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

White light is passed through a double slit and interference is observed on a screen \(1.5 \,{m}\) away. The separation between the slits is \(0.3 \,{mm}\). The first violet and red fringes are formed \(2.0 \,{mm}\) and \(3.5\, {mm}\) away from the central white fringes. The difference in wavelengths of red and voilet light is \(.... \,{nm} .\)

Energy released when two deuterons \(\left({ }_1 \mathrm{H}^2\right)\) fuse to form a helium nucleus \(\left({ }_2 \mathrm{He}^4\right)\) is :
(Given : Binding energy per nucleon of \({ }_1 \mathrm{H}^2=1.1 \mathrm{MeV}\) and binding energy per nucleon of \({ }_2 \mathrm{He}^4=7.0 \mathrm{MeV}\))

A deuteron and an alpha particle having equal kinetic energy enter perpendicular into a magnetic field. Let \(r_{d}\) and \(r_{\alpha}\) be their respective radii of circular path. The value of \(\frac{r_{d}}{r_{\alpha}}\) is equal to

Velocity \((v)\) and acceleration \((a)\) in two systems of units \(1\) and \(2\) are related as \(V _{2}=\frac{ n }{ m ^{2}} v _{1}\) and \(a_{2}=\frac{a_{1}}{m n}\) respectively. Here \(m\) and \(n\) are constants. The relations for distance and time in two systems respectively are

Two resistors of \(200 \, \Omega\) and \(400 \, \Omega\) are connected in series with a battery of \(100\) V. A bulb rated at \(200\) V, \(100\) W is connected across the \(400 \, \Omega\) resistance. The potential drop across the bulb is _______ V.

In a resonance tube experiment when the tube is filled with water up to height of \(17.0\, cm\) from bottom, it resonates with a given tuning fork. When the water level is raised the next resonance with the same tuning fork occurs at a height of \(24.5\, cm\). If the velocity of sound in air is \(330\, m / s ,\) the tuning fork frequency is\(......Hz\)

An open glass tube is immersed in mercury in such a way that a length of \(8\ cm\) extends above the mercury level. The open end of the tube is then closed and sealed and the tube is raised vertically up by additional \(46\ cm\). What will be length of the air column above mercury in the tube now ....... \(cm\) ? ( Atmospheric pressure = \(76\ cm\) of \(Hg\))

A particle \(P\) is formed due to a completely inelastic collision of particles \(x\) and \(y\) having de -Broglie wavelengths \({\lambda _x}\) and \({\lambda _y}\) respectively. If \(x\) and \(y\) were moving in opposite directions, then the de -Broglie wavelength of \(P\) is

Three Carnot engines operate in series between a heat source at a temperature \(T_1\) and a heat sink at temperature \(T_4\) (see figure). There are two other reservoirs at temperature \(T_2\) and \(T_3\), as shown, with \(T_1 > T_2 > T_3 > T_4\). The three engines are equally efficient if