JEE Advanced 2016 Paper-2

The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by
$$E = \frac{3}{5}\frac{Z(Z - 1)e^2}{4 \pi ε_0 R}$$
The measured masses of the neutron, $$^1_1H, ^{15}_7N$$ and $$^{15}_8O$$ are 1.008665 u, 1.007825 u, 15.000109 u and 15.003065 u,respectively. Given that the radii of both the $$^{15}_7N$$ and $$^{15}_8O$$ nuclei are same, $$1 u = 931.5 MeV/c^2$$ (c is the speed oflight) and $$\frac{e^2}{(4 \pi ε_0)} = 1.44 MeV fm$$. Assumingthat the difference between the binding energies of $$^{15}_7 N$$ and $$^{15}_8 O$$ is purely due to the electrostatic energy, the radius of either of the nuclei is$$(1 fm = 10^{-15}m)$$

An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?

A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure $$P_i = 10^5 Pa$$ and volume $$V_i = 10^{-3} m^3$$ changes to a final state at $$P_f = \left(\frac{1}{32}\right) \times 10^5 Pa$$ and $$V_f = 8 \times 10^{-3} m^3$$ in an adiabatic quasi-static process, such that $$P^3V^5$$ = constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at $$P_i$$. followed by an isochoric (isovolumetric) process at volume $$V_f$$. The amount of heat supplied to the system in the two-step process is approximately

The ends Q and of two thin wires, PQ and RS, are soldered (joined) together. Initially each of the wires has a length of 1 m at $$10^\circ C$$. Now the end P is maintained at $$10^\circ C$$, while the end S is heated and maintained at $$400^\circ C$$. The system is thermally insulated from its surroundings. If the thermal conductivity of wire PQ is twice that of the wire RS and the coefficient of linear thermal expansion of PQ is $$1.2 \times 10^{-5}K^{-1}$$ the change in length of the wire PQ is

A small object is placed 50 cm to the left of a thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle $$\theta = 30^\circ$$ to the axis of the lens, as shown in the figure.

If the origin of the coordinate system is taken to be at the center of the lens, the coordinates (in cm) of the point (x,y) at which the image is formed are

There are two Vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. The Vernier scale of one of the calipers $$(C_1)$$ has 10 equal divisions that correspond to 9 main scale divisions. The Vernier scale of the other caliper $$(C_2)$$ has 10 equal divisions that correspond to 11 main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in cm) by calipers $$C_1$$ and $$C_2$$, respectively, are

Two thin circular discs of mass m and 4m, having radii of a and 2a, respectively, are rigidly fixed by a mass less, rigid rod of length $$l = 24\sqrt{24} a$$ through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $$\omega$$. The angular momentum of the entire assembly about the point ‘O’ is $$\overrightarrow{L}$$ (see the figure). Which of the following statement(s) is(are) true?

Light of wave length $$\lambda_{ph}$$ falls on a cathode plate inside a vacuumtube as shown in the figure. The work function of the cathode surface is $$\phi$$ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is $$\lambda_{e}$$, which of the following statement(s) is(are) true?

In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a periodic motion is $$T = 2 \pi \sqrt{\frac{7(R - r)}{5g}}$$. The values of R and r are measured to be $$(60 \pm 1)$$ mm and $$(10 \pm 1)$$ mm, respectively. In five successive measurements, the time period is found to be 0.52 s, 0.56 s, 0.57 s, 0.54 s and 0.59 s. The least count of the watch used for the measurement of time period is 0.01 s. Which of the following statement(s) is(are) true?

Consider two identical galvanometers and two identical resistors with resistance R. If the internal resistance of the galvanometers $$R_c < \frac{R}{2}$$, which of the following statement(s) about any one of the galvanometers is(are) true?

In the circuit shown below, the key is pressed at time t=0. Which of the following statement(s) is(are) true?

A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position $$x_0$$. Consider two cases:
(i) when the block is at $$x_0$$; and
(ii) when the block is at $$x = x_0 + A$$.
In both thecases, a particle with mass m(< M) is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass mis placed on the mass M ?

While conducting the Young’s double slit experiment, a student replaced the two slits with a large opaque plate in the x-y plane containing two small holes that act as two coherent point sources $$(S_1, S_2)$$ emitting light of wavelength 600 nm. The student mistakenly placed the screen parallel to the x-z plane (for z > 0) at a distance D = 3 m from the mid-point of S,S,, as shown schematically in the figure. The distance between the sources d = 0.6003 mm. The origin O is at the intersection of the screen and the line joining $$S_1, S_2$$. Which of the following is(are) true of the intensity pattern on the screen?

A rigid wire loop of square shape having side of length L and resistance R is moving along the x-axis with a constant velocity $$v_0$$ in the plane of the paper. At t = 0, the right edge of the loop enters a region of length 3L where there is a uniform magnetic field $$B_0$$, into the plane of the paper, as shown in the figure. For sufficiently large $$v_0$$, the loop eventually crosses the region. Let x be the location of the right edge of the loop. Let v(x), I(x) and F(x) represent the velocity of the loop, current in the loop, and force on the loop, respectively, as a function of x. Counter-clockwise current is taken as positive.

Which of the following schematic plot(s) is(are) correct? (Ignore gravity)

The distance r of the block at time t is

The net reaction of the disc on the block is

Which one of the following statementsis correct?

The average current in the steady state registered by the ammeter in the circuit will be

For the following electrochemical cell at 298 K,
$$Pt(s) \mid H_2(g, 1 bar) \mid H^+(aq, 1M) \parallel M^{4+}(aq), M^{2+}(aq) \mid Pt(s)$$
$$E_{cell} = 0.092 V$$ and when $$\frac{[M^{2+}(aq)]}{[M^{4+}(aq)]} = 10^x$$.
Given : $$E_{\frac{M^{4+}}{M^{2+}}}^{0} = 0.15 V; 2.303 \frac{RT}{F} = 0.059 V$$
The value of x is

The qualitative sketches I, II and III given below show the variation of surface tension with molar concentration of three different aqueous solutions of KCl, $$CH_3OH$$ and $$CH_3(CH_2)_{11}OSO_3{^-Na^+}$$ at room temperature. The correct assignment of the sketches is

In the following reaction sequence in aqueous solution, the species X, Y and Z, respectively, are

The geometries of the ammonia complexes of $$Ni^{2+}, Pt^{2+}$$ and $$Zn^{2+}$$, respectively, are

The correct order of acidity for the following compounds is

The major product of the following reaction sequence is

According to Molecular Orbital Theory,

Mixture(s) showing positive deviation from Raoult’s law at $$35^\circ C$$ is(are)

The CORRECT statement(s) for cubic close packed (ccp) three dimensional structure is(are)

Extraction of copper from copper pyrite $$(CuFeS_2)$$ involves

The nitrogen containing compound producedin the reaction of $$HNO_3$$ with $$P_4O_{10}$$

For ‘invert sugar’, the correct statement(s) is(are)
(Given: specific rotations of( +)-sucrose, (+)-maltose, L-(—)-glucose and L-(+)-fructose in aqueous solution are $$+66^\circ, +140^\circ, -52^\circ$$ and $$+92^\circ$$, respectively)

Reagent(s) which can be used to bring about the following transformation is(are)

Among the following, reaction(s) which gives(give) tert-butyl benzene as the major product is(are)

The equilibrium constant $$K_p$$ for this reaction at 298 K, in terms of $$\beta_{equilibrium}$$, is

The INCORRECT statement amongthe following, for this reaction, is

The compound R is

The compound T is

Let $$\begin{bmatrix}1 & 0 & 0 \\4 & 1 & 0 \\16 & 4 & 1 \end{bmatrix}$$ and I be the identity matrix of order 3. If $$Q = [q_{ij}]$$ is a matrix such that $$P^{50} - Q = I$$, then $$\frac{q_{31} + q_{32}}{q_{21}}$$ equals

Let $$b_i > 1$$ for i = 1, 2, ......, 101. Suppose $$\log_e b_1, \log_e b_2, ......., \log_e b_{101}$$ are in Arithmetic Progression (A.P.) with the common difference $$\log_e 2$$. Suppose $$a_1, a_2, ........, a_{101}$$ are in A.P. such that $$a_1 = b_1$$ and $$a_{51} = b_{51}$$. If $$t = b_1 + b_2 + ....... + b_{51}$$ and $$s = a_1 + a_2 + ..... + a_{51}$$, then

The value of $$\sum_{k = 1}^{13} \frac{1}{\sin\left(\frac{\pi}{4} + \frac{(k - 1)\pi}{6}\right)\sin\left(\frac{\pi}{4} + \frac{k \pi}{6}\right)}$$ is equal to

The value of $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^2 \cos x}{1 + e^x}dx$$ is equal to

Area of the region $$\left\{(x, y) \epsilon R^2 : y \geq \sqrt{\mid x + 3 \mid}, 5y \leq x + 9 \leq 15\right\}$$ is equal to

Let P be the image of the point (3,1,7) with respect to the plane $$x - y + z = 3$$. Then the equation of the plane passing through P and containing the straight line $$\frac{x}{1} = \frac{y}{2} = \frac{z}{1}$$ is

Let $$f(x) = \lim_{n \rightarrow \infty}\left(\frac{n^n(x + n)\left(x + \frac{n}{2}\right).........\left(x + \frac{n}{n}\right)}{n!(x^2 + n^2)\left(x^2 + \frac{n^2}{4}\right).........\left(x^2 + \frac{n^2}{n^2}\right)}\right)^{\frac{x}{n}}$$, for all x > 0. Then

Let $$a, b \epsilon R$$ and $$f: R \rightarrow R$$ be bedefined by $$f(x) = a \cos (\mid x^3 - x \mid) + b \mid x \mid \sin(\mid x^3 + x \mid)$$. Then f is

Let $$f : R \rightarrow (0, \infty)$$ and $$g : R \rightarrow R$$ be be twice differentiable functions such that $$f''$$ and $$g''$$ are continuous functions on R. Suppose $$f'(2) = g(2) = 0, f''(2) \neq 0$$ and $$g'(2) \neq 0$$. If $$\lim_{x \rightarrow 2}\frac{f(x)g(x)}{f'(x)g'(x)} = 1$$, then

Let $$f:\left[-\frac{1}{2}, 2\right] \rightarrow R$$ and $$g:\left[-\frac{1}{2}, 2\right] \rightarrow R$$ be functions defined by $$f(x) = [x^2 - 3]$$ and $$g(x) = \mid x \mid f(x) + \mid 4x - 7 \mid f(x)$$, where $$[y]$$ denotes the greatest integer less than or equal to y for $$y \epsilon R$$. Then

Let $$a, b \epsilon R$$ and $$a^2 + b^2 \neq 0$$. Suppose $$S = \left\{z \epsilon C : z = \frac{1}{a + ibt}, t \epsilon R, t \neq 0\right\}$$, where $$i = \sqrt{-1}$$. If $$z = x + iy$$ and $$z \epsilon S$$, then (x, y) lies on

Let P be the point on the parabola $$y^2 = 4x$$ which is at the shortest distance from the center S of the circle $$x^2 + y^2 - 4x - 16y + 64 = 0$$. Let Q be the pomt on the circle dividing the line segment SP internally. Then

Let $$a, \lambda, \mu \epsilon R$$. Consider the system of linear equations
$$ax + 2y = \lambda$$
$$3x - 2y = \mu$$
Which of the following statement(s) is(are) correct?

Let $$\hat{u} = u_1\hat{i} + u_2\hat{j} + u_3\hat{k}$$ be a unit vector in $$R^3$$ and $$\hat{\omega} = \frac{1}{\sqrt{6}}(\hat{i} + \hat{j} + 2 \hat{k})$$. Given that there exists a vector $$\overrightarrow{v}$$ in $$R^3$$ such that $$\mid \hat{u} \times \overrightarrow{v} \mid = 1$$ and $$\overrightarrow{\omega} . (\hat{u} \times \overrightarrow{v}) = 1$$. Which of the following statement(s) is(are) correct?

$$P(X > Y)$$ is

$$P(X = Y)$$ is

The orthocentre of the triangle $$F_1MN$$ is

If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral $$MF_1NF_2$$ is