JEE (Advanced) 2019 Paper-1

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?

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

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

A block of weight 100 N is suspended by copper and steel wires of same cross sectional area $$0.5 cm^2$$ 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 copperand steel wires with ceiling are $$30^\circ and 60^\circ$$, respectively. If elongation in copper wire is $$(\triangle l_C)$$ and elongation in steel wire is $$(\triangle l_S)$$, then the ratio $$\frac{\triangle l_C}{\triangle l_S}$$ is.
[Young’s modulus for copper andsteel are $$1 \times 10^{11} N/m^2 and 2 \times 10^{11} N/m^2$$, respectively.]

A tram 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]

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 $$\delta = \frac{d}{N}$$. The dielectric constant of the $$m^{th}$$ layer is $$K_m = K \left(1 + \frac {m}{N}\right)$$, For a very large $$N (> 10^3)$$, the capacitance C is $$\alpha \left(\frac{K \epsilon_0A}{d ln 2}\right)$$. The value of $$\alpha$$ will be _________.
[$$\epsilon_0$$ is the permittivity of free space]

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 planar structure of length £ and width W is made of two different optical media of refractive indices $$n_1 = 1.5 and n_2 = 1.44$$ as shown in figure. If $$L \gg W$$, a ray entering from end AB will emerge from end 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 array to exit the plane CD is $$t \times 10^{-9} s$$, where t is _______.
[ Speed of light $$c = 3 \times 10^8 m/s$$]

The green colour produced in the borax beadtest of a chromium(III) salt is due to

Calamine, malachite, magnetite and cryolite, respectively, are