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A shell fired from the base of a mountain just clears it. If $\mathrm{\alpha }$ is the angle of projection, then the angular elevation of the summit $\mathrm{\beta }$ is

1. $\frac{\mathrm{\alpha }}{2}$
2. ${\tan }^{-1}\left(\frac{1}{2}\right)$
3. ${\tan }^{-1}\left(\frac{\tan \alpha }{2}\right)$
4. ${\tan }^{-1}\left(2\tan \alpha \right)$

A particle is projected with a speed u at an angle $\mathrm{\theta }$ to the horizontal. Find the radius of curvature at highest point of its trajectory-
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. $\frac{\sqrt{3}{\mathrm{u}}^2{\cos }^2\mathrm{\theta }}{\mathrm{g}}$

In fig. the time taken by the projectile to reach from A to B is t. Then the distance AB is equal to 

1. $\frac{\mathrm{ut}}{\sqrt{3}}$
2. $\frac{\sqrt{3}\mathrm{ut}}{2}$
3. $\sqrt{3}ut$
4.  2ut

A ball rolls off to the top of a staircase with a horizontal velocity u m/s. If the steps are h metre high and b metre wide, the ball will hit the edge of the nth step, if
1. $\mathrm{n}=\frac{2\mathrm{hu}}{{\mathrm{gb}}^2}$
2. $\mathrm{n}=\frac{2{\mathrm{hu}}^2}{\mathrm{gb}}$
3. $\mathrm{n}=\frac{2{\mathrm{hu}}^2}{{\mathrm{gb}}^2}$
4. $\mathrm{n}=\frac{{\mathrm{hu}}^2}{{\mathrm{gb}}^2}$

If $\overrightarrow{\mathrm{A}}=\overrightarrow{B}+\overrightarrow{C}$, and the magnitudes of $\overrightarrow{\mathrm{A}},\overrightarrow{B},\overrightarrow{C}$ are 5,4 and 3 units, then the angle between $\overrightarrow{\mathrm{A}} and \overrightarrow{C}$ is
1. ${\cos }^{-1}\left(\frac{3}{5}\right)$
2. ${\cos }^{-1}\left(\frac{4}{5}\right)$
3. ${\sin }^{-1}\left(\frac{3}{4}\right)$
4. $\frac{\mathrm{\pi }}{2}$

A tube of length L is filled completely with an incompressible liquid of mass M and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity $\mathrm{\omega }$. The force exerted by the liquid at the other end is 
1. $\frac{{\mathrm{ML\omega }}^2}{2}$
2. ${\mathrm{ML\omega }}^2$
3. $\frac{{\mathrm{ML\omega }}^2}{4}$
4. $\frac{{\mathrm{ML}}^2{\mathrm{\omega }}^2}{2}$

Four persons K,L,M and N are initially at the corners of a square of side of length d. If every person starts moving, such that K is always headed towards L,L towards M,M is headed directly towards N and N towards K, then the four persons will meet after 
1. $\frac{\mathrm{d}}{\mathrm{v}} sec$
2. $\frac{\sqrt{2\mathrm{d}}}{\mathrm{v}} sec$
3. $\frac{d}{\sqrt{2\mathrm{v}}} sec$
4. $\frac{\mathrm{d}}{2\mathrm{v}} sec$

A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of 'P' is such that it sweeps out a length $\mathrm{s}={\mathrm{t}}^3+5$, where s is in metres and t is in seconds. The radius of the path is 20 m. The acceleration of 'P' when t=2 s is nearly 

1. 14 $\mathrm{m}/{\mathrm{s}}^2$
2. 13 $\mathrm{m}/{\mathrm{s}}^2$
3. 12 $\mathrm{m}/{\mathrm{s}}^2$
4. 7.2 $\mathrm{m}/{\mathrm{s}}^2$

A particle originally at rest at the highest point of a smooth vertical circle is slightly displaced. It will leave the circle at a vertical distance h below the highest point such that 

1. h = R
2. $\mathrm{h}=\frac{\mathrm{R}}{3}$
3. $\mathrm{h}=\frac{\mathrm{R}}{2}$
4. $\mathrm{h}=\frac{2\mathrm{R}}{3}$

A ball is projected with kinetic energy E at an angle of 45$°$ to the horizontal. At the highest point during its flight, its kinetic energy will be
1. zero
2. E/2
3. $\mathrm{E}/\sqrt{2}$
4. E

Given that centripetal force $\mathrm{F}=-\mathrm{k}/{\mathrm{r}}^2$. The total energy is
1. $-\mathrm{k}/{\mathrm{r}}^2$
2. -k/r
3. $-\mathrm{k}/2{\mathrm{r}}^2$
4. -k/2r

A particle is projected from the bottom of an inclined plane of inclination 30$°$. At what angle $\mathrm{\theta }$ (from the horizontal should the particle be projected to get maximum range on the inclined plane :
1. 45$°$
2. 53$°$
3. 76$°$
4. 60$°$

For a particle in uniform circular motion the acceleration a at a point P(R,$\mathrm{\theta }$) on the circle of radius R is (here $\mathrm{\theta }$ is measured from the x-axis)
1. $-\frac{{\mathrm{v}}^2}{\mathrm{R}}\mathrm{cos\theta }\hat{\mathrm{i}}+\frac{{\mathrm{v}}^2}{\mathrm{R}}\mathrm{sin\theta }\hat{\mathrm{j}}$
2. $-\frac{{\mathrm{v}}^2}{\mathrm{R}}\mathrm{sin\theta }\hat{\mathrm{i}}+\frac{{\mathrm{v}}^2}{\mathrm{R}}\mathrm{cos\theta }\hat{\mathrm{j}}$
3. $-\frac{{\mathrm{v}}^2}{\mathrm{R}}\mathrm{cos\theta }\hat{\mathrm{i}}-\frac{{\mathrm{v}}^2}{\mathrm{R}}\mathrm{sin\theta }\hat{\mathrm{j}}$
4. $\frac{{\mathrm{v}}^2}{\mathrm{R}}\hat{\mathrm{i}}+\frac{{\mathrm{v}}^2}{\mathrm{R}}\hat{\mathrm{j}}$

1. \(\frac{x\sin\theta_2}{u_1}\)
2. \(\frac{x\cos\theta_2}{u_1}\)
3. \(\frac{x\sin\theta_2}{u_1(\sin(\theta_2-\theta_1))}\)
4. \(\frac{x\sin\theta_2}{u_2(\sin(\theta_2-\theta_1))}\)

A man can row a boat with 8 km/hr in still water. If he is crossing a river where the current is 4 km/hr, in what direction should be head the boat to cross the river in shortest time and in what minimum time he will cross?
(Width of river is 8 km)
1. $\mathrm{}$\(30^\circ,\) \(23\) hr
2. \(60^\circ,\) \(23\) hr
3. \(90^\circ,\) \(1\) hr
4. \(120^\circ,\) \(43\) hr