A vector $\overrightarrow{a}$ is turned without a change in its length through a small angle $d\theta .$ The value of $\left|\Delta \overrightarrow{a}\right|$ and $\Delta a$ are, respectively:

1.	\(0, ad\theta\)	2.	\(d\theta, 0\)
3.	\(0,0\)	4.	None of these

A particle is moving such that its position coordinates \((x,y)\) are \((2\) m, \(3\) m) at time \(t=0,\)  \((6\) m, \(7\) m) at time \(t=2\) s and   \((13\) m, \(14\) m) at time \(t=5\) s. Average velocity vector \((v_{avg})\) from \(t=0\) to \(t=5\) s is:

The angle turned by a body undergoing circular motion depends on the time as given by the equation, \(\theta = \theta_{0} + \theta_{1} t + \theta_{2} t^{2}\). It can be deduced that the angular acceleration of the body is? 
1. \(\theta_1\)
2. \(\theta_2\)
3. \(2\theta_1\)
4. \(2\theta_2\)

Certain neutron stars are believed to be rotating at about \(1\) rev/s. If such a star has a radius of \(20\) km, the acceleration of an object on the equator of the star will be:

Three particles are moving with constant velocities \(v_1 ,v_2\) and \(v\) respectively as given in the figure. After some time, if all the three particles are in the same line, then the relation among \(v_1 ,v_2\) and \(v\) is:
                                    
1. \(v =v_1+v_2\)
2. \(v= \sqrt{v_{1} v_{2}}\)
3. \(v = \frac{v_{1} v_{2}}{v_{1} + v_{2}}\)
4. \(v=\frac{\sqrt{2} v_{1} v_{2}}{v_{1} + v_{2}}\)

A body is projected with velocity \(\overrightarrow{v} = \left( \alpha \hat{i} + \beta \hat{j} \right)\) m/s. The time of flight of the body is: [considering \(x\) as horizontal and \(y\) as vertical axis and \(g\) is acceleration due to gravity]
1. \(\frac{2 \beta}{g}\)
2. \(\frac{2 \alpha}{g}\)
3. \(\frac{2 \alpha \beta}{g}\)
4. \(\frac{2 \alpha}{g \beta}\)$\frac{}{}$

A car moves on a circular path such that its speed is given by \(v= Kt\), where \(K\) = constant and \(t\) is time. Also given: radius of the circular path is \(r\). The net acceleration of the car at time \(t\) will be:
1. \(\sqrt{K^{2} +\left(\frac{K^{2} t^{2}}{r}\right)^{2}}\)
2. \(2K\)
3. \(K\)
4. \(\sqrt{K^{2}   +   K^{2} t^{2}}\)

Raindrops are falling with speed \(v\) vertically downwards and a man is running on a horizontal road with speed \(u.\) The magnitude of the velocity of the raindrops with respect to the man is:
1. \(v-u\)
2. \(v+u\)
3. \(\sqrt{{v}^2 + {u}^2 \over 2}\)
4. \(\sqrt{{v}^2 + {u}^2}\)

The equation of trajectory of a projectile is given by \(y = x-10x^{2}\). Its speed of projection is: (\(g =1 0\) m/${\mathrm{s}}^2$)
1. \(1\) m/s
2. \(2\) m/s
3. \(3\) m/s
4. \(4\) m/s

A particle is thrown obliquely at \(t=0\). The particle has the same K.E. at \(t=5\) seconds and at \(t=9\) seconds. The particle attains maximum altitude at:
1. \(t=6\) s
2. \(t=7\) s
3. \(t=8\) s
4. \(t=14\) s