JEE (Advanced) 2016 Paper-1

A conducting loop in the shape of a right angled isosceles triangle of height 10 cm is kept such that the $$90^\circ$$ 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 counter clockwise direction and increased at a constant rate of 10 As$$^{-1}$$. Which of the following statement(s) is(are) true?

The position vector $$\overrightarrow{r}$$ of a particle of mass m is given by the following equation 

$$\overrightarrow{r}(t) = \alpha t^3 \hat{i} + \beta t^2 \hat{j}$$
where $$\alpha = \frac{10}{3} ms^{-3}, \beta = 5 ms^{-2}$$ and m=0.1 kg. At t=1 s, which of the following statement(s) is(are) true about the particle?

A transparent slab of thickness d has a refractive index n(z) that increases with z. Here z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices $$n_1$$ and $$n_2(> n_1)$$, as shown in the figure. A ray of light is incident with angle $$\theta_i$$ from medium 1 and emerges in medium 2 with refraction angle $$\theta_f$$, with a lateral displacement l.

Which of the following statement(s) is(are) true?

A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated (P) by the metal. The sensor hasa scale that displays log, $$\left(\frac{P}{P_0}\right)$$, where $$P_0$$ is a constant. When the metal surface is at a temperature of $$487^\circ C$$, the sensor shows a value 1. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to $$2767^\circ C$$?

The isotope $$_{5}^{12}B$$ having a mass 12.014 u undergoes $$\beta$$- decay to $$_{6}^{12}C$$, $$_{6}^{12}C$$ has an excited state of the nucleus ($$_{6}^{12}C^*$$) at 4.041 MeV above its ground state. If $$_{5}^{12}B$$ decays to $$_{6}^{12}C^*$$, the maximumkinetic energy of the $$\beta$$- particle in units of MeV is
(1 u = 931.5 MeV/c$$^2$$, where c is the speed of light in vacuum).

A hydrogen atom in its ground state is irradiated by light of wavelength 970 A. Taking $$hc/e = 1.237 \times 10^{-6} eV$$ m and the ground state energy of hydrogen atom as —13.6 eV, the numberof lines present in the emission spectrum is

Consider two solid spheres P and Q each of density 8 gm cm$$^{-3}$$ and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm cm$$^{-3}$$ and viscosity $$\eta = 3$$ poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm cm$$^{-3}$$ and viscosity $$\eta = 2$$ poiseulles. The ratio of the terminal velocities of P and Q is

Two inductors $$L_1$$(inductance 1 mH, internal resistance 3 Ω) and $$L_2$$ (inductance 2 mH, internal resistance 4 Ω), and a resistor R (resistance 12 Ω) are all connected in parallel across a 5 V battery. The circuit is switched on at time ¢=0. The ratio of the maximum to the minimum current $$\left(\frac{I_{max}}{I_{min}}\right)$$ drawn from the battery is

P is the probability of finding the 1s electron of hydrogen atom in a spherical shell of infinitesimal thickness, dr, at a distance r from the nucleus. The volumeof this shell is $$4 \pi r^2 dr$$. The qualitative sketch of the dependence of P on is

One mole of an ideal gas at 300 K in thermal contact with surroundings expands isothermally from 1.0 L to 2.0 L against a constant pressure of 3.0 atm. In this process, the
changein entropy of surroundings $$(\triangle S_{surr})$$ in $$JK^{-1}$$ is (1 Latm = 101.3 d)