Q. No.Trains travel between station \(A\) and station \(B\): on the way up (from \(A~\text{to}~B\)) - they travel at a speed of \(80\) km/h, while on the return trip the trains travel at twice that speed. The services are maintained round the clock. Trains leave station \(A\) every \(30\) min for station \(B\) and reach \(B\) in \(2\) hrs. All trains operate continuously, without any rest at \(A\) or \(B\).
| 1. | the frequency of trains leaving \(B\) must be twice as much as \(A\). |
| 2. | the frequency of trains leaving \(B\) must be half as much as \(A\). |
| 3. | the frequency of trains leaving \(B\) is equal to that at \(A\). |
| 4. | the situation is impossible to maintain unless larger number of trains are provided at \(A\). |
A simple pendulum of time period \(T_0\) is taken in a rocket which is accelerating upwards initially and then, after some time, it moves with uniform velocity upward. The time period of the pendulum is observed within the rocket and is found to be \(2T_0\). The rocket, at that time, must be at a distance (above the earth's surface) of (radius of earth = \(R\))
| 1. | \(\dfrac{R}{2}\) | 2. | \(\dfrac{R}{4}\) |
| 3. | \(R\) | 4. | \(4R\) |
The energy of the block is \(E\), and the plane is smooth, the wall at the end \(B\) is smooth. Collisions with walls are elastic. The distance \(AB=l\), the spring is ideal and the spring constant is \(k\). The time period of the motion is:
| 1. | \(2\pi\sqrt{\dfrac{m}{k}}\) |
| 2. | \(\pi\sqrt{\dfrac{m}{k}}+l\sqrt{\dfrac{2m}{E}}\) |
| 3. | \(2\pi\sqrt{\dfrac{m}{k}}+2l\sqrt{\dfrac{2m}{E}}\) |
| 4. | \(\pi\sqrt{\dfrac{m}{k}}+l\sqrt{\dfrac{m}{2E}}\) |
The equation of motion of a particle that starts moving at \(t=0\) s is given by \({x}=5 \sin \left(\dfrac{\pi t}{2}+\dfrac{\pi}{3}\right) \) where \(x\) is in cm and time \(t\) is in second. The time, when the particle first comes to rest, is:
| 1. | \(\dfrac{1}{3}\) s | 2. | \(\dfrac{7}{6}\) s |
| 3. | \(\dfrac{2}{3}\) s | 4. | \(\dfrac{13}{6}\) s |
A uniform rod of length \(l\) is suspended by an end and is made to undergo small oscillations. The time period of small oscillation is \(T\). Then, the acceleration due to gravity at this place is:
| 1. | \(4\pi^2\dfrac{l}{T^2}\) | 2. | \(\dfrac{4\pi^2}{3}\dfrac{l}{T^2}\) |
| 3. | \(\dfrac{8\pi^2}{3}\dfrac{l}{T^2}\) | 4. | \(\dfrac{12\pi^2l}{T^2}\) |
A particle moves in the x-y plane according to the equation
\(x = A \cos^2 \omega t\) and \(y = A \sin^2 \omega t\)
Then, the particle undergoes:
| 1. | uniform motion along the line \(x + y = A\) |
| 2. | uniform circular motion along \(x^2 + y^2 = A^2\) |
| 3. | SHM along the line \(x + y = A\) |
| 4. | SHM along the circle \(x^2 + y^2 = A^2\) |
A particle of mass \(m\) executes SHM along a straight line with an amplitude \(A\) and frequency \(f.\)
| Assertion (A): | The kinetic energy of the particle undergoes oscillation with a frequency \(2f.\) |
| Reason (R): | Velocity of the particle, \(v = {\dfrac{dx}{dt}}\), its kinetic energy equals \({\dfrac 12}mv^2\) and the particle oscillates sinusoidally with a frequency \(f\). |
| 1. | Both (A) and (R) are True and (R) is the correct explanation of (A). |
| 2. | Both (A) and (R) are True but (R) is not the correct explanation of (A). |
| 3. | (A) is True but (R) is False. |
| 4. | (A) is False but (R) is True. |
| 1. | the amplitude increases. |
| 2. | the amplitude decreases. |
| 3. | the frequency increases. |
| 4. | the frequency decreases. |
| 1. | \(2\pi\sqrt{\dfrac{L}{g}}\) | 2. | \(2\pi\sqrt{\dfrac{2L}{g}}\) |
| 3. | \(2\pi\sqrt{\dfrac{L}{2g}}\) | 4. | \(2\pi\sqrt{\dfrac{2L}{\sqrt3g}}\) |
| 1. | \(T\) | 2. | \(\pi T\) |
| 3. | \(\pi\sqrt2T\) | 4. | \(\dfrac{\pi}{\sqrt 2}T\) |
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