An ideal cell of emf \(10\, V\) is connected in circuit shown in figure. Each resistance is \(2\, \Omega .\) The potential difference (in \(V\)) across the capacitor when it is fully charged is

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 \(.......cm\)

\(27\) identical drops are charged at \(22\, V\,\,each.\) They combine to form a bigger drop. The potential of the bigger drop will be............ \(V.\)

Match List-I with List-II.

List-I (Relation)		List-II (Law)
A.	\(\oint \overrightarrow{ E } \cdot \overrightarrow{ d } l =-\frac{ d }{ dt } \oint \overrightarrow{ B } \cdot \overrightarrow{ d a}\)	I.	Ampere's circuital law.
B.	\(\oint\) B. \(\vec{d} l=\mu_0\left(1+\epsilon_0 \frac{d \phi_E}{d t}\right)\)	II.	Faraday's laws of electromagnetic induction.
C.	\(\oint \overrightarrow{ E } . \overrightarrow{ d a}=\frac{1}{\epsilon_0} \int \rho dv\)	III.	Ampere-Maxwell law
D.	\(\oint \overrightarrow{ B } \cdot \overrightarrow{ d } l =\mu_0 I\)	IV.	Gauss's law of electrostatics

Choose the correct answer from the options given below :

A water drop of radius \(1\,cm\) is broken into \(729\) equal droplets. If surface tension of water is \(75\,dyne / cm\), then the gain in surface energy upto first decimal place will be\(...\times 10^{-4}\,J\)

Given below are two statements: Statement \(I:\) For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases. Statement \(II:\) Escape velocity is independent of the radius of the planet. In the light of above statements, choose the most appropriate answer from the options given below on :

Two point charges \(A\) and \(B\) of magnitude \(+8 \times 10^{-6}\,C\) and \(-8 \times 10^{-6}\,C\) respectively are placed at a distance \(d\) apart. The electric field at the middle point \(O\) between the charges is \(6.4 \times 10^{4}\,NC ^{-1}\). The distance ' \(d\) ' between the point charges \(A\) and \(B\) is..............\(m\)

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group \(A\) and the remaining 3 from group \(B\), is equal to :

Four resistances \(40 \ \Omega, 60\  \Omega, 90\  \Omega\) and \(110\  \Omega\) make the arms of a quadrilateral \(A,B,C,D\). Across \(AC\) is a battery of emf \(40\, V\) and internal resistance negligible. The potential difference across \(BD\) is \(V\) is...... 

Let \(A=\left[a_{i j}\right]\) be a square matrix of order 2 with entries either 0 or 1 . Let \(E\) be the event that \(A\) is an invertible matrix. Then the probability \(\mathrm{P}(\mathrm{E})\) is :

The value of the integral \(\int_{-1}^2 \log _e\left(x+\sqrt{x^2+1}\right) d x\) is :

If the real part of the complex number \(z=\frac{3+2 i \cos \theta}{1-3 i \cos \theta}, \theta \in\left(0, \frac{\pi}{2}\right)\) is zero, then the value of \(\sin ^{2} 3 \theta+\cos ^{2} \theta\) is equal to \(.....\)

An example of a perfect diamagnet is a superconductor. This implies that when a superconductor is put in a magnetic field of intensity \(B\), the magnetic field \(B_s\) inside the superconductor will be such that