Two plane mirrors \({M}_{1}\) and \({M}_{2}\) are at right angle to each other shown. A point source \('P'\) is placed at \('a'\) and \('2a'\) meter away from \({M}_{1}\) and \({M}_{2}\) respectively. The longest distance between the images thus formed is : (Take \(\sqrt{5}=2.3\) )

A particle is released on a vertical smooth semicircular track from point \(X\) so that \(OX\) makes angle \(\theta \) from the vertical ( see figure). The normal reaction of the track on the particle vanishes at point \(Y\) where \(OY\) makes angle \(\phi \) with the horizontal. Then

The velocity (\(v\)) versus time (\(t\)) plot of a particle is shown in the figure, for a time interval of \(40\text{ s}\). The total distance travelled by the particle and the average velocity during this period are, respectively _______.

The radius of \(R\) of a nucleus of mass number \(A\) can be estimated by the formula \(R =\left(1.3 \times 10^{-15}\right) A ^{1 / 3}\, m .\) It follows that the mass density of a nucleus is of the order of \(\left( M _{\text {prot. }} \cong M _{\text {neut. }}=1.67 \times 10^{-27} kg \right)\)

A \(220 \; V , 50 \; Hz\) AC source is connected to a \(25 \; V\), \(5 \; W\) lamp and an additional resistance \(R\) in series (as shown in figure) to run the lamp at its peak brightness, then the value of \(R\) (in ohm) will be

The half life period of a radioactive substance is 60 days. The time taken for \(\frac{7}{8}\) th of its original mass to disintegrate will be \(......days\) 

Calculate the time (in \(minutes\)) interval between \(33 \,\%\) decay and \(67\, \%\) decay if half-life of a substance is \(20\, minutes.\)

In an experment ot verify Newton's law of cooling, a graph is plotted between, the temperature difference \((\Delta T )\) of the water and surroundings and time as shown in figure. The initial temperature of water is taken as \(80^{\circ} \,C\). The value of \(t _{2}\) as mentioned in the graph will be...........

An ac current is represented as
\(\mathrm{i}=5 \sqrt{2}+10 \cos \left(650 \pi \mathrm{t}+\frac{\pi}{6}\right) \mathrm{Amp}\)
The r.m.s value of the current is

A thin circular ring of mass \(M\) and radius \(R\) is rotating with a constant angular velocity \(2 \; rads ^{-1}\) in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass \(m\) be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in \(rads ^{-1}\) )