JEE Advanced 2016 Paper-1

In a historical experiment to determine Planck’s constant, a metal surface was irradiated with light of different wavelengths. The emitted photo electron energies were measured by applying a stopping potential. The relevant data for the wavelength ($$\lambda$$) of incident light and the corresponding stopping potential ($$V_0$$) are given below :

Given that $$c = 3 \times 10^8 ms^{-1}$$ and $$e = 1.6 \times 10^{-19} C$$, Planck’s constant (in units of J s) found from such an experiment is

A uniform wooden stick of mass 1.6 kg and length / rests in an inclined manneron a smooth, vertical wall of height A(</) such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of $$30^\circ$$ with the wall and the bottom of the stick is on a roughfloor. The reaction of the wall on the stick is equal in magnitudeto the reaction of the floor on the stick. The ratio h/l and thefrictional force f at the bottom of the stick are (g = 10 ms$$^2$$)

A water cooler of storage capacity 120 litres can cool water at a constant rate of P watts. In a closed circulation system (as shown schematically in the figure), the water from the cooler is used to cool an external device that generates constantly 3 kW of heat (thermal load). The temperature of water fed into the device cannot exceed $$30^\circ C$$ and the entire stored 120 litres of water is initially cooled to $$10^\circ C$$. The entire system is thermally insulated. The minimum value of P (in watts) for which the device can be operated for 3 hours is

(Specific heat of water is 4.2 KJ kg$$^{-1}$$ K$$^{-1}$$ and the density of water is 1000 kg m$$^{-3}$$)

A parallel beam of light is incident from air at an angle a on the side PQ of a right angled triangular prism of refractive index $$n = \sqrt{2}$$. Light undergoes total internal reflection in the prism at the face PR when a has a minimum value of $$45^\circ$$. The angle $$\theta$$ of the prism is

An infinite line charge of uniform electric charge density $$\lambda$$ lies along the axis of an electrically conducting infinite cylindrical shell of radius R. At time t = 0, the space inside the cylinder is filled with a material of permittivity ε and electrical conductivity $$\sigma$$. The electrical conduction in the material follows Ohm’s law. Which one of the following graphs best describes the subsequent variation of the magnitude of current density j(t) at any point in the material?

Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where n >> 1. Which of the
following statement(s) is(are) true?

Two loudspeakers M and WN are located 20 m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A caris initially at a point P , 1800 m away from the midpoint Q
of the line MN and moves towards @ constantly at 60 km/hr along the perpendicular bisector of MN. It crosses Q and eventually reaches a point R, 1800 m away from Q. Let v(t) represent the beat frequency measured by a person sitting in the car at time t. Let $$V_P, V_Q$$ and $$V_R$$ be the beat frequencies measured at locations P, Q and R, respectively. The speed of sound in air is 330 ms$$^{-1}$$ Which of the following statement(s) is(are) true regarding the sound heard bythe person?

An incandescent bulb has a thin filament of tungsten that is heated to high temperature by passing an electric current. The hot filament emits black-body radiation. The filament is observed to break up at randomlocations after a sufficiently long time of operation due to non-uniform evaporation of tungsten from the filament. If the bulb is powered at constant voltage, which of the following statement(s) is(are) true?

A plano-convex lens is made ofa material of refractive index n. When a smallobject is placed 30 cm awayin front of the curved surface of the lens, an image of double the size of the object is produced. Dueto reflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm away from the lens. Which of the following statement(s)
is(are) true?

A length-scale (J) depends on the permittivity ($$ε$$) of a dielectric material, Boltzmann constant ($$k_B$$), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles. Which of the following expression(s) for l is(are) dimensionally correct?

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)

The increasing order of atomic radii of the following Group 13 elements is

Among $$[Ni (CO)_4], [NiCl_4]^{2-}, [Co(NH_3)_4Cl_2]Cl, Na_3[CoF_6], Na_2O_2$$ and $$CsO_2$$, the total numberof paramagnetic compounds is

On complete hydrogenation, natural rubber produces

According to the Arrhenius equation,

A plot of the number of neutrons (NV) against the number of protons (P) of stable nuclei exhibits upward deviation from linearity for atomic number, Z > 20. For an unstable nucleus having N/P ratio less than 1, the possible mode(s) of decay is(are)

The crystalline form of borax has

The compound(s) with TWO lonepairs of electrons on the central atom is(are)

The reagent(s) that can selectively precipitate $$S^{2-}$$ from a mixture of $$S^{2-}$$ and $$SO_{4}^{2-}$$ in aqueous solution is(are)

Positive Tollen’s test is observed for

The product(s) of the following reaction sequence is(are)

The correct statement(s) about the following reaction sequence is(are)

The mole fraction of a solute in a solution is 0.1. At 298 K, molarity of this solution is the sameas its molality. Density of this solution at 298 K is 2.0 gcm$$^{-3}$$. Theratio of the molecular weights of the solute and solvent, $$\left(\frac{MW_{solute}}{MW_{solvent}}\right)$$, is

The diffusion coefficient of an ideal gas is proportional to its mean free path and meanspeed. The absolute temperature of an ideal gas is increased 4 times and its pressure is increased 2 times. As a result, the diffusion coefficient of this gas increases x times. The value of x is

In neutral or faintly alkaline solution, 8 moles of permanganate anion quantitatively oxidize thiosulphate anions to produce X moles of a sulphur containing product. The magnitude of X is

The number of geometric isomers possible for the complex $$[CoL_2Cl_2]^- (L = H_2NCH_2CH_2O^-)$$ is

In the following monobromination reaction, the number of possible chiral products is

Let $$-\frac{\pi}{6} < \theta < \frac{\pi}{12}$$ Suppose $$\alpha_1$$ and $$\beta_1$$ are the roots of the equation $$x^2 - 2x \sec \theta + 1$$ and $$\alpha_2$$ and $$\beta_2$$ are the roots of the equation $$x^2 + 2x \tan \theta - 1 = 0$$. If $$\alpha_1 > \beta_1$$ and $$\alpha_2 > \beta_2$$ then $$\alpha_1 + \beta_2$$ equals

A debate club consists of 6 girls and 4 boys. A team of 4 membersis to be selected from this club including the selection of a captain (from among these 4 members) for the team.If the team hasto include at most one boy, then the numberof waysof selecting the team is

Let $$ S = \left\{x \epsilon (-\pi, \pi): x \neq 0, \pm \frac{\pi}{2}\right\}$$ The sum of all distinct solutions of the equation $$\sqrt{3} \sec x + \cosec x + 2(\tan x - \cot x) = 0$$ in the set S is equal to

A computer producing factory has only two plants $$T_1$$ and $$T_2$$ Plant $$T_1$$ produces 20% and plant $$T_2$$ produces 80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that P (computer turns out to be defective given that it is produced in plant $$T_1$$) = 10P (computer turns out to be defective given that it is produced in plant $$T_2$$), where P(E) denotes the probability of an event E. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is producedin plant $$T_2$$ is

The least value of $$\alpha \in R$$ for which $$4 \alpha x^2 + \frac{1}{x} \geq 1$$, for all x > 0, is

Consider a pyramid OPQRS located in the first octant $$(x \geq 0, y \geq 0, z \geq 0)$$ with O as origin, and OP and ORalongthe x-axis and the y-axis, respectively. The base OPQR of the pyramid is a square with OP =3. Thepoint S is directly above the mid-point T of diagonal OQ such that TS = 3. Then

Let $$f : (0, \infty) \rightarrow R$$ be a differentiable function such that $$f'(x) = 2 - \frac{f(x)}{x}$$ for all $$x \in (0, \infty)$$ and $$f(x) \neq 1$$. Then

Let $$P = \begin{bmatrix}3 & -1 & -2 \\2 & 0 & \alpha \\3 & -5 & 0 \end{bmatrix}$$ where $$\alpha \in R$$. Suppose $$Q = [q_{ij}]$$ is a matrix such that $$PQ = kI$$, where $$k \in R, k \neq 0$$ and $$I$$ is the identity matrix of order 3. If $$q_{23} = -\frac{k}{8}$$ and $$det(Q) = \frac{k^2}{2}$$, then

In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X, Y, Z, respectively, and $$2s = x + y + z$$. If $$\frac{s - x}{4} = \frac{s - y}{3} = \frac{s - z}{2}$$ andareaofincircle of the triangle XYZ is $$\frac{8 \pi}{3}$$, Then

A solution curve of the differential equation $$(x^2 + xy + 4x + 2y + 4)\frac{dy}{dx} - y^2 = 0, x > 0$$, passes through the point (1, 3). Then the solution curve

Let $$f : R \rightarrow R, g : R \rightarrow R$$ and $$h : R \rightarrow R$$ be differentiable functions such that $$f(x) = x^3 + 3x + 2, g(f(x)) = x$$ and $$h(g(g(x))) = x$$ for all $$x \in R$$. Then

The circle $$C_1: x^2 + y^2 = 3$$, with centre at O, intersects the parabola $$x^2 = 2y$$ at the point P in the first quadrant. Let the tangent to the circle $$C_1$$, at P touches other two circles $$C_2$$ and $$C_3$$ at $$R_2$$ and $$R_3$$, respectively. Suppose $$C_2$$ and $$C_3$$ have equal radii $$2\sqrt{3}$$ and centres $$Q_2$$, and $$Q_3$$, respectively. If $$Q_2$$ and $$Q_3$$ lie on the y-axis, then

Let RS be the diameter of the circle $$x^2 + y^2 = 1$$, where S is the point (1,0). Let P be a variable point (other than R and S) on thecircle and tangents to the circle at S and P meet at the point Q@. The normal to the circle at P intersects a line drawn through Q parallel to RS at point £. Then thelocus of £ passes through the point(s)

The total number of distinct $$x \in R$$ for which

Let m be the smallest positive integer such that the coefficient of $$x^2$$ in the expansion of $$(1 + x)^2 + (1 + x)^3 + ....... + (1 + x)^{49} + (1 + mx)^{50}$$ is $$(3n + 1)^{51}C_3$$ for some positive integer n. Then the value of n is

The total number of distinct $$x \in [0, 1]$$ for which $$\int_{0}^{x} \frac{t^2}{1 + t^4}dt = 2x - 1$$ is

Let $$\alpha, \beta \in R$$ be such that $$ \lim_{x \rightarrow 0}\frac{x^2 \sin (\beta x)}{\alpha x - \sin x} = 1$$. Then $$6(\alpha + \beta)$$ equals

Let $$z = \frac{-1 + \sqrt{3}i}{2}$$, where $$i = \sqrt{-1}$$, and $$r, s \in \left\{1, 2, 3\right\}$$. Let $$P = \begin{bmatrix}(-z)^r & z^{2s} \\z^{2s} & z^r \end{bmatrix}$$ and I be the identity matrix of order 2. Then the total number of ordered pairs $$(r, s)$$ for which $$P^2 = -I$$ is