JEE Advanced 2016 Paper 2 Question Paper

Question 1

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

Accepted Answer

(B) 103

Answer

(A) 52

Answer

(C) 201

Answer

(D) 205

Question 2

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

Accepted Answer

(C) $\frac{3}{2}$

Answer

(A) $\frac{1}{6}$

Answer

(B) $\frac{4}{3}$

Answer

(D) $\frac{5}{3}$

Question 3

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

Accepted Answer

(C) $2\left( \sqrt{3} - 1 \right)$

Answer

(A) $3 - \sqrt{3}$

Answer

(B) $2\left( 3 - \sqrt{3} \right)$

Answer

(D) $2\left( 2 + \sqrt{3} \right)$

Question 4

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

Accepted Answer

(B) $s > t$ and $a_{101} < b_{101}$

Answer

(A) $s > t$ and $a_{101} > b_{101}$

Answer

(C) $s < t$ and $a_{101} > b_{101}$

Answer

(D) $s < t$ and $a_{101} < b_{101}$

Question 5

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

Accepted Answer

(A) $\frac{\pi^{2}}{4} - 2$

Answer

(B) $\frac{\pi^{2}}{4} + 2$

Answer

(C) $\pi^{2} - e^{\frac{\pi}{2}}$

Answer

(D) $\pi^{2} + e^{\frac{\pi}{2}}$

Question 6

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

Accepted Answer

(C) $x - 4y + 7z = 0$

Answer

(A) $x + y - 3z = 0$

Answer

(B) $3x + z = 0$

Answer

(D) $2x - y = 0$

Question 7

Let $a, b \in \mathbb{R}$ and $f : \mathbb{R} 	o \mathbb{R}$ be defined by $f(x) = a \cos(|x^{3} - x|) + b|x| \sin(|x^{3} + x|)$. Then f is

Accepted Answer

(A) differentiable at $x = 0$ if $a = 0$ and $b = 1$

Accepted Answer

(B) differentiable at $x = 1$ if $a = 1$ and $b = 0$

Answer

(C) NOT differentiable at $x = 0$ if $a = 1$ and $b = 0$

Answer

(D) NOT differentiable at $x = 1$ if $a = 1$ and $b = 1$

Question 8

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

Accepted Answer

(B) $f\left( \frac{1}{3} \right) \leq f\left( \frac{2}{3} \right)$

Accepted Answer

(C) $f'(2) \leq 0$

Answer

(A) $f\left( \frac{1}{2} \right) \geq f(1)$

Answer

(D) $\frac{f'(3)}{f(3)} \geq \frac{f'(2)}{f(2)}$

Question 9

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

Accepted Answer

(A) f has a local minimum at $x = 2$

Accepted Answer

(D) $f(x) - f''(x) = 0$ for at least one $x \in \mathbb{R}$

Answer

(B) f has a local maximum at $x = 2$

Answer

(C) $f''(2) > f(2)$

Question 10

Let $\hat{u} = u_{1}\hat{i} + u_{2}\hat{j} + u_{3}\hat{k}$ be a unit vector in $\mathbb{R}^{3}$ and $\hat{w} = \frac{1}{\sqrt{6}} \left( \hat{i} + \hat{j} + 2\hat{k} ight)$. Given that there exists a vector $\vec{v}$ in $\mathbb{R}^{3}$ such that $|\hat{u} 	imes \vec{v}| = 1$ and $\hat{w} \cdot (\hat{u} 	imes \vec{v}) = 1$. Which of the following statement(s) is(are) correct?

Accepted Answer

(B) There are infinitely many choices for such $\vec{v}$

Accepted Answer

(C) If $\hat{u}$ lies in the xy-plane then $|u_{1}| = |u_{2}|$

Answer

(A) There is exactly one choice for such $\vec{v}$

Answer

(D) If $\hat{u}$ lies in the xz-plane then $2|u_{1}| = |u_{3}|$

Question 11

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 point on the circle dividing the line segment SP internally. Then

Accepted Answer

(A) $SP = 2\sqrt{5}$

Accepted Answer

(C) the x-intercept of the normal to the parabola at P is 6

Accepted Answer

(D) the slope of the tangent to the circle at Q is $\frac{1}{2}$

Answer

(B) $SQ : QP = \left( \sqrt{5} + 1 \right) : 2$

Question 12

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

Accepted Answer

(A) the circle with radius $\frac{1}{2a}$ and centre $\left( \frac{1}{2a}, 0 ight)$ for $a > 0, b 
eq 0$

Accepted Answer

(C) the x-axis for $a 
eq 0, b = 0$

Accepted Answer

(D) the y-axis for $a = 0, b 
eq 0$

Answer

(B) the circle with radius $-\frac{1}{2a}$ and centre $\left( -\frac{1}{2a}, 0 ight)$ for $a < 0, b 
eq 0$

Question 13

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

Accepted Answer

(B) If $a 
eq -3$, then the system has a unique solution for all values of $\lambda$ and $\mu$

Accepted Answer

(C) If $\lambda + \mu = 0$, then the system has infinitely many solutions for $a = -3$

Accepted Answer

(D) If $\lambda + \mu 
eq 0$, then the system has no solution for $a = -3$

Answer

(A) If $a = -3$, then the system has infinitely many solutions for all values of $\lambda$ and $\mu$

Question 14

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

Accepted Answer

(B) f is discontinuous exactly at four points in $\left[ -\frac{1}{2}, 2 \right]$

Accepted Answer

(C) g is NOT differentiable exactly at four points in $\left( -\frac{1}{2}, 2 \right)$

Answer

(A) f is discontinuous exactly at three points in $\left[ -\frac{1}{2}, 2 \right]$

Answer

(D) g is NOT differentiable exactly at five points in $\left( -\frac{1}{2}, 2 \right)$

Question 15

Football teams $T_{1}$ and $T_{2}$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_{1}$ winning, drawing and losing a game against $T_{2}$ are $\frac{1}{2}$, $\frac{1}{6}$ and $\frac{1}{3}$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams $T_{1}$ and $T_{2}$, respectively, after two games.

$P(X > Y)$ is

Accepted Answer

(B) $\frac{5}{12}$

Answer

(A) $\frac{1}{4}$

Answer

(C) $\frac{1}{2}$

Answer

(D) $\frac{7}{12}$

Question 16

Football teams $T_{1}$ and $T_{2}$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_{1}$ winning, drawing and losing a game against $T_{2}$ are $\frac{1}{2}$, $\frac{1}{6}$ and $\frac{1}{3}$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams $T_{1}$ and $T_{2}$, respectively, after two games.

$P(X = Y)$ is

Accepted Answer

(C) $\frac{13}{36}$

Answer

(A) $\frac{11}{36}$

Answer

(B) $\frac{1}{3}$

Answer

(D) $\frac{1}{2}$

Question 17

Let $F_{1}(x_{1}, 0)$ and $F_{2}(x_{2}, 0)$, for $x_{1} < 0$ and $x_{2} > 0$, be the foci of the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{8} = 1$. Suppose a parabola having vertex at the origin and focus at $F_{2}$ intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant.

The orthocentre of the triangle $F_{1}MN$ is

Accepted Answer

(A) $\left( -\frac{9}{10}, 0 \right)$

Answer

(B) $\left( \frac{2}{3}, 0 \right)$

Answer

(C) $\left( \frac{9}{10}, 0 \right)$

Answer

(D) $\left( \frac{2}{3}, \sqrt{6} \right)$

Question 18

Let $F_{1}(x_{1}, 0)$ and $F_{2}(x_{2}, 0)$, for $x_{1} < 0$ and $x_{2} > 0$, be the foci of the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{8} = 1$. Suppose a parabola having vertex at the origin and focus at $F_{2}$ intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant.

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 $MF_{1}NF_{2}$ is

Accepted Answer

(C) 5 : 8

Answer

(A) 3 : 4

Answer

(B) 4 : 5

Answer

(D) 2 : 3

Question 19

The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by

$E = \frac{3}{5}\,\frac{Z(Z-1)e^{2}}{4\pi\varepsilon_{0}R}$

The measured masses of the neutron, $^{1}_{1}	ext{H}$, $^{15}_{7}	ext{N}$ and $^{15}_{8}	ext{O}$ are 1.008665 u, 1.007825 u, 15.000109 u and 15.003065u, respectively. Given that the radii of both the $^{15}_{7}	ext{N}$ and $^{15}_{8}	ext{O}$ nuclei are same, 1 u = 931.5 MeV/$c^{2}$ ( c is the speed of light) and $e^{2}/(4\pi\varepsilon_{0})$ = 1.44 MeV fm. Assuming that the difference between the binding energies of $^{15}_{7}	ext{N}$ and $^{15}_{8}	ext{O}$ is purely due to the electrostatic energy, the radius of either of the nuclei is
(1 fm = $10^{-15}$m)

Accepted Answer

(C) 3.42 fm

Answer

(A) 2.85 fm

Answer

(B) 3.03 fm

Answer

(D) 3.80 fm

Question 20

The ends Q and R of two thin wires, PQ and RS, are soldered (joined) together. Initially each of the wires has a length of 1 m at 10 $^{0}$C. Now the end P is maintained at 10 $^{0}$C, while the end S is heated and maintained at 400 $^{0}$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 $1.2 	imes 10^{-5}\ 	ext{K}^{-1}$, the change in length of the wire PQ is

Accepted Answer

(A) 0.78 mm

Answer

(B) 0.90 mm

Answer

(C) 1.56 mm

Answer

(D) 2.34 mm

Question 21

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?

Accepted Answer

(C) 108

Answer

(A) 64

Answer

(B) 90

Answer

(D) 120

Question 22

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

Accepted Answer

(C) 2.87 and 2.83

Answer

(A) 2.87 and 2.86

Answer

(B) 2.87 and 2.87

Answer

(D) 2.85 and 2.82

Question 23

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 $V_{i} = 10^{-3}\ 	ext{m}^{3}$ changes to a final state at $P_{f} = (1/32) 	imes 10^{5}$ Pa and $V_{f} = 8 	imes 10^{-3}\ 	ext{m}^{3}$ in an adiabatic quasi-static process, such that $P^{3}V^{5}$ = 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

Accepted Answer

(C) 588 J

Answer

(A) 112 J

Answer

(B) 294 J

Answer

(D) 813 J

Question 24

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 $	heta = 30^{0}$ to the axis of the lens, as shown in the figure.

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

Accepted Answer

(A) $(25,\ 25\sqrt{3})$

Answer

(B) $(125/3,\ 25/\sqrt{3})$

Answer

(C) $(50 - 25\sqrt{3},\ 25)$

Answer

(D) $(0,\ 0)$

Question 25

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 $S_{1}S_{2}$, 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?

Accepted Answer

(B) Semi circular bright and dark bands centred at point O

Accepted Answer

(C) The region very close to the point O will be dark

Answer

(A) Hyperbolic bright and dark bands with foci symmetrically placed about O in the x-direction

Answer

(D) Straight bright and dark bands parallel to the x-axis

Question 26

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 ± 1) mm and (10 ± 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?

Accepted Answer

(A) The error in the measurement of r is 10%

Accepted Answer

(B) The error in the measurement of T is 3.57%

Accepted Answer

(D) The error in the determined value of g is 11%

Answer

(C) The error in the measurement of T is 2%

Question 27

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)

Accepted Answer

(C) (C)

Accepted Answer

(D) (D)

Answer

(A) (A)

Answer

(B) (B)

Question 28

Light of wavelength $\lambda_{ph}$ falls on a cathode plate inside a vacuum tube 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?

Accepted Answer

(A) For large potential difference (V >> $\phi/e$), $\lambda_{e}$ is approximately halved if V is made four times

Answer

(B) $\lambda_{e}$ increases at the same rate as $\lambda_{ph}$ for $\lambda_{ph} < hc/\phi$

Answer

(C) $\lambda_{e}$ is approximately halved, if d is doubled

Answer

(D) $\lambda_{e}$ decreases with increase in $\phi$ and $\lambda_{ph}$

Question 29

Two thin circular discs of mass m and 4m, having radii of a and 2a, respectively, are rigidly fixed by a massless, rigid rod of length $\ell = \sqrt{24}\,a$ 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 $\vec{L}$ (see the figure). Which of the following statement(s) is(are) true?

Accepted Answer

(A) The magnitude of angular momentum of the assembly about its center of mass is $17\,ma^{2}\omega/2$

Accepted Answer

(D) The center of mass of the assembly rotates about the z-axis with an angular speed of $\omega/5$

Answer

(B) The magnitude of the z-component of $\vec{L}$ is $55\,ma^{2}\omega$

Answer

(C) The magnitude of angular momentum of center of mass of the assembly about the point O is $81\,ma^{2}\omega$

Question 30

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

Accepted Answer

(B) The maximum voltage range is obtained when the two resistors and one galvanometer are connected in series, and the second galvanometer is connected in parallel to the first galvanometer

Accepted Answer

(C) The maximum current range is obtained when all the components are connected in parallel

Answer

(A) The maximum voltage range is obtained when all the components are connected in series

Answer

(D) The maximum current range is obtained when the two galvanometers are connected in series and the combination is connected in parallel with both the resistors

Question 31

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

Accepted Answer

(A) The voltmeter displays – 5 V as soon as the key is pressed, and displays +5 V after a long time

Accepted Answer

(B) The voltmeter will display 0 V at time $t = \ln 2$ seconds

Accepted Answer

(C) The current in the ammeter becomes $1/e$ of the initial value after 1 second

Accepted Answer

(D) The current in the ammeter becomes zero after a long time

Question 32

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 the cases, 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 m is placed on the mass M?

Accepted Answer

(A) The amplitude of oscillation in the first case changes by a factor of $\sqrt{\frac{M}{m+M}}$, whereas in the second case it remains unchanged

Accepted Answer

(B) The final time period of oscillation in both the cases is same

Accepted Answer

(D) The instantaneous speed at $x_{0}$ of the combined masses decreases in both the cases

Answer

(C) The total energy decreases in both the cases

Question 33

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity $\omega$ is an example of a non-inertial frame of reference. The relationship between the force $\vec{F}_{rot}$ experienced by a particle of mass m moving on the rotating disc and the force $\vec{F}_{in}$ experienced by the particle in an inertial frame of reference is

$\vec{F}_{rot} = \vec{F}_{in} + 2m\left(\vec{v}_{rot} 	imes \vec{\omega}ight) + m\left(\vec{\omega} 	imes \vec{r}ight) 	imes \vec{\omega},$

where $\vec{v}_{rot}$ is the velocity of the particle in the rotating frame of reference and $\vec{r}$ is the position vector of the particle with respect to the centre of the disc.
Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed $\omega$ about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis $\left(\vec{\omega} = \omega\hat{k}ight)$. A small block of mass m is gently placed in the slot at $\vec{r} = (R/2)\hat{i}$ at t = 0 and is constrained to move only along the slot.

The distance r of the block at time t is

Accepted Answer

(D) $\frac{R}{4}\left(e^{\omega t} + e^{-\omega t}\right)$

Answer

(A) $\frac{R}{4}\left(e^{2\omega t} + e^{-2\omega t}\right)$

Answer

(B) $\frac{R}{2}\cos 2\omega t$

Answer

(C) $\frac{R}{2}\cos \omega t$

Question 34

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity $\omega$ is an example of a non-inertial frame of reference. The relationship between the force $\vec{F}_{rot}$ experienced by a particle of mass m moving on the rotating disc and the force $\vec{F}_{in}$ experienced by the particle in an inertial frame of reference is

$\vec{F}_{rot} = \vec{F}_{in} + 2m\left(\vec{v}_{rot} 	imes \vec{\omega}ight) + m\left(\vec{\omega} 	imes \vec{r}ight) 	imes \vec{\omega},$

where $\vec{v}_{rot}$ is the velocity of the particle in the rotating frame of reference and $\vec{r}$ is the position vector of the particle with respect to the centre of the disc.
Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed $\omega$ about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis $\left(\vec{\omega} = \omega\hat{k}ight)$. A small block of mass m is gently placed in the slot at $\vec{r} = (R/2)\hat{i}$ at t = 0 and is constrained to move only along the slot.

The net reaction of the disc on the block is

Accepted Answer

(C) $\frac{1}{2}m\omega^{2}R\left(e^{\omega t} - e^{-\omega t}\right)\hat{j} + mg\hat{k}$

Answer

(A) $-m\omega^{2}R\cos \omega t\,\hat{j} - mg\hat{k}$

Answer

(B) $m\omega^{2}R\sin \omega t\,\hat{j} - mg\hat{k}$

Answer

(D) $\frac{1}{2}m\omega^{2}R\left(e^{2\omega t} - e^{-2\omega t}\right)\hat{j} + mg\hat{k}$

Question 35

Consider an evacuated cylindrical chamber of height h having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius r << h. Now a high voltage source (HV) is connected across the conducting plates such that the bottom plate is at $+V_{0}$ and the top plate at $-V_{0}$. Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)

Which one of the following statements is correct?

Accepted Answer

(A) The balls will bounce back to the bottom plate carrying the opposite charge they went up with

Answer

(B) The balls will execute simple harmonic motion between the two plates

Answer

(C) The balls will bounce back to the bottom plate carrying the same charge they went up with

Answer

(D) The balls will stick to the top plate and remain there

Question 36

Consider an evacuated cylindrical chamber of height h having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius r << h. Now a high voltage source (HV) is connected across the conducting plates such that the bottom plate is at $+V_{0}$ and the top plate at $-V_{0}$. Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)

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

Accepted Answer

(B) proportional to $V_{0}^{2}$

Answer

(A) proportional to $V_{0}^{1/2}$

Answer

(C) proportional to the potential $V_{0}$

Answer

(D) zero

Question 37

The correct order of acidity for the following compounds is

Accepted Answer

(A) I > II > III > IV

Answer

(B) III > I > II > IV

Answer

(C) III > IV > II > I

Answer

(D) I > III > IV > II

Question 38

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

Accepted Answer

(A) octahedral, square planar and tetrahedral

Answer

(B) square planar, octahedral and tetrahedral

Answer

(C) tetrahedral, square planar and octahedral

Answer

(D) octahedral, tetrahedral and square planar

Question 39

For the following electrochemical cell at 298 K,
$Pt(s)\,|\,H_{2}(g, 1\ bar)\,|\,H^{+}(aq, 1\ M)\,||\,M^{4+}(aq), M^{2+}(aq)\,|\,Pt(s)$
$E_{cell} = 0.092$ V when $\dfrac{[M^{2+}(aq)]}{[M^{4+}(aq)]} = 10^{x}$
Given: $E^{0}_{M^{4+}/M^{2+}} = 0.151$ V; $2.303\dfrac{RT}{F} = 0.059$ V
The value of $x$ is

Accepted Answer

(D) 2

Answer

(A) $-2$

Answer

(B) $-1$

Answer

(C) 1

Question 40

The major product of the following reaction sequence is

Accepted Answer

(A) (A)

Answer

(B) (B)

Answer

(C) (C)

Answer

(D) (D)

Question 41

In the following reaction sequence in aqueous solution, the species X, Y and Z, respectively, are
$S_{2}O_{3}^{2-} \xrightarrow{Ag^{+}} \underset{	ext{clear solution}}{X} \xrightarrow{Ag^{+}} \underset{	ext{white precipitate}}{Y} \xrightarrow{	ext{with time}} \underset{	ext{black precipitate}}{Z}$

Accepted Answer

(A) $[Ag(S_{2}O_{3})_{2}]^{3-}$, $Ag_{2}S_{2}O_{3}$, $Ag_{2}S$

Answer

(B) $[Ag(S_{2}O_{3})_{3}]^{5-}$, $Ag_{2}SO_{3}$, $Ag_{2}S$

Answer

(C) $[Ag(SO_{3})_{2}]^{3-}$, $Ag_{2}S_{2}O_{3}$, $Ag$

Answer

(D) $[Ag(SO_{3})_{3}]^{3-}$, $Ag_{2}SO_{4}$, $Ag$

Question 42

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, $CH_{3}OH$ and $CH_{3}(CH_{2})_{11}OSO_{3}^{-}Na^{+}$ at room temperature. The correct assignment of the sketches is

Accepted Answer

(D) I : $CH_{3}OH$; II : KCl; III : $CH_{3}(CH_{2})_{11}OSO_{3}^{-}Na^{+}$

Answer

(A) I : KCl; II : $CH_{3}OH$; III : $CH_{3}(CH_{2})_{11}OSO_{3}^{-}Na^{+}$

Answer

(B) I : $CH_{3}(CH_{2})_{11}OSO_{3}^{-}Na^{+}$; II : $CH_{3}OH$; III : KCl

Answer

(C) I : KCl; II : $CH_{3}(CH_{2})_{11}OSO_{3}^{-}Na^{+}$; III : $CH_{3}OH$

Question 43

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°, +140°, –52° and +92°, respectively)

Accepted Answer

(B) ‘invert sugar’ is an equimolar mixture of D-(+)-glucose and D-(–)-fructose

Accepted Answer

(C) specific rotation of ‘invert sugar’ is –20°

Answer

(A) ‘invert sugar’ is prepared by acid catalyzed hydrolysis of maltose

Answer

(D) on reaction with $Br_{2}$ water, ‘invert sugar’ forms saccharic acid as one of the products

Question 44

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

Accepted Answer

(B) (B)

Accepted Answer

(C) (C)

Accepted Answer

(D) (D)

Answer

(A) (A)

Question 45

Extraction of copper from copper pyrite ($CuFeS_{2}$) involves

Accepted Answer

(A) crushing followed by concentration of the ore by froth flotation

Accepted Answer

(B) removal of iron as slag

Accepted Answer

(C) self-reduction step to produce ‘blister copper’ following evolution of $SO_{2}$

Answer

(D) refining of ‘blister copper’ by carbon reduction

Question 46

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

Accepted Answer

(B) The efficiency of atom packing is 74%

Accepted Answer

(C) The number of octahedral and tetrahedral voids per atom are 1 and 2, respectively

Accepted Answer

(D) The unit cell edge length is $2\sqrt{2}$ times the radius of the atom

Answer

(A) The number of the nearest neighbours of an atom present in the topmost layer is 12

Question 47

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

Accepted Answer

(C) $NaBH_{4}$ in $C_{2}H_{5}OH$

Accepted Answer

(D) Raney Ni/$H_{2}$ in THF

Answer

(A) $LiAlH_{4}$ in $(C_{2}H_{5})_{2}O$

Answer

(B) $BH_{3}$ in THF

Question 48

Mixture (s) showing positive deviation from Raoult’s law at 35°C is (are)

Accepted Answer

(A) carbon tetrachloride + methanol

Accepted Answer

(B) carbon disulphide + acetone

Answer

(C) benzene + toluene

Answer

(D) phenol + aniline

Question 49

The nitrogen containing compound produced in the reaction of $HNO_{3}$ with $P_{4}O_{10}$

Accepted Answer

(B) is diamagnetic

Accepted Answer

(D) reacts with Na metal producing a brown gas

Answer

(A) can also be prepared by reaction of $P_{4}$ and $HNO_{3}$

Answer

(C) contains one N – N bond

Question 50

According to Molecular Orbital Theory

Accepted Answer

(A) $C_{2}^{2-}$ is expected to be diamagnetic

Accepted Answer

(C) $N_{2}^{+}$ and $N_{2}^{-}$ have the same bond order

Answer

(B) $O_{2}^{2+}$ expected to have a longer bond length than $O_{2}$

Answer

(D) $He_{2}^{+}$ has the same energy as two isolated He atoms

Question 51

Thermal decomposition of gaseous $X_{2}$ to gaseous X at 298 K takes place according to the following equation:
$X_{2}(g) \rightleftharpoons 2X(g)$
The standard reaction Gibbs energy, $\Delta_{r}G^{0}$, of this reaction is positive. At the start of the reaction, there is one mole of $X_{2}$ and no X. As the reaction proceeds, the number of moles of X formed is given by β. Thus, $\beta_{equilibrium}$ is the number of moles of X formed at equilibrium. The reaction is carried out at a constant total pressure of 2 bar. Consider the gases to behave ideally. (Given: R = 0.083 L bar $K^{-1}$ $mol^{-1}$)

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

Accepted Answer

(B) $\dfrac{8\beta_{equilibrium}^{2}}{4-\beta_{equilibrium}^{2}}$

Answer

(A) $\dfrac{8\beta_{equilibrium}^{2}}{2-\beta_{equilibrium}}$

Answer

(C) $\dfrac{4\beta_{equilibrium}^{2}}{2-\beta_{equilibrium}}$

Answer

(D) $\dfrac{4\beta_{equilibrium}^{2}}{4-\beta_{equilibrium}^{2}}$

Question 52

Thermal decomposition of gaseous $X_{2}$ to gaseous X at 298 K takes place according to the following equation:
$X_{2}(g) \rightleftharpoons 2X(g)$
The standard reaction Gibbs energy, $\Delta_{r}G^{0}$, of this reaction is positive. At the start of the reaction, there is one mole of $X_{2}$ and no X. As the reaction proceeds, the number of moles of X formed is given by β. Thus, $\beta_{equilibrium}$ is the number of moles of X formed at equilibrium. The reaction is carried out at a constant total pressure of 2 bar. Consider the gases to behave ideally. (Given: R = 0.083 L bar $K^{-1}$ $mol^{-1}$)

The INCORRECT statement among the following, for this reaction is

Accepted Answer

(C) $\beta_{equilibrium} = 0.7$

Answer

(A) Decrease in the total pressure will result in formation of more moles of gaseous X

Answer

(B) At the start of the reaction, dissociation of gaseous $X_{2}$ takes place spontaneously

Answer

(D) $K_{c} < 1$

Question 53

Treatment of compound O with $KMnO_{4}/H^{+}$ gave P, which on heating with ammonia gave Q. The compound Q on treatment with $Br_{2}$/NaOH produced R. On strong heating, Q gave S, which on further treatment with ethyl 2-bromopropanoate in the presence of KOH followed by acidification, gave a compound T.

The compound R is

Accepted Answer

(A) (A)

Answer

(B) (B)

Answer

(C) (C)

Answer

(D) (D)

Question 54

Treatment of compound O with $KMnO_{4}/H^{+}$ gave P, which on heating with ammonia gave Q. The compound Q on treatment with $Br_{2}$/NaOH produced R. On strong heating, Q gave S, which on further treatment with ethyl 2-bromopropanoate in the presence of KOH followed by acidification, gave a compound T.

The compound T is

Accepted Answer

(B) alanine

Answer

(A) glycine

Answer

(C) valine

Answer

(D) serine

JEE Advanced 2016 Paper 2 Question Paper with Answers

Physics — JEE Advanced 2016 Paper 2

Chemistry — JEE Advanced 2016 Paper 2

Mathematics — JEE Advanced 2016 Paper 2

Chapters tested in this paper

Attempt JEE Advanced 2016 Paper 2 under exam timing.