The surface tension of a soap bubble is \(0.03\) N/m. The work done in increasing the diameter of bubble from \(2\) cm to \(6\) cm is \(\alpha \pi \times 10^{-4}\) J. The value of \(\alpha\) is _______. (Take \(\pi = 3.14\))

A granite rod of \(60\ cm\) length is clamped at its middle point and is set into longitudinal vibrations. The density of granite is \(2.7 \times 10^3 \) \(kg/m^3\) and its Young's modulus is \(9.27 \times 10^{10}\) \(Pa\) What will be the fundamental frequency of the longitudinal vibrations .... \(kHz\) ?

Different combination of \(3\) resistors of equal resistance \(R\) are shown in the figures.
The increasing order for power dissipation is:

In \(Li^{++},\) electron in first Bohr orbit is excited to a level by a radiation of wavelength \(\lambda .\) When the ion gets deexcited to the ground state in all possible ways(including intermediate emission) a total of six spectral lines are observed. What is the value of \(\lambda \)?.....\(nm\) (Given: \(h = 6.63\times 10^{34}\,js; e = 3 \times 10^8\,ms^{-1}\) )

A block of mass \(1 \mathrm{~kg}\) is pushed up a surface inclined to horizontal at an angle of \(60^{\circ}\) by a force of \(10 \mathrm{~N}\) parallel to the inclined surface as shown in figure. When the block is pushed up by \(10 \mathrm{~m}\) along inclined surface, the work done against frictional force is _______. \(\left[\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\right]\)

A two point charges \(4 q\) and \(-q\) are fixed on the \(x-\)axis at \(x=-\frac{d}{2}\) and \(x=\frac{d}{2},\) respectively. If a third point charge \('q'\) is taken from the origin to \(x = d\) along the semicircle as shown in the figure, the energy of the charge will

Match List\(-I\) with List\(-II\)

List\(-I\)	List\(-II\)
\((a)\) Phase difference between current and voltage in a purely resistive \(AC\) circuit	\((i)\) \(\frac{\pi}{2} ;\) current leads voltage
\((b)\) Phase difference between current and voltage in a pure inductive \(AC\) circuit	\((ii)\) zero
\((c)\) Phase difference between current and voltage in a pure capacitive \(AC\) circuit	\((iii)\) \(\frac{\pi}{2} ;\) current lags voltage
\((d)\) Phase difference between current and voltage in an \(LCR\) series circuit	\((iv)\) \(\tan ^{-1}\left(\frac{X_{C}-X_{L}}{R}\right)\)

Choose the most appropriate answer from the options given below:

To establish an instantaneous current of \(2\,A\) through a \(1\,\mu F\) capacitor ; the potential difference across the capacitor plates should be changed at the rate of

Two wires \(A\) and \(B\) are made up of the same material and have the same mass. Wire A has radius of \(2.0 \mathrm{~mm}\) and wire \(B\) has radius of \(4.0 \mathrm{~mm}\). The resistance of wire B is \(2 \Omega\). The resistance of wire \(\mathrm{A}\) is _______ \(\Omega\).

For hydrogen atom, \(\lambda_1\) and \(\lambda_2\) are the wavelengths corresponding to the transitions \(1\) and \(2\) respectively as shown in figure. The ratio of \(\lambda_1\) and \(\lambda_2\) is \(\frac{x}{32}\). The value of \(x\) is \(..........\)

Which of the following points lies on the tangent to the curve \(x^{4} e^{y}+2 \sqrt{y+1}=3\) at the point \((1,0) ?\)

For a \(3 \times 3\) matrix \(M\), let trace \((M)\) denote the sum of all the diagonal elements of \(M\). Let \(A\) be a \(3 \times 3\) matrix such that \(|A|=\frac{1}{2}\) and trace \((A)=3\). If \(B=\operatorname{adj}(\operatorname{adj}(2 A))\), then the value of \(|B|+\) trace (B) equals :

In a Vernier Calipers. \(10\) divisions of Vernier scale is equal to the \(9\) divisions of main scale. When both jaws of Vernier calipers touch each other, the zero of the Vernier scale is shifted to the left of \(zero\) of the main scale and \(4^{\text {th }}\) Vernier scale division exactly coincides with the main scale reading. One main scale division is equal to \(1\,mm\). While measuring diameter of a spherical body, the body is held between two jaws. It is now observed that zero of the Vernier scale lies between \(30\) and \(31\) divisions of main scale reading and \(6^{\text {th }}\) Vernier scale division exactly. coincides with the main scale reading. The diameter of the spherical body will be \(.......cm\)