In a radioactive sample, $${}_{19}^{40}K$$ nuclei either decay into stable $${}_{20}^{40}Ca$$ nuclei with decay constant 4.5 $$ \times $$ 10^-10 per year or into stable $${}_{18}^{40}Ar$$ nuclei with decay constant 0.5 $$ \times $$ 10^-10 per year. Given that in this sample all the stable $${}_{20}^{40}Ca$$ and $${}_{18}^{40}Ar$$ nuclei are produced by the $${}_{19}^{40}K$$ nuclei only. In time t $$ \times $$ 10⁹ years, if the ratio of the sum of stable $${}_{20}^{40}Ca$$ and $${}_{18}^{40}Ar$$ nuclei to the radioactive $${}_{19}^{40}K$$ nuclei is 99, the value of t will be

[Given : In 10 = 2.3]

A current carrying wire heats a metal rod. The wire provides a constant power (P) to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature (T) in the metal rod changes with time (t) as $$T(t) = {T_0}\left( {1 + \beta {t^{{1 \over 4}}}} \right)$$, where $$\beta $$ is a constant with appropriate dimension while T₀ is a constant with dimension of temperature. The heat capacity of the metal is

A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V₀. A hole with a small area $$\alpha $$4$$\pi $$R²($$\alpha $$ << 1) is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

Consider a spherical gaseous cloud of mass density $$\rho $$(r) in free space where r is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If $$\rho $$(r) is constant in time, the particle number density n(r) = $$\rho $$(r)/m is [G is universal gravitational constant]

A cylindrical capillary tube of 0.2 mm radius is made by joining two capillaries T₁ and T₂ of different materials having water contact angles of 0$$^\circ $$ and 60$$^\circ $$, respectively. The capillary tube is dipped vertically in water in two different configurations, case I and II as shown in figure. Which of the following option(s) is (are) correct? [Surface tension of water = 0.075 N/m, density of water = 1000 kg/m³, take g = 10 m/s²]

A conducting wire of parabolic shape, initially y = x², is moving with velocity $$v = {v_0}\widehat i$$ in a non-uniform magnetic field $$B = {B_0}\left( {1 + {{\left( {{y \over L}} \right)}^\beta }} \right)\widehat k$$, as shown in figure. If V₀, B₀, L and $$\beta $$ are positive constants and $$\Delta $$$$\phi $$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are

A thin convex lens is made of two materials with refractive indices n₁ and n₂, as shown in the figure. The radius of curvature of the left and right spherical surfaces are equal. f is the focal length of the lens when n₁ = n₂ = n. The focal length is f + $$\Delta $$f when n₁ = n and n₂ = n + $$\Delta $$n.

Assuming $$\Delta $$n << (n - 1) and 1 < n < 2, the correct statement(s) is/are

One mole of a monatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature (V-T) diagram. The correct statement(s) is/are [R is the gas constant]

In the circuit shown, initially there is no charge on the capacitors and keys S₁ and S₂ are open. The values of the capacitors are C₁ = 10$$\mu $$F, C₂ = 30$$\mu $$F and C₃ = C₄ = 80$$\mu $$F.



Which of the statement(s) is/are correct?

A charged shell of radius R carries a total charge Q. Given $$\phi $$ as the flux of electric field through a closed cylindrical surface of height h, radius r and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct?

[$$ \in $$₀ is the permittivity of free space]

Two identical moving coil galvanometers have 10$$\Omega $$ resistance and full scale deflection at 2$$\mu $$A current. One of them is converted into a voltmeter of 100 mV full scale reading and the other into an ammeter of 1 mA full scale current using appropriate resistors. These are then used to measure the voltage and current in the Ohm's law experiment with R = 1000$$\Omega $$ resistor by using an ideal cell. Which of the following statement(s) is/are correct?

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of L, which of the following statement(s) is/are correct?

A parallel plate capacitor of capacitance C has spacing d between two plates having area A. The region between the plates is filled with N dielectric layers, parallel to its plates, each with thickness, $$\delta = {d \over N}$$. The dielectric constant of the m^th layer is $${K_m} = K\left( {1 + {m \over N}} \right)$$. For a very large N(>10³), the capacitance C is $$\alpha \left( {{{K{\varepsilon _0}A} \over {d\ln 2}}} \right)$$

The value of $$\alpha $$ will be ..................

[$$ \in $$₀ is the permittivity of free space.]

A planar structure of length L and width W is made of two different optical media of refractive indices n₁ = 1.5 and n₂ = 1.44 as shown in figure. If L >> W, a ray entering from end AB will emerge from end CD. CD only if the total internal reflection condition is met inside the structure. For L = 9.6 m, if the incident angle $$\theta $$ is varied, the maximum time taken by a ray to exit the plane CD is t $$ \times $$ 10^-9 s, where, t is ................

[Speed of light, c = 3 $$ \times $$ 10⁸ m/s]

A block of weight 100 N is suspended by copper and steel wires of same cross-sectional area 0.5 cm² and length $$\sqrt 3 $$ m and 1 m, respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are 30$$^\circ $$ and 60$$^\circ $$, respectively. If elongation in copper wire is ($$\Delta {l_c}$$) and elongation in steel wire is ($$\Delta {l_s}$$), then the ratio $${{\Delta {l_c}} \over {\Delta {l_s}}}$$ is .............. .

[Young's modulus for copper and steel are 1 $$ \times $$ 10¹¹ N/m² and 2 $$ \times $$ 10¹¹ N/m² respectively.]

A liquid at 30$$^\circ $$C is poured very slowly into a Calorimeter that is at temperature of 110$$^\circ $$C. The boiling temperature of the liquid is 80$$^\circ $$C. It is found that the first 5 gm of the liquid completely evaporates. After pouring another 80 gm of the liquid the equilibrium temperature is found to be 50$$^\circ $$C. The ratio of the latent heat of the liquid to its specific heat will be ...........$$^\circ $$C.

[Neglect the heat exchange with surrounding]

A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force $$F = (\alpha y\widehat i + 2\alpha x\widehat j)$$ N, where x and y are in meter and $$\alpha $$ = $$ - $$1 Nm^-1. The work done on the particle by this force F will be ............... Joule.

A train S1, moving with a uniform velocity of 108 km/h, approaches another train S2 standing on a platform. An observer O moves with a uniform velocity of 36 km/h towards S2, as shown in figure.



Both the trains are blowing whistles of same frequency 120 Hz. When O is 600 m away from S2 and distance between S1 and S2 is 800 m, the number of beats heard by O is ............ . [Speed of the sound = 330 m/s ............ .]