When the \(rms\) voltages \(V_L, V_C\) and \(V_R\) are measured respectively across the inductor \(L\), the capacitor \(C\) and the resistor \(R\) in a series \(LCR\) circuit connected to an \(AC\) source, it is found that the ratio \(V_L : V_C : V_R = 1 : 2 : 3\). If the \(rms\) voltage of the \(AC\) sources is \(100\, V\), the \(V_R\) is close to........\(V\)

Consider a completely full cylindrical water tank of height 1.6 m and cross-sectional area \(0.5 \mathrm{~m}^2\). It has a small hole in its side at a height 90 cm from the bottom. Assume, the cross-sectional area of the hole to be negligibly small as compared to that of the water tank. If a load 50 kg is applied at the top surface of the water in the tank then the velocity of the water coming out at the instant when the hole is opened is : \(\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\right)\)

The energy required to remove the electron from a singly ionized Helium atom is \(2.2\, times\) the energy required to remove an electron from Helium atom . The total energy required to ionize the Helium atom completely is......\(eV\)

For \(a\) gas \(C_{p}-C_{V}=R\) in a state \(P\) and \(C_{p}-C_{V}=1.10 R\) in a state \(Q, T_{p}\) and \(T_{Q}\) are the temperatures in two different states \(P\) and \(Q\) respectively. Then

In the given circuit, \(C _1=2\,\mu F , C _2=0.2\,\mu F\), \(C _3=2\,\mu F , C _4=4\,\mu F,\) \(C _5=2\,\mu F , C _6=2\,\mu F\), the charge stored on capacitor \(C _4\) is \(.....\mu C\).

The magnetic field in a region is given by \(\overrightarrow{ B }= B _{0}\left(\frac{ x }{ a }\right) \,\hat{ k }\). A square loop of side \(d\) is placed with its edges along the \(x\) and \(y\) axes. The loop is moved with a constant velocity \(\overrightarrow{ v }= v _{0} \hat{ i }\). The emf induced in the loop is:

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : The density of the copper \(\left({ }_{29}^{64} \mathrm{Cu}\right)\) nucleus is greater than that of the carbon \(\left({ }_6^{12} \mathrm{C}\right)\) nucleus.
Reason (R): The nucleus of mass number A has a radius proportional to \(\mathrm{A}^{1 / 3}\).
In the light of the above statements, choose the most appropriate answer from the options given below :

In the given figure of meter bridge experiment, the balancing length \(AC\) corresponding to null deflection of the galvanometer is \(40\,cm\). The balancing length, if the radius of the wire \(AB\) is doubled, will be \(....cm\)

A radioactive sample disintegrates via two independent decay processes having half lives \(T _{1 / 2}^{(1)}\) and \(T _{1 / 2}^{(2)}\) respectively. The effective half- life \(T _{1 / 2}\) of the nuclei is

A paralle plate capacitor is made up of stair like structure with a palte area \(A\) of each stair and that is connected with a wire of length \(b\), as shown in the figure. The capacitance of the arrangement is \(\frac{ x }{15} \frac{\varepsilon_{0} A }{ b }\). The value of \(x\) is ............