JEE Advanced 2019 Paper-2

A thin and uniform rod of mass M and length L is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle $$60^\circ$$ with vertical?
[g is the acceleration due to gravity]

A block of mass 2M is attached to a massless spring with spring-constant k. This block is connected to two other blocks of masses M and 2M using two massless pulleys and strings. The accelerations of the blocks are $$a_1, a_2$$ and $$a_3$$ as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $$x_0$$. Which of the following option(s) is/are correct?
[g is the acceleration due to gravity. Neglect friction]

A small particle of mass m moving inside a heavy, hollow and straight tube along the tube axis undergoeselastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other endis fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is $$L = L_0$$ the particle speed is $$v = v_0$$. The piston is moved inward at a very low speed V such that $$V \ll \frac{dL}{L} v_0,$$ where dL is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?

An electric dipole with dipole moment $$\frac{p_o}{\sqrt 2} (\widehat{l}\widehat{J})$$ is held fixed at the origin O in the presence of an uniform electric field of magnitude $$E_0$$. If the potential is constant on a circle of radius R centered at the origin as shown in figure, then the correct statement(s) is/are:
($$\epsilon_0$$ is permittivity of free space. $$ R \gg$$ dipole size)

A mixture of ideal gas containing 5 moles of monatomic gas and 1 mole of rigid diatomic gas is initially at pressure $$P_0$$, volume $$V_0$$, and temperature $$T_0$$. If the gas mixture is adiabatically compressed to a volume $$\frac{V_0}{4}$$, then the correct statement(s)is/are,
(Given $$2^{1.2} = 2.3; 2^{3.2} = 9.2;$$ R is gas constant)

Three glass cylinders of equal height H = 30 cm and same refractive index n = 1.5 are placed on a horizontal surface as shown in figure. Cylinder I has a flat top, cylinder II has a convex top and cylinder III has a concave top. The radu of curvature of the two curved tops are same (R = 3 m). If $$H_1, H_2$$ and $$H_3$$ are the apparent depths of a point_X on the bottom of the three cylinders, respectively, the correct statement(s) is/are:

In a Young’s double slit experiment, the slit separation d is 0.3 mm and the screen distance D is 1 m. A parallel beam of light of wavelength 600 nm is incident on the slits at angle $$\alpha$$ as shown in figure. On the screen, the point O is equidistant from the slits and distance PO is 11.0 mm. Which of the following statement(s) is/are correct?

A free hydrogen atom after absorbing a photon of wavelength $$\lambda_a$$ gets excited from the state n = 1 to the state n = 4. Immediately after that the electron jumps to n = m state by emitting a photon of wavelength $$\lambda_e$$. Let the change in momentum of atom due to the absorption and the emission are $$\triangle p_a$$ and $$\triangle p_e,$$ respectively. If $$\frac{\lambda_a}{\lambda_e} = \frac{1}{5}$$ which of the option(s) is/are correct?
[Use hc = 1242 eV nm ; 1 nm = $$10^{-9}$$ m, h and c are Planck’s constant and speed of light, respectively]

A ball is thrown from ground at an angle $$\theta$$ with horizontal and with aninitial speed $$\mu_0$$. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is $$V_1$$. After hitting the ground, the ball rebounds at the same angle $$\theta$$ but with a reduced speed of $$\frac{\mu_0}{\alpha}$$. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is $$0.8 V_1,$$ the value of $$\alpha$$ is __________ .

A 10 cm long perfectly conducting wire PQ is moving with a velocity 1 cm/s on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor L = 1 mH and a resistance $$R = 1 Ω$$ as shown in figure. The horizontal rails, L and R lie in the same plane with a uniform magnetic field B = 1 T perpendicularto the plane. If the key S is closed at certain instant, the current in the circuit after 1 millisecond is $$x \times 10^{-3} A$$, where the value of x is ___________ .
[Assume the velocity of wire PQ remains constant (1 cm/s) after key S is closed. Given: $$e^{-1} = 0.37$$, where e is base of the natural logarithm ]

A monochromatic light is incident from air on a refracting surface of a prism of angle $$75^\circ$$ and refractive index $$n_0 = \sqrt 3$$. The other refracting surface of the prism is coated by a thin film of material of refractive index n as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of $$\theta \leq 60^\circ$$. The value of $$n^2$$ is ___________ .

A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency $$Ω$$ such that $$\frac{4 \pi M Ω}{h} = 10^{24} m^{-2}$$ with h as Planck’s constant. N photons of wavelength $$\lambda = 8 \pi \times 10^{-6} m$$ strike the mirror simultaneously at normal incidence such that the mirror gets displaced by $$1 \mu m.$$ If the value of N is $$x \times 10^{12},$$ then the value of x is ________ .
[Consider the spring as mass less]

Suppose a $$_{88}^{226} Ra$$ nucleus at rest and in ground state undergoes $$\alpha-$$decay to a $$_{86}^{222} Rn$$ nucleus in its excited state. The kinetic energy of the emitted $$\alpha$$ particle is found to be 4.44 MeV. $$_{86}^{222} Rn$$ nucleus then goes to its ground state by $$\gamma-$$decay. The energy of the emitted $$\gamma$$ photon is ________ ke V.
[Given: atomic mass of $$_{88}^{226} Ra = 226.005 u,$$ atomic mass of $$_{86}^{222} Rn = 222.000 u,$$ atomic mass of $$\alpha$$ particle = 4.000 u, $$1 u = 931 Me V/c^2,$$ c is speed of the light]

An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at 135 cm mark. In this experiment, the percentage error in the measurement ofthe focal length ofthe lens is _____________ .

If the tension in each string is $$T_0$$, the correct match for the highest fundamental frequency in $$f_0$$ units will be,

The length of the strings 1, 2, 3 and 4 are kept fixed at $$L_0, \frac{3L_0}{2}, \frac{5L_0}{4}$$ and $$\frac{7L_0}{4}$$, respectively. Strings 1, 2, 3, and 4 are vibrated at their $$1^{st}, 3^{rd}, 5^{th}$$ and $$14^{th}$$ harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of $$T_0$$ will be,

If the process carried out on one mole of monatomic ideal gas 1s as shown in figure in the PV -diagram with $$P_0 V_0 = \frac{1}{3} RT_0$$ the correct match is

If the process on one mole of monatomic ideal gas 1s as shown in the TV-diagram with $$P_0 V_0 = \frac{1}{3} RT_0$$ the correct match is,

The cyanide process of gold extraction involves leaching out gold from its ore with $$CN^{-}$$ in the presence of Q in water to form R. Subsequently, R is treated with T to obtain Au and Z. Choose the correct option(s)

With reference to aqua regia, choose the correct option(s)

Consider the following reactions (unbalanced)
Zn + hot conc.$$H_2SO_4 \rightarrow G + R + X$$
Zn + conc.$$NaOH \rightarrow T + Q$$
G + $$H_2S + NH_4OH \rightarrow Z$$ (a precipitate) + X + Y
Choose the correct option(s)

The ground state energy of hydrogen atom is —13.6 eV. Consider an electronic state $$\psi$$ of $$He^+$$ whose energy, azimuthal quantum number and magnetic quantum number are —3.4 eV, 2 and 0, respectively. Which of the following statement(s) is(are) true for the state $$\psi$$?

Which of the following reactions produce(s) propane as a major product?

Choose the correct option(s) that give(s) an aromatic compound as the major product

Choose the correct option(s) for the following reaction sequence

Consider Q, R and S as major products

Choose the correct option(s) from the following

The amount of water produced (in g) in the oxidation of 1 mole of rhombic sulphur by conc. $$HNO_3$$ to a compound with the highest oxidation state of sulphur is ___________ .
(Given data: Molar mass of water = $$18 g mol^{-1}$$)

Total number of c is N-Mn-Cl bond angles (that is, Mn-N and Mn-Cl bonds in c is positions) present in a molecule of c is-$$[Mn(en)_2 Cl_2]$$ complex is (en = $$NH_2CH_2CH_2NH_2)$$

The decomposition reaction

is started in a closed cylinder under isothermal isochoric condition at an initial pressure of 1 atm. After $$Y \times 10^3 s$$, the pressure inside the cylinder is found to be 1.45 atm. If the rate constant of the reaction is $$5 \times 10^{-4} s^{-1}$$, assuming ideal gas behavior, the value of Y is __________ .

The mole fraction of urea in an aqueous urea solution containing 900 g of water is 0.05. If the density of the solution is $$1.2 g cm^{-3}, the molarity of urea solution is
(Given data: Molar masses of urea and water are $$60 g mol^{-1}$$ and $$18 g mol^{-1},$$ respectively)

Total number of hydroxyl groups present in a molecule of the major product P is ______ .

Total number of isomers, considering both structural and stereoisomers, of cyclic ethers with the molecular formula $$C_4H_8O$$ is

Which of the following options has the correct combination considering List-I and List-II?

Which of the following options has the correct combination considering List-I and List-II?

Which of the following options has correct combination considering List-I and List-II?

Which of the following options has correct combination considering List-I and List-II?

Let
$$P_1 = I = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}, P_2 = \begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1\\0 & 1 & 0\end{bmatrix}, P_3 = \begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0\\0 & 0 & 1\end{bmatrix},$$
$$P_4 = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1\\1 & 0 & 0\end{bmatrix}, P_5 = \begin{bmatrix}0 & 0 & 1 \\1 & 0 & 0\\0 & 1 & 0\end{bmatrix}, P_6 = \begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0\\1 & 0 & 0\end{bmatrix},$$
and $$X = \sum_{k = 1}^6 P_k \begin{bmatrix}2 & 1 & 3 \\1 & 0 & 2\\3 & 2 & 1\end{bmatrix} P_k^T$$
where $$P_k^T$$ denotes the transpose of the matrix $$P_k$$. Then which of the following options is/are correct?

Let $$x \in R$$ and let
$$P = \begin{bmatrix}1 & 1 & 1 \\0 & 2 & 2\\0 & 0 & 3\end{bmatrix}, Q = \begin{bmatrix}2 & x & x \\0 & 4 & 0\\x & x & 6\end{bmatrix}$$ and $$R = PQP^{-1}$$
Then which of the following options is/are correct?

For non-negative integers n,let
$$f(n) = \frac{{\sum_{k = 0}^n}\sin \left(\frac{k + 1}{n + 2}\pi\right) \sin \left(\frac{k + 2}{n + 2}\pi\right)}{\sum_{k = 0}^n \sin^2 \left(\frac{k + 1}{n + 2}\pi\right)} $$
Assuming $$\cos^{-1}x$$ takes values in $$[0,\pi],$$ which of the following options is/are correct?

Let $$f: R \rightarrow R$$ bea function. We say that f has
PROPERTY 1 if $$\lim_{h \rightarrow 0} \frac{f(h) - f(0)}{\sqrt{|h|}}$$ exists and is finite, and
PROPERTY 2 if $$\lim_{h \rightarrow 0} \frac{f(h) - f(0)}{h^2}$$ exists and is finite.
Then which of the following options is/are correct?

Let
$$f(x) = \frac{\sin \pi x}{x^2}, x > 0.$$
Let $$x_1 < x_2 < x_3 < ... < x_n < ...$$ be all the points of local maximum off
and $$y_1 < y_2 < y_3 < ... < y_n < ...$$ be all the points of local minimum off.
Then which of the following options is/are correct?

For $$a \in R, |a| > 1,$$ let
$$\lim_{n \rightarrow \infty} \left(\frac{1 + \sqrt[3]{2} + ... + \sqrt[3]{n}}{n^{\frac{7}{3} \left(\frac{1}{\left(an + 1\right)^2} + \frac{1}{\left(an + 2\right)^2} + ... + \frac{1}{\left(an + n\right)^2} \right)}}\right) = 54.$$
Then the possible value(s) of a is/are

Let $$f : R \rightarrow R$$ be given by $$f(x) = (x - 1)(x - 2)(x - 5).$$ Define
$$F(x) = \int_{0}^{x} f(t) dt, x > 0.$$
Then which of the following options is/are correct?

Three lines
$$L_1 : \overrightarrow{r} = \lambda \widehat{i}, \lambda \in R$$
$$L_2: \overrightarrow{r} = \widehat{k} + \mu \widehat{j}, \mu \in R$$ and
$$L_3: \overrightarrow{r} = \widehat{i} + \widehat{j} + v \widehat{k}, v \in R$$
are given. For which point(s) Q on $$L_2$$ can wefind a point P on $$L_1$$ and a point R on $$L_3$$ so that P,Q and R are collinear?

Suppose
$$det\begin{bmatrix} \sum_{k = 0}^nk & \sum_{k = 0}^n {^nC_k k^2} \\\sum_{k = 0}^n {^n C_k k} & \sum_{k = 0}^n {^n C_k 3^k} \end{bmatrix} = 0$$
holds for somepositive integer n. Then $$\sum_{k = 0}^n \frac {^nC_k}{k + 1}$$ Equals __________

Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is__

Let |X| denote the number of elements in a set X. Let S = {1, 2,3, 4,5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such that $$1 \leq |B| < |A|$$, equals ________

The value of
$$\sec^{-1} \left(\frac{1}{4}\sum_{k = 0}^{10} \sec \left(\frac{7\pi}{12} + \frac{k\pi}{2}\right) \sec \left(\frac{7\pi}{12} + \frac{\left(k + 1\right)\pi}{2}\right) \right)$$
in the interval $$\begin{bmatrix}-\frac {\pi}{4}, & \frac{3\pi}{4} \end{bmatrix}$$ equals _______

The value of the integral
$$\int_{0}^{\pi/2} \frac{3 \sqrt{\cos \theta}}{\left(\sqrt{\cos \theta} + \sqrt{\sin \theta}\right)^5} d \theta$$
equals ______

Let $$\overrightarrow{a} = 2\widehat{i} + \widehat{j} - \widehat{k}$$ and $$\overrightarrow{b} = \widehat{i} + 2\widehat{j} + \widehat{k}$$ be two vectors. Consider a vector $$\overrightarrow{c} = \alpha \overrightarrow{a} + \beta \overrightarrow{b}, \alpha, \beta \in R.$$ If the projection of $$\overrightarrow{c}$$ on the vector $$\left(\overrightarrow{a} + \overrightarrow{b}\right)$$ is $$3\sqrt{2},$$ then the minimum value of $$\left(\overrightarrow{c} - \left(\overrightarrow{a} \times \overrightarrow{b}\right)\right) . \overrightarrow{c}$$ equals _________ .

Which of the following is the only CORRECT combination?

Which ofthe following is the only CORRECT combination?

Which of the following is the only CORRECT combination?

Which of the following is the only INCORRECT combination?