The fraction of the original number of radioactive atoms that disintegrates (decays) during the average lifetime of a radioactive substance will be:
1. \(\frac{1}{e}\)
2. \(\frac{1}{1+e}\)
3. \(\frac{e-1}{e+1}\)
4. \(\frac{e-1}{e}\)
The figure given below shows the displacement and time, \((x\text -t)\) graph of a particle moving along a straight line:
The correct statement, about the motion of the particle, is:
| 1. | the particle moves at a constant velocity up to a time \(t_0\) and then stops. |
| 2. | the particle is accelerated throughout its motion. |
| 3. | the particle is accelerated continuously for time \(t_0\) then moves with constant velocity. |
| 4. | the particle is at rest. |
Air is pushed carefully into a soap bubble of radius \(r\) to double its radius. If the surface tension of the soap solution is \(T,\) then the work done in the process is:
| 1. | \(12\pi r^2T\) | 2. | \(24\pi r^2T \) |
| 3. | \(4\pi r^2T\) | 4. | \(8\pi r^2T\) |
| Statement I: | The magnetic field of a circular loop at very far away point on the axial line varies with distance as like that of a magnetic dipole. |
| Statement II: | The magnetic field due to magnetic dipole varies inversely with the square of the distance from the centre on the axial line. |
| 1. | Statement I is correct and Statement II is incorrect. |
| 2. | Statement I is incorrect and Statement II is correct. |
| 3. | Both Statement I and Statement II are correct. |
| 4. | Both Statement I and Statement II are incorrect. |
A positively charged particle \(+q\) is projected with speed \(v\) toward a fixed charge \(+Q,\) and rebounds after reaching a minimum distance \(r.\) What will be the new closest distance of approach if its initial velocity is doubled to \(2v\text{?}\)
| 1. | \(\dfrac{r}{4}\) | 2. | \(\dfrac{r}{2}\) |
| 3. | \(\dfrac{r}{16}\) | 4. | \(\dfrac{r}{8}\) |
The determination of the value of acceleration due to gravity \((g)\) by simple pendulum method employs the formula,
\(g=4\pi^2\dfrac{L}{T^2}\)
The expression for the relative error in the value of \(g\) is:
| 1. | \(\dfrac{\Delta g}{g}=\dfrac{\Delta L}{L}+2\Big(\dfrac{\Delta T}{T}\Big)\) |
| 2. | \(\dfrac{\Delta g}{g}=4\pi^2\Big[\dfrac{\Delta L}{L}-2\dfrac{\Delta T}{T}\Big]\) |
| 3. | \(\dfrac{\Delta g}{g}=4\pi^2\Big[\dfrac{\Delta L}{L}+2\dfrac{\Delta T}{T}\Big]\) |
| 4. | \(\dfrac{\Delta g}{g}=\dfrac{\Delta L}{L}-2\Big(\dfrac{\Delta T}{T}\Big)\) |
A monochromatic light of frequency \(500~\text{THz}\) is incident on the slits of Young's double slit experiment. If the distance between the slits is \(0.2~\text{mm}\) and the screen is placed at a distance \(1~\text{m}\) from the slits, the width of \(10\) fringes will be:
| 1. | \(1.5~\text{mm}\) | 2. | \(15~\text{mm}\) |
| 3. | \(30~\text{mm}\) | 4. | \(3~\text{mm}\) |
In a meter bridge experiment, the null point is at a distance of \(30~\text{cm}\) from \(A.\) If a resistance of \(16~\Omega\) is connected in parallel with resistance \(Y\), the null point occurs at \(50~\text{cm}\) from \(A.\) The value of the resistance \(Y\) is:
| 1. | \(\dfrac{112}{3}~\Omega\) | 2. | \(\dfrac{40}{3}~\Omega\) |
| 3. | \(\dfrac{64}{3}~\Omega\) | 4. | \(\dfrac{48}{3}~\Omega\) |
The temperature at which the RMS speed of atoms in neon gas is equal to the RMS speed of hydrogen molecules at \(15^{\circ} \text{C}\) is:
(the atomic mass of neon \(=20.2~\text u,\) molecular mass of hydrogen \(=2~\text u\))
1. \(2.9\times10^{3}~\text K\)
2. \(2.9~\text K\)
3. \(0.15\times10^{3}~\text K\)
4. \(0.29\times10^{3}~\text K\)
Two planets orbit a star in circular paths with radii \(R\) and \(4R,\) respectively. At a specific time, the two planets and the star are aligned in a straight line. If the orbital period of the planet closest to the star is \(T,\) what is the minimum time after which the star and the planets will again be aligned in a straight line?
| 1. | \((4)^2T\) | 2. | \((4)^{\frac13}T\) |
| 3. | \(2T\) | 4. | \(8T\) |
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