JEE Advanced 2019 Question Paper with Solutions

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 T 0 is a constant with dimension of temperature. The heat capacity of the metal is

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 cylindrical capillary tube of 0.2 mm radius is made by joining two capillaries T 1 and T 2 of different materials having water contact angles of 0 ∘ and 60 ∘ , 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/m 3 , take g = 10 m/s 2 ]

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?

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]

Which of the following statement(s) is(are) correct regarding the root mean square speed (U rms ) and average translational kinetic energy (E av ) of a molecule in a gas at equilibrium?

For the following reaction, the equilibrium constant K c at 298 K is 1.6 $\times$ 10 17 . Fe 2+ (aq) + S 2- (aq) ⇌ FeS(s) When equal volumes of 0.06 M Fe 2+ (aq) and 0.2 M S 2 $-$ (aq) solutions are mixed, the equilibrium concentration of Fe 2+ (aq) is found by Y $\times$ 10 $-$ 17 M. The value of Y is .................

Choose the reaction(s) from the following options, for which the standard enthalpy of reaction of equal to the standard enthalpy of formation.

Each of the following options contains a set of four molecules. Identify the option(s) where all four molecules posses permanent dipole moment at room temperature.

Choose the correct option(s) for the following set of reactions.

Define the collections {E 1 , E 2 , E 3 , ...} of ellipses and {R 1 , R 2 , R 3 .....} of rectangles as follows : ${E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$ R 1 : rectangle of largest area, with sides parallel to the axes, inscribed in E 1 ; En : ellipse ${{{x^2}} \over {a_n^2}} + {{{y^2}} \over {b_n^2}} = 1$ of the largest area inscribed in ${R_{n - 1}},n > 1$ ; R n : rectangle of largest area, with sides parallel to the axes, inscribed in E n , n > 1. Then which of the following options is/are correct?

Let $\omega \ne 1$ be a cube root of unity. Then the minimum of the set $\{ {\left| {a + b\omega + c{\omega ^2}} \right|^2}:a,b,c$ distinct non-zero integers} equals ..................

A line y = mx + 1 intersects the circle ${(x - 3)^2} + {(y + 2)^2}$ = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate $- {3 \over 5}$ , then which one of the following options is correct?

Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If A P ( 1 ; 3 ) ∩ A P ( 2 ; 5 ) ∩ A P ( 3 ; 7 ) = AP(a ; d), then a + d equals ..............

Let S be the set of all complex numbers z satisfying $\left| {z - 2 + i} \right| \ge \sqrt 5$ . If the complex number z 0 is such that ${1 \over {\left| {{z_0} - 1} \right|}}$ is the maximum of the set $\left\{ {{1 \over {\left| {{z_0} - 1} \right|}}:z \in S} \right\}$ , then the principal argument of ${{4 - {z_0} - {{\overline z }_0}} \over {{z_0} - {{\overline z }_0} + 2i}}$ is

A thin convex lens is made of two materials with refractive indices n 1 and n 2 , 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 n 1 = n 2 = n. The focal length is f + $\Delta$ f when n 1 = n and n 2 = n + $\Delta$ n. Assuming $\Delta$ n << (n - 1) and 1 < n < 2, the correct statement(s) is/are

A planar structure of length L 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 >> 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 $\times$ 10 8 m/s]

A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V 0 . A hole with a small area $\alpha$ 4 $\pi$ R 2 ( $\alpha$ << 1) is made on the shell without affecting the rest of the shell. Which one of the following statements is 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?

The area of the region {(x, y) : xy ≤ 8, 1 ≤ y ≤ x 2 } is

Let $\Gamma$ denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to I` at a point P intersect the y-axis at Y P . If PY P has length 1 for each point P on I`, then which of the following options is/are correct?

There are three bags B 1 , B 2 and B 3 . The bag B 1 contains 5 red and 5 green balls, B 2 contains 3 red and 5 green balls, and B 3 contains 5 red and 3 green balls. Bags B 1 , B 2 and B 3 have probabilities ${3 \over {10}}$ , ${3 \over {10}}$ and ${4 \over {10}}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

Let S be the sample space of all 3 $\times$ 3 matrices with entries from the set {0, 1}. Let the events E 1 and E 2 be given by E 1 = {A $\in$ S : det A = 0} and E 2 = {A $\in$ S : sum of entries of A is 7}. If a matrix is chosen at random from S, then the conditional probability P(E 1 | E 2 ) equals ...............

If $I = {2 \over \pi }\int\limits_{ - \pi /4}^{\pi /4} {{{dx} \over {(1 + {e^{\sin x}})(2 - \cos 2x)}}}$ , then 27I 2 equals .................

Fusion of MnO 2 with KOH in presence of O 2 produces a salt W. Alkaline solution of W upon electrolytic oxidation yields another salt X. The manganese containing ions present in W and X, respectively, are Y and Z. Correct statement(s) is (are)

In the decay sequence. x 1 , x 2 , x 3 and x 4 are particles/radiation emitted by the respective isotopes. The correct option(s) is(are)

Calamine, malachite, magnetic and cryolite, respectively, are

A tin chloride Q undergoes the following reactions (not balanced) $Q + C{l^ - } \to X$ $Q + M{e_3}N \to Y$ $Q + CuC{l_2} \to Z + CuCl$ X is a monoanion having pyramidal geometry. Both Y and Z are neutral compounds. Choose the correct option(s).

Molar conductivity ( $\Lambda$ m ) of aqueous solution of sodium stearate, which behaves as a strong electrolyte, is recorded at varying concentrations (C) of sodium stearate. Which one of the following plots provides the correct representation of micelle formation in the solution? (critical micelle concentration (CMC) is marked with an arrow in the figures)