\({T}_{0}\) is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to \(\frac{1}{16}\) times of its initial value, the modified time

If \(1000\) droplets of water of surface tension \(0.07\,N / m\). having same radius \(1\,mm\) each, combine to from a single drop. In the process the released surface energy is \(\left(\text { Take } \pi=\frac{22}{7}\right)\)

The planet Mars has two moons, if one of them has a period \(7\, hours,\, 30\, minutes\) and an orbital radius of \(9.0 \times 10^{3}\, {km} .\) Find the mass of Mars. \(\left\{\operatorname{Given} \frac{4 \pi^{2}}{G}=6 \times 10^{11} {N}^{-1} {m}^{-2} {kg}^{2}\right\}\)

Which of the following Maxwell's equations is valid for time varying conditions but not valid for static conditions

A thin bar of length \(L\) has a mass per unit length \(\lambda \), that increases linearly with distance from one end. If its total mass is \(M\) and its mass per unit length at the lighter end is \(\lambda_0\), then the distance of the centre of mass from the lighter end is

The equation of a wave travelling on a string is \(\mathrm{y}=\sin [20 \pi \mathrm{x}+10 \pi \mathrm{t}]\), where x and t are distance and time in SI units. The minimum distance between two points having the same oscillating speed is :

Which of the following combinations has the dimension of electrical resistance ( \({ \varepsilon _0}\) is the permittivity of vacuum and \({\mu _0}\) is the permeability of vacuum) ?

The pressure and volume of an ideal gas are related as \(\mathrm{PV}^{3 / 2}=\mathrm{K}\) (Constant). The work done when the gas is taken from state \(A\left(P_1, V_1, T_1\right)\) to state \(\mathrm{B}\left(\mathrm{P}_2, \mathrm{~V}_2, \mathrm{~T}_2\right)\) is _______.

The vector \(\left( {\hat i \times \vec a.\vec b} \right)\hat i + \left( {\hat j \times \vec a.\vec b} \right)\hat j + \left( {\hat k \times \vec a.\vec b} \right)\hat k\) is equal to

A bead \(P\) sliding on a frictionless semi-circular string \((A C B)\) and it is at point \(S\) at \(t =0\) and at this instant the horizontal component of its velocity is \(v\). Another bead \(Q\) of the same mass as \(P\) is ejected from point \(A\) at \(t =0\) along the horizontal string \(A B\), with the speed \(v\), friction between the beads and the respective strings may be neglected in both cases. Let \(t_p\) and \(t_Q\) be the respective times taken by beads \(P\) and \(Q\) to reach the point \(B\), then the relation between \(t_p\) and \(t_Q\) is

Let \(\mathrm{f}:[0,3] \rightarrow \mathrm{R}\) be defined by \(f(x)=\min \{x-[x], 1+[x]-x\}\) where \([\mathrm{x}]\) is the greatest integer less than or equal to \(\mathrm{x}\). Let \(\mathrm{P}\) denote the set containing all \(x \in[0,3]\) where \(f\) is discontinuous, and \(Q\) denote the set containing all \(x \in(0,3)\) where \(f\) is not differentiable. Then the sum of number of elements in \(\mathrm{P}\) and \(\mathrm{Q}\) is equal to \(......\)

Effective capacitance of parallel combination of two capacitors \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\) is \(10\; \mu \mathrm{F}\). When these capacitors are individually connected to a voltage source of \(1\; \mathrm{V},\) the energy stored in the capacitor \(\mathrm{C}_{2}\) is \(4\) times that of \(\mathrm{C}_{1}\). If these capacitors are connected in series, their effective capacitance will be

All five letter words are made using all the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and arranged as in an English dictionary with serial numbers. Let the word at serial number \(n\) be denoted by \(W_n\). Let the probability \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)\) of choosing the word \(\mathrm{W}_{\mathrm{n}}\) satisfy \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)=2 \mathrm{P}\left(\mathrm{W}_{\mathrm{n}-1}\right), \mathrm{n} \gt 1\).
If \(\mathrm{P}(\mathrm{CDBEA})=\frac{2^\alpha}{2^\beta-1}, \alpha, \beta \in \mathbb{N}\), then \(\alpha+\beta\) is equal to : _______