The displacement \(x\) of a particle varies with time \(𝑡\) as \(x=ae^{\alpha t}+be^{\beta t}\) where \(𝑎,\ 𝑏,\ 𝛼\) and \(\beta\) are positive constants. The velocity of the particle will:

1. Go on decreasing with time
2. Be independent of \(𝛼\) and \(\beta\)
3. Drop to zero when \(𝛼 = 𝛽\)
4. Go on increasing with time

A car, starting from rest, accelerates at the rate \(f\) through a distance \(S\), then continues at a constant speed for time \(t\) and then decelerates at the rate \(\frac f2\) to come to rest. If the total distance traversed is \(15\ \text{S}\), then:

1. \(S = \frac{1}{2}ft^2\)
2. \(S = \frac{1}{4}ft^2\)
3. \(S = \frac{1}{72}ft^2\)
4. \(S = \frac{1}{6}ft^2\)

A man is \(45\ \text{m}\) behind the bus when the bus starts accelerating from rest with an acceleration of \(2.5\ \text{m/s}^2\). With what minimum velocity should the man start running to catch the bus?

1.  \(12\ \text{m/s}\)
2.  \(14\ \text{m/s}\)
3.  \(15\ \text{m/s}\)
4.  \(16\ \text{m/s}\)

A particle moves along the x-axis as \({x}=4({t}-2)+{a}({t}-2)^2.\)Which of the following is true? 

A body starting from rest moves with constant acceleration. The ratio of the distance covered by the body during the \(5^{th}\) second to that covered in \(5\) second is:

1. \(9/25\)
2. \(3/5\)
3. \(25/9\)
4. \(1/25\)

Two trains, each \(50\) m long, are travelling in the opposite direction with velocities \(10\) m/s and \(15\) m/s. The time of crossing is:
1. \(10\) sec
2. \(4\) sec
3. \(2\sqrt{3}\) sec
4. \(4\sqrt{3}\) sec

A \(120\ \text{m}\) long train is moving in a direction with speed \(20\ \text{m/s}\). A train \(B\), moving with \(30\ \text{m/s}\) in the opposite direction and \(130\ \text{m}\) long, crosses the first train in a time:

1.  \(4\ \text{s}\)
2.  \(36\ \text{s}\)
3.  \(38\ \text{s}\)
4.  \(5\ \text{s}\)

A \(210\) meter long train is moving due north at a speed of \(25\ \text{m/s}\). A small bird is flying due South a little above the train with a speed of \(5\ \text{m/s}\). The time taken by the bird to cross the train is:

1. \(6\ \text{s}\)
2. \(7\ \text{s}\)
3. \(9\ \text{s}\)
4. \(10\ \text{s}\)

A police jeep is chasing with a velocity of \(45\text{ km/h}\) a thief in another jeep moving with a velocity of \(153\text{ km/h}.\) Police fire a bullet with a muzzle velocity of \(180\text{ m/s}.\) The velocity with which it will strike the car of the thief is:

1.  \(150\text{ m/s}\)
2.  \(27\text{ m/s}\)
3.  \(450\text{ m/s}\)
4.  \(250\text{ m/s}\)

A train of \(150\) meter length is going towards the north direction at a speed of \(10\ \text{m/s}\). A parrot flies at the speed of \(5\ \text{m/s}\) towards the south direction parallel to the railway track. The time taken by the parrot to cross the train is:

1. \(12\) seconds

2. \(8\) seconds

3. \(15\) seconds

4. \(10\) seconds