A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position x₀. Consider two cases:
(i) when the block is at x₀; and
(ii) when the block is at x = x₀ + A.
In both cases, a particle with mass m( < M) is softly placed on the block after which they stick on each other. Which of the following statement(s) is(are) true about the motion after the mass m is placed on the mass M?

An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?

The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by $$E = {3 \over 5}{{Z(Z - 1){e^2}} \over {4\pi {\varepsilon _0}R}}$$

The measured masses of the neutron, $$_1^1H$$, $$_7^{15}N$$ and $$_8^{15}O$$ are 1.008665u, 1.007825u, 15.000109u and 15.003065u, respectively. Given that the radii of both the $$_7^{15}N$$ and $$_8^{15}O$$ nuclei are same, 1 u = 931.5 MeV/c² (c is the speed of light) and e²/(4$$\pi$$$${{\varepsilon _0}}$$) = 1.44 MeV fm. Assuming that the difference between the binding energies of $$_7^{15}N$$ and $$_8^{15}O$$ is purely due to the electrostatic energy, the radius of either of the nuclei is (1 fm = 10^$$-$$15 m)

A small object is placed 50 cm to the left of a thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle $$\theta$$ = 30$$^\circ$$ to the axis of the lens, as shown in the figure.


If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point (x, y) at which the image is formed are