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

1.	The acceleration of the particle is zero.
2.	The acceleration of the particle is increasing.
3.	The acceleration of the particle is necessarily in the plane of motion.
4.	The particle must be undergoing a uniform circular motion.

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

The speed of a projectile at its maximum height is half of its initial speed. The angle of projection is:
1. \(15^{\circ}\)
2. \(30^{\circ}\)
3. \(45^{\circ}\)
4. \(60^{\circ}\)

A particle moves in the \((x\text-y)\) plane according to the rule \(x = a \sin (\omega t)\) and \(y = a \cos (\omega t)\). The particle follows:

A car is moving at a speed of \(40\) m/s on a circular track of radius \(400\) m. This speed is increasing at the rate of \(3\) m/s². The acceleration of the car is:
1. \(4\) m/s²
2. \(7\) m/s²
3. \(5\) m/s²
4. \(3\) m/s²

When a particle is projected at some angle to the horizontal, it has a range \(R\) and time of flight \(t_1\). If the same particle is projected with the same speed at some other angle to have the same range, its time of flight is \(t_2\), then:
1. \(t_{1} + t_{2} = \frac{2 R}{g}\)
2. \(t_{1} - t_{2} = \frac{R}{g}\)
3. \(t_{1} t_{2} = \frac{2 R}{g}\)
4. \(t_{1} t_{2} = \frac{R}{g}\)

The equation of a projectile is \(y = ax -bx^{2}\). Its horizontal range is?
1. \(\frac{a}{b}\)
2. \(\frac{b}{a}\)
3. \(a+b\)
4. \(b-a\)

The velocity at the maximum height of a projectile is \(\frac{\sqrt{3}}{2}\) times its initial velocity of projection \((u)\). Its range on the horizontal plane is:
1. \(\frac{\sqrt{3} u^{2}}{2 g}\)
2. \(\frac{3 u^{2}}{2 g}\)
3. \(\frac{3 u^{2}}{ g}\)
4. \(\frac{u^{2}}{2 g}\)

The position coordinates of a projectile projected from ground on a certain planet (with no atmosphere) are given by
\(y =4 t - 2 t^{2}~ \text{m}\)  and \(x =3t\) metre, where \(t\) is in seconds and point of projection is taken as the origin. The angle of projection of projectile with vertical is:
1. \(30^{\circ}\)
2. \(37^{\circ}\)
3. \(45^{\circ}\)
4. \(60^{\circ}\)

A particle projected from origin moves in the \(x\text-y\) plane with a velocity \(\overrightarrow{v} = 3 \hat{i} + 6 x \hat{j}\)$\stackrel{}{\mathrm{}}$, where \(\hat i\) and \(\hat j\) are the unit vectors along the \(x\) and \(y\text-\)axis. The equation of path followed by the particle is:
1. \(y=x^2\)
2. \(y=\frac{1}{x^2}\)
3. \(y=2x^2\)
4. \(y=\frac{1}{x}\)