JEE Advanced 2019 Paper-1

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 ofparticles 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 numberdensity n(r) = $$\rho$$(r)/m is
[G is universal gravitational constant]

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 $$\alpha 4 \pi R^2 (\alpha \ll 1)$$ is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

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) inthe metal rod changes with time (t) as
$$T(t) = T_0 (1 + \beta t^{\frac{1}{4}}),$$
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,

In a radioactive sample, $$_{19}^{40}K$$ nuclei either decay into stable $$_{20}^{40}Ca$$ nuclei with decay constant $$4.5 \times 10^{-19}$$ 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 $$t \times 10^9$$ 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: $$\ln 10 = 2.3$$]

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 = $$1000 kg/m^3$$, take $$g = 10 m/s^2$$]

A conducting wire of parabolic shape, initially $$y = x^2$$, is moving with velocity $$\overrightarrow{V} = V_0 \widehat{l}$$ in a non-uniform magnetic field $$\overrightarrow{B} = B_0 \left( 1 + \left(\frac {y}{L}\right)^\beta \right) \widehat{k}$$ as shown in figure. If $$V_0, B_0, L$$ and $$\beta$$ are positive constants and $$\triangle\phi$$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

In the circuit shown,initially there is no charge on capacitors and keys $$S_1 and S_2$$ are open. The values of the capacitors are $$C_1 = 10 \mu F, C_2 = 30 \mu F and C_3 = C_4 = 80 \mu F.$$

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?
[$$\epsilon_0$$ is the permittivity of free space]

One mole of a monatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature (V-7 ) diagram. The correct statement(s) is/are:
[R is the gas constant]

A thin convex lens is made of two materials with refractive indices $$n_1 and n_2$$ as shown in 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 $$f + \triangle f$$ when $$n_1 = n$$ and $$n_2 = n + \triangle n$$. Assuming $$\triangle n \ll (n — 1)$$ and 1 < n < 2, the correct statement(s) is/are

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

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

The correct order of acid strength of the following carboxylic acids is

A tin chloride Q undergoes the following reactions (not balanced)
$$Q + Cl^- \rightarrow X$$
$$Q + Me_3N \rightarrow Y$$
$$Q + CuCl_2 \rightarrow Z + CuCl$$
X is a monoanion having pyramidal geometry. Both Y and Z are neutral compounds. Choose the correct option(s)

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)

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

Which of the following statement(s) is (are) correct regarding the root mean square speed $$(\mu_{rms})$$ and average translational kinetic energy $$(\epsilon_{av})$$ of a molecule in a gas at equilibrium ?

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

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)

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

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

Among $$B_2H_6, B_3N_3H_6, N_2O, N_2O_4, H_2S_2O_3 and H_2S_2O_8$$, the total number of molecules containing covalent bond between two atoms of the same kind is ___________

At 143 K,the reaction of $$XeF_4$$ with $$O_2F_2$$ produces a xenon compound Y. The total number of lone pair(s) of electrons present on the whole molecule of Y is

For the following reaction, the equilibrium constant $$K_$$ at 298 K is $$1.6 \times 10^{17}$$
$$Fe^{2+} (aq) + S^{2-} (aq) \rightleftharpoons FeS (s)$$
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 to be $$Y \times 10^{-17} M$$. The value of Y is _________ .

On dissolving 0.5 g of a non-volatile non-ionic solute to 39 g of benzene, its vapor pressure decreases from 650 mm Hg to 640 mm Hg. The depression of freezing point of benzene (in K) upon addition of the solute is ___________ .
(Given data: Molar mass and the molal freezing point depression constant of benzene are $$78 g mol^{-1} and 5.12 K kg mol^{-1}$$ respectively)

Consider the kinetic data given in the following table for the reaction A + B + C $$\rightarrow$$ Product.

The rate of the reaction for $$[A] = 0.15 mol dm^{-3}, [B] = 0.25 mol dm^{-3} and [C] = 0.15 mol dm^{-3}$$ is found to be $$Y \times 10^{-5} mol dm^{-3} s^{-1}$$. The value of Y is _______ .

Schemes 1 and 2 describe the conversion of P to Q and to S, respectively. Scheme 3 describes the synthesis of T from Q and S. The total number of Br atoms in a molecule of T is _________ .

Let S be the set of all complex numbers z satisfying $$|z - 2 + i | \geq \sqrt 5$$ If the complex number $$Z_0$$ is such that $$\frac{1}{|z_0 - 1|}$$ is the maximum of the set $$\left\{\frac{1}{|z - 1|}: z \in S\right\}$$ then the principal argument of $$\frac{4 - z_0 - \overline{z_0}}{z_0 - \overline{z_0} + 2i}$$ is

Let
$$M = \begin{bmatrix}\sin^4\theta & -1-\sin^2\theta \\1 + \cos^2\theta & \cos^4\theta \end{bmatrix} = \alpha I + \beta M^{-1}$$
where $$\alpha = \alpha (\theta) and \beta = \beta (\theta)$$ are real numbers, and I is the $$2 \times 2$$ identity matrix. If
$$\alpha^*$$ is the minimum of the set $$\left\{\alpha (\theta): \theta \in [0, 2\pi)\right\}$$ and
$$\beta^*$$ is the minimum of the set $$\left\{\beta (\theta): \theta \in [0, 2\pi)\right\}$$,
then the value of $$\alpha^* + \beta^*$$ is

A line y = mx + 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 $$-\frac{3}{5},$$ then which one of the following options is correct?

The area of the region $${(x,y): xy \leq 8, 1 \leq y \leq x^2}$$ is

Let $$\alpha and \beta$$ be the roots of $$x^2 — x — 1 = 0,$$ with \alpha > \beta$$. For all positive integers n, define
$$a_n = \frac{\alpha^n - \beta^n}{\alpha - \beta},$$ $$n \geq 1,$$
$$b_1 = 1$$ and $$b_n = a_{n - 1} + a_{n + 1},$$ $$n \geq 2.$$
Then which of the following options is/are correct?

Let
$$M = \begin{bmatrix}0 & 1 & a \\1 & 2 & 3 \\3 & b & 1 \end{bmatrix}$$ and $$adj M = \begin{bmatrix}-1 & 1 & -1 \\8 & -6 & 2 \\-5 & 3 & -1 \end{bmatrix}$$
where a and b are real numbers. 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 $$\frac{3}{10}, \frac{3}{10}$$ and $$\frac{4}{10}$$ respectively of being chosen. A bag is selected at random anda ball is chosen at random from the bag. Then which of the following options is/are correct?

In a non-right-angled triangle $$\triangle PQR$$,let p, g,r denote the lengths of the sides opposite to the angles at P,Q,R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at FE, and RS and PE intersect at O. If $$p = \sqrt 3, q = 1,$$ and the radius of the circumcircle of the $$\triangle PQR$$ equals 1, then which of the following options is/are correct?

Define the collections {E_1, E_2, E_3, ... } of ellipses and {R_1, R_2, R_3, ... } of rectangles as follows:
$$E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1$$;
$$R_1$$: rectangle of largest area, with sides parallel to the axes, inscribed in $$E_1$$;
$$E_n$$: ellipse $$\frac{x^2}{a_n^2} + \frac{y^2}{b_n^2} = 1$$ of 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 $$f:R \rightarrow R$$ be given by
$$f(x) = \begin{cases}x^5 + 5x^4 + 10x^3 + 10x^2 + 3x +1 & x < 0;\\x^2 - x + 1, & 0 \leq x \leq 1; \\\frac{2}{3}x^3 - 4x^2 + 7x - \frac{8}{3}, & 1 \leq x < 3;\\ (x - 2) \log_e(x - 2) - x + \frac{10}{3}, & x \geq 3. \end{cases}$$
Then which of the following options is/are correct?

Let $$\lceil$$ 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 $$\lceil$$ at a point P intersect the y-axis at $$Y_P$$. If $$PY_P$$ has length 1 for each point P on $$\lceil$$ then which of the following options is/are correct?

Let $$L_1$$ and $$L_2$$ denote the lines
$$\overrightarrow{r} = \widehat{i} + \lambda (-\widehat{i} + 2\widehat{j} + 2\widehat{k}), \lambda \in R$$ and
$$\overrightarrow{r} = \mu (2\widehat{i} - \widehat{j} + 2\widehat{k}), \mu \in R$$
respectively. If $$L_3$$ is a line which is perpendicular to both $$L_1$$ and $$L_2$$ and cuts both of them, then which of the following options describe(s) $$L_3$$?

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

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
$$AP(1; 3) \cap AP(2; 5) \cap AP(3; 7) = AP(a; d)$$
then a + d equals __________

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
$$E_1 = \left\{A \in S : \det A = 0\right\}$$ 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 _______

Let the point B be the reflection of the point A(2, 3) with respect to the line 8x — 6y — 23 = 0. Let $$\lceil_A$$ and $$\lceil_B$$ be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles $$\lceil_A$$ and $$\lceil_B$$ such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing through A and B, then the length of the line segment AC is __________.

If
$$I = \frac{2}{\pi} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac {1}{(1 + e^{\sin x})(2 - \cos 2x)}$$
then 27 $$I^2$$ equals _______ .

Three lines are given by
$$\overrightarrow{r} = \lambda \widehat{i}, \lambda \in R$$
$$\overrightarrow{r} = \mu (\widehat{i} + \widehat{j}), \mu \in R$$ and
$$\overrightarrow{r} = v (\widehat{i} + \widehat{j} + \widehat{k} ), v \in R$$.
Let the lines cut the plane x + y + z = 1 at the points A, B and C respectively. If the area of the triangle ABC is $$\triangle$$ then the value of $$(6\triangle)^2$$ equals__________ .