A few electric field lines for a system of two charges \(Q_1\) and \(Q_2\) fixed at two different points on the \(x\)-axis are shown in the figure. These lines suggest that

A \(2 \mu \mathrm{F}\) capacitor is charged as shown in the figure. The percentage of its stored energy dissipated after the switch \(S\) is turned to position 2 , is

Two wires each carrying a steady current \(I\) are shown in four configurations in Column I. Some of the resulting effects are described in Column II. Match the statements in Column I with the statements in Column II and indicate your answer by darkening appropriate bubbles in the \(4 \times 4\) matrix given in the ORS.

Paragraph:
The figure shows a surface \(X Y\) separating two transparent media, medium-1 and medium-2. The lines \(a b\) and \(c d\) represent wavefronts of a light wave travelling in medium-1 and incident on \(X Y\). The lines ef and \(g h\) represent wavefronts of the light wave in medium-2 after refraction.
Question:
The phases of the light wave at \(c, d, e\) and \(f\) are \(\phi_c, \phi_d, \phi_e\) and \(\phi_f\), respectively. It is given that \(\phi_c \neq \phi_f\).

Consider a star of mass \(m_2 \mathrm{~kg}\) revolving in a circular orbit around another star of mass \(m_1 \mathrm{~kg}\) with \(m_1 \gg m_2\). The heavier star slowly acquires mass from the lighter star at a constant rate of \(\gamma \mathrm{kg} / \mathrm{s}\). In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is \(r\), then its relative rate of change \(\frac{1}{r} \frac{\mathrm{~d} r}{\mathrm{~d} t}\left(\right.\) in s \(\left.^{-1}\right)\) is given by:
[Note: The original question was awarded bonus marks in the JEE Advanced 2025 exam. We have changed the options to make the question valid and solvable.]

A heavy nucleus $N$, at rest, undergoes fission $N\to P+Q,$ where $P$ and $Q$ are two lighter nuclei. Let $\delta =M_N-M_P-M_Q$, where $M_P, M_Q$ and $M_N$ are the masses of $P,Q$ and $N$, respectively. $E_P$ and $E_Q$ are the kinetic energies of $P$ and $Q$, respectively. The speeds of $P$ and $Q$ are $v_P$ and $v_Q$, respectively. If $c$ is the speed of light, which of the following statement(s) is (are) correct?

A conducting loop in the shape of right angled isosceles triangle of height 10 cm is kept such that the ${90}^o$ vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire. The current in the triangular loop is in counterclockwise direction and increased at constant rate of $10A{s}^{-1}.$ Which of the following statement(s) is (are) true?

A thin convex lens is made of two materials with refractive indices ${\mathrm{n}}_1$ and ${\mathrm{n}}_2,$ as shown in figure. The radius of curvature of the left and right spherical surfaces are equal. $\mathrm{f}$ is the focal length of the lens when ${\mathrm{n}}_1={\mathrm{n}}_2=\mathrm{n}.$ The focal length is $\mathrm{f}+\mathrm{\Delta }\mathrm{f}$ when ${\mathrm{n}}_1=\mathrm{n}$ and ${\mathrm{n}}_2=\mathrm{n}+\mathrm{\Delta }\mathrm{n}.$ Assuming $\mathrm{\Delta }\mathrm{n}<<\left(\mathrm{n}-1\right)$ and $1<\mathrm{n}<2,$ the correct statement(s) is/are:

The circular scale of a screw gauge has 50 divisions and pitch of \(0.5 \mathrm{~mm}\). Find the diameter of sphere. Main scale reading is 2 .

If \(a, b\) and \(c\) are the sides of a \(\triangle A B C\) such that \(x^2-2(a+b+c) x+3 \lambda(a b+b c+c a)=0\) has real roots, then

\[
\text { Match the statements given in Column I with the values given in Column II. }
\]

Statement 1 The plot of atomic number ( \(Y\)-axis) versus number of neutrons ( \(x\)-axis) for stable nuclei shows a curvature towards \(X\)-axis from the line of \(45^{\circ}\) slope as the atomic number is increased.
Statement 2 Proton-proton electrostatic repulsions begin to overcome attractive forces involving protons and neutrons in heavier nuclides.

Two spherical stars $A$ and $B$ have densities ${\rho }_A$ and ${\rho }_B$, respectively. $A$ and $B$ have the same radius, and their masses $M_A$ and $M_B$ are related by $M_B=2M_A$. Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains ${\rho }_A$. The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being ${\rho }_A$. If $v_A$ and $v_B$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\frac{v_B}{v_A}=\sqrt{\frac{10n}{{15}^{{\displaystyle \frac{1}{3}}}}}$. The value of $n$ is