The following are four different relations about displacement, velocity and acceleration for the motion of a particle in general.

(a)	\(v_{a v}=1 / 2\left[v\left(t_1\right)+v\left(t_2\right)\right]\)
(b)	\(v_{{av}}={r}\left({t}_2\right)-{r}\left({t}_1\right) / {t}_2-{t}_1\)
(c)	\(r=1 / 2\left[v\left(t_2\right)-v\left(t_1\right)\right]\left({t}_2-{t}_1\right)\)
(d)	\({a}_{{av}}=v\left({t}_2\right)-v\left({t}_1\right) / {t}_2-{t}_1\)

The incorrect options is/are:

1.	(a) and (d) only	2.	(a) and (c) only
3.	(b) and (c) only	4.	(a) and (b) only

For a particle performing uniform circular motion,

Choose the correct statement/s:

Two particles are projected in the air with speed \(v_0\), at angles \(\theta_1\) and \(\theta_2\) to the horizontal, respectively. If the height reached by the first particle is greater than that of the second, then:

Choose the correct option:
1. (a), (c), (d)
2. (a), (c)
3. (b), (c), (d)
4. (a), (b)

A particle slides down a frictionless parabolic track starting from rest at point \(A\). Point \(B\) is at the vertex of the parabola and point \(C\) is at a height less than that of point \(A\). After \(C\), the particle moves freely in the air as a projectile. If the particle reaches the highest point at \(P\), then,

In a two-dimensional motion, instantaneous speed \(v_0\) is a positive constant. Then which of the following is necessarily true?

In a two-dimensional motion, instantaneous speed \(v_0\) is a positive constant. Then which of the following is necessarily true?

The horizontal range of a projectile fired at an angle of \(15^\circ\) is \(50~\text m.\) If it is fired with the same speed at an angle of \(45^\circ,\) its range will be:
1. \(60~\text m\)
2. \(71~\text m\)
3. \(100~\text m\)
4. \(141~\text m\) 

A person is sitting in a traveling train and facing the engine. He tosses up a coin and the coin falls behind him. It can be concluded that the train is:

1. Moving forward and gaining speed.

2. Moving forward and losing speed.

3. Moving forward with uniform speed.

4. Moving backward with uniform speed.

For a particle performing uniform circular motion, choose the correct statement(s) from the following:

1. Magnitude of particle velocity (speed) remains variable.

2. Particle velocity remains directed parallel to radius vector.

3. Direction of acceleration keeps changing as particle moves.

4. Angular momentum is constant in magnitude but direction keeps changing.

The horizontal range of a projectile fired at an angle of 15^o is 50 m. If it is fired with the same speed at an angle of 45^o, its range will be:

1. 60 m

2. 71 m

3. 100 m

4. 141 m