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Six charges are placed around a regular hexagon of side length 𝑎 as shown in the figure. Five of them have charge 𝑞, and the remaining one has charge 𝑥. The perpendicular from each charge to the nearest hexagon side passes through the center O of the hexagon and is bisected by the side. Which of the following statement(s) is(are) correct in SI units? (A) When 𝑥=𝑞, the magnitude of the electric field at O is zero. (B) When 𝑥=−𝑞, the magnitude of the electric field at O is � �� �� �� . (C) When 𝑥=2𝑞, the potential at O is �� �√����� . (D) When 𝑥=−3𝑞, the potential at O is − �� �√����� .

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The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be 𝐸� � and the binding energy of a neutron be 𝐸� � in the nucleus. Which of the following statement(s) is(are) correct? (A) 𝐸� �−𝐸� � is proportional to 𝑍(𝑍−1) where Z is the atomic number of the nucleus. (B) 𝐸� �−𝐸�� is proportional to 𝐴� � � where A is the mass number of the nucleus. (C) 𝐸� �−𝐸�� is positive. (D) 𝐸� � increases if the nucleus undergoes a beta decay emitting a positron.

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Three plane mirrors form an equilateral triangle with each side of length 𝐿. There is a small hole at a distance 𝑙 > 0 from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle 𝜃 and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane. Which of the following statement(s) is(are) correct? (A) The ray of light will come out for 𝜃 = 30∘, for 0 < 𝑙 < 𝐿. (B) There is an angle for 𝑙 = � � at which the ray of light will come out after two reflections. (C) The ray of light will NEVER come out for 𝜃 = 60∘, and 𝑙 = � � (D) The ray of light will come out for 𝜃 = 60∘, and 0 < 𝑙 < � � . after six reflections.

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An ideal gas of density 𝜌 = 0.2 𝑘𝑔 𝑚�� enters a chimney of height ℎ at the rate of 𝛼 = 0.8 𝑘𝑔 𝑠�� from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is 𝐴� = 0.1 𝑚� and the upper end is 𝐴� = 0.4 𝑚�. The pressure and the temperature of the gas at the lower end are 600 𝑃𝑎 and 300 𝐾, respectively, while its temperature at the upper end is 150 𝐾. The chimney is heat insulated so that the gas undergoes adiabatic expansion. Take 𝑔 = 10 𝑚 𝑠�� and the ratio of specific heats of the gas � � =2. Ignore atmospheric pressure. Which of the following statement(s) is(are) correct? (A) The pressure of the gas at the upper end of the chimney is 300 𝑃𝑎. (B) The velocity of the gas at the lower end of the chimney is 40 𝑚 𝑠�� and at the upper end is 20 𝑚 𝑠��. (C) The height of the chimney is 590 𝑚. (D) The density of the gas at the upper end is 0.05 𝑘𝑔 𝑚��.

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The figure shows a circuit having eight resistances of 1 Ω each, labelled 𝑅� to 𝑅�, and two ideal batteries with voltages ℰ� = 12 𝑉 and ℰ� = 6 𝑉. � �� � �� 𝑅� ℰ� � �� ℰ� � �� � �� Which of the following statement(s) is(are) correct? � �� 𝑅� (A) The magnitude of current flowing through 𝑅� is 7.2 A. (B) The magnitude of current flowing through 𝑅� is 1.2 A. (C) The magnitude of current flowing through 𝑅� is 4.8 A. (D) The magnitude of current flowing through 𝑅� is 2.4 A.

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A medium having dielectric constant K >1 fills the space between the plates of a parallel plate capacitor. The plates have large area, and the distance between them is d. The capacitor is connected to a battery of voltage V, as shown in Figure (a). Now, both the plates are moved by a distance of � � from their original positions, as shown in Figure (b). d 𝑑 2 V d � � 2 V Figure (a) Figure (b) In the process of going from the configuration depicted in Figure (a) to that in Figure (b), which of the following statement(s) is(are) correct? (A) The electric field inside the dielectric material is reduced by a factor of 2𝐾. (B) The capacitance is decreased by a factor of � ��� . (C) The voltage between the capacitor plates is increased by a factor of (𝐾 + 1). (D) The work done in the process DOES NOT depend on the presence of the dielectric material.

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A projectile is fired from horizontal ground with speed 𝑣 and projection angle 𝜃. When the acceleration due to gravity is 𝑔, the range of the projectile is 𝑑. If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is � �� = � �.�� , then the new range is 𝑑� = 𝑛𝑑. The value of n is _____.

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Consider an LC circuit, with inductance 𝐿 = 0.1 𝐻 and capacitance 𝐶 = 10��𝐹, kept on a plane. The area of the circuit is 1 𝑚�. It is placed in a constant magnetic field of strength 𝐵� which is perpendicular to the plane of the circuit. At time 𝑡 = 0, the magnetic field strength starts increasing linearly as 𝐵 = 𝐵� + 𝛽𝑡 with 𝛽 = 0.04 𝑇𝑠��. The maximum magnitude of the current in the circuit is _____ mA.

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A solid sphere of mass 1 kg and radius 1 m rolls without slipping on a fixed inclined plane with an angle of inclination 𝜃 = 30° from the horizontal. Two forces of magnitude 1 N each, parallel to the incline, act on the sphere, both at distance 𝑟 = 0.5 m from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is ____ 𝑚𝑠��. (Take 𝑔 = 10 𝑚 𝑠��.)

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A 2 cm rod is inclined at 2π/3 rad to the principal axis of a convex lens (f = 10 cm). The lens is placed 40/3 cm from the object. The image height is (30√3)/13 cm, and its angle with the principal axis is α = π/n rad. Find n.

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At time 𝑡 = 0, a disk of radius 1 m starts to roll without slipping on a horizontal plane with an angular acceleration of 𝛼 = � � 𝑟𝑎𝑑 𝑠��. A small stone is stuck to the disk. At 𝑡 = 0, it is at the contact point of the disk and the plane. Later, at time 𝑡 = √𝜋 𝑠, the stone detaches itself and flies off tangentially from the disk. The maximum height (in m) reached by the stone measured from the plane is � � + � �� . The value of x is _____. [Take 𝑔 = 10 𝑚 𝑠��.]

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In the circuit below, C₁ = 12 μF, C₂ = C₃ = 4 μF, C₄ = C₅ = 2 μF. Find the charge (in μC) stored in C₃.

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Find the minimum kinetic energy (in MeV) required for an alpha particle to cause the nuclear reaction: ¹⁶₇N + ⁴₂He → ¹₁H + ¹⁹₈O Given: Masses - ¹⁶₇N: 16.006 u, ⁴₂He: 4.003 u, ¹₁H: 1.008 u, ¹⁹₈O: 19.003 u 1 u = 930 MeV/c².

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Two spherical stars A and B have densities ρ_A and ρ_B, respectively. They have the same radius, and their masses M_A and M_B satisfy M_B = 2M_A. Star A loses mass such that its radius halves while ρ_A remains constant. The lost mass forms a spherical shell around B with density ρ_A. If v_A and v_B are the escape velocities after this process, and (v_B/v_A) = √(10n/15^(1/3)), find n.

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