In a potentiometer arrangement, a cell of emf \(1.20\, V\) gives a balance point at \(36\, cm\) length of wire. This cell is now replaced by another cell of emf \(1.80\, V\). The difference in balancing length of potentiometer wire in above conditions will be \(....cm\).

As per the given figure, two plates \(A\) and \(B\) of thermal conductivity \(K\) and \(2 K\) are joined together to form a compound plate. The thickness of plates are \(4.0 \,cm\) and \(2.5 \,cm\) respectively and the area of cross-section is \(120 \,cm ^{2}\) for each plate. The equivalent thermal conductivity of the compound plate is \(\left(1+\frac{5}{\alpha}\right) K\), then the value of \(\alpha\) will be_______

A set of \(20\) tuning forks is arranged in a series of increasing frequencies. If each fork gives \(4\) beats with respect to the preceding fork and the frequency of the last fork is twice the frequency of the first, then the frequency of last fork is \(\dots \; Hz\).

An ideal gas undergoes a quasi static, reversible process in which its molar heat capacity \(C\) remains constant. If during this process the relation of pressure \(P\) and volume \(V\) is given by \(PV^n =\) constant, then n is given by (Here \(C_p\) and \(C_v\) are molar specific heat at constant pressure and constant volume, respectively) :

The electric field of a plane electromagnetic wave is given by \(\overrightarrow{\mathrm{E}}=\mathrm{E}_{0} \frac{\hat{\mathrm{i}}+\hat{\mathrm{j}}}{\sqrt{2}} \cos (\mathrm{kz}+\omega \mathrm{t})\) At \(\mathrm{t}=0,\) a positively charged particle is at the point \((\mathrm{x}, \mathrm{y}, \mathrm{z})=\left(0,0, \frac{\pi}{\mathrm{k}}\right) .\) If its instantaneous velocity at \((t=0)\) is \(v_{0} \hat{\mathrm{k}},\) the force acting on it due to the wave is

A series \(LCR\) circuit driven by \(300\, {V}\) at a frequency of \(50 \,{Hz}\) contains a resistance \({R}=3 \,{k} \Omega\), an inductor of inductive reactance \(X_{L}=250 \,\pi \Omega\) and an unknown capacitor. The value of capacitance to maximize the average power should be : (Take \(\pi^{2}=10\) ) (in \(\mu {F}\))

Two liquids of densities \(\rho_{1}\) and \(\rho_{2}\left(\rho_{2}=2 \rho_{1}\right)\) are filled up behind a square wall of side \(10\; \mathrm{m}\) as shown in figure. Each liquid has a height of \(5 \;\mathrm{m} .\) The ratio of the forces due to these liquids exerted on upper part \(MN\) to that at the lower pait \(NO\) is (Assume that the liquids are not mixing)

The potential energy function (in J) of a particle in a region of space is given as \(U=\left(2 x^2+3 y^3+2 z\right)\). Here \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\) are in meter. The magnitude of \(\mathrm{x}\) - component of force (in \(\mathrm{N}\) ) acting on the particle at point \(\mathrm{P}(1,2,3)\ \mathrm{m}\) is:

Both the diodes used in the circuit shown are assumed to be ideal and have negligible resistance when these are forward biased. Built in potential in each diode is \(0.7\; \mathrm{V} .\) For the input voltages shown in the figure, the voltage (in \(Volts\)) at point \(A\) is