0.17em}}\mathit{=}\mathit{\hspace{0.17em}}\frac{{\mathit{V}}_{\mathit{r}\mathit{w}}}{\sqrt{\mathit{V}{\mathit{r}}^{\mathit{2}}\mathit{\hspace{0.17em}}\mathit{+}\mathit{\hspace{0.17em}}\mathit{V}{\mathit{w}}^{\mathit{2}}}}$

4. $\tan <mspace\; width="1em"></mspace>\theta \mathit{\hspace{0.17em}}\mathit{=}\mathit{\hspace{0.17em}}\frac{{\mathit{V}}_{\mathit{r}}}{\sqrt{\mathit{V}{\mathit{r}}^{\mathit{2}}\mathit{\hspace{0.17em}}\mathit{+}\mathit{\hspace{0.17em}}\mathit{V}{\mathit{w}}^{\mathit{2}}}}$

The component of vector $\overrightarrow{A}=a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$ along the direction of $\hat{i}-\hat{j}$ is

(1) $a_x-a_y+a_z$

(2) $a_x-a_y$

(3) $(a_x-a_y)/\sqrt{2}$

(4) $(a_x+a_y+a_z)$

A force is \(60^{\circ}\) inclined to the horizontal. If its rectangular component in the horizontal direction is \(50\) N, then the magnitude of the force in the vertical direction is:

Calculate the electric field on the axis of a very long uniformly charged, thin rod at a distance r from one end. The charge per unit length of the rod is $\lambda$. [This question includes concepts from 12th syllabus]

1. $\frac{2k\lambda }{r}$

2. $\frac{k\lambda }{r}$

3. $\frac{k\lambda }{2r}$

4. $\frac{k\lambda }{4r}$

Electric field is given by E $=\frac{100}{x^2}$ . Find the potential difference between x= 10 and x= 20 m. [This question includes concepts from 12th syllabus]

1. 5 V

2. 10 V

3. 15 V

4. 20 V

The gravitational field due to a mass distribution is given by $\overrightarrow{I}=<mspace\; width="1em"></mspace>\frac{k}{x^2}\hat{i}$, where k is a constant. Assuming the potential to be zero at infinity, find the potential at a point x = a.[This question includes concepts from Gravitation chapter]

1. $\frac{k}{a^2}$

2. $\frac{-k}{a^2}$

3. $\frac{k}{a}$

4. $\frac{-k}{a}$

A constant force F is applied on a body. The power (P) generated is related to the time elapsed (t) as [This question includes concepts from Work, Energy and Power chapter]

1. $P<mspace\; width="1em"></mspace>\propto <mspace\; width="1em"></mspace>t^2$

2. $P<mspace\; width="1em"></mspace>\propto <mspace\; width="1em"></mspace>t$

3. $P<mspace\; width="1em"></mspace>\propto \sqrt{<mspace\; width="1em"></mspace>t}$

4. $P<mspace\; width="1em"></mspace>\propto <mspace\; width="1em"></mspace>t^{3/2}$

The graph between the resistive force F acting on a body and the distance covered by the body is shown in the figure. The mass of the body is 25 kg and initial velocity is 2 m/s. When the distance covered by the body is 4 m, its kinetic energy would be  [This question includes concepts from Work, Energy & Power chapter]

1. 50 J

2. 40 J

3. 20 J

4. 10 J

The relationship between force and position is shown in the figure given (in one dimensional case). Find the work done by the force in displaying a body from x= 1 cm to x= 5cm is [This question includes concepts from Work, Energy and Power chapter]

1. 10 erg

2. 20 erg

3. 30 erg

4. 40 erg

The 9 kg block is moving to the right with a velocity of 0.6 m/s on a horizontal surface when a force F, whose time variation is shown in the graph, is applied to it at time t = 0. Calculate the velocity v of the block when t= 0.4s. The coefficient of kinetic fricton is ${\mu }_k=0.3$. [This question includes concepts from Work, Energy & Power chapter]

1. 0.6 m/s

2. 1.2 m/s

3. 1.8 m/s

4. 2.4 m/s