**JEE Advanced 2016 Paper 2**

Q. 1 The electrostatic energy of Z protons uniformly distributed uniformly throughout a spherical nucleus of radius R is given in the picture. Thea measured masses of the neutron, ¹H₁, ¹⁵N₇ and ¹⁵O₈ are 1.008665 u, 1.0077825 u, 15.000109 u and 15.003065 u, respectively. Given that the radii of both the ¹⁵N₇ and ¹⁵O₈ are same, 1 u = 931.5 MeV/c² (c is the speed of light) and e²/(4Πε0) = 1.44 MeV fm. Assuming that the difference between the binding energies of ¹⁵N₇ ¹⁵O₈ is purely due to the electrostatic energy, the radius of either of the nuclei is (1 fm = 10⁻¹⁵ m)

E=35Z (Z-1) e24 0R

A. 2.85 fm

B. 3.03 fm

C. 3.42 fm

D. 3.80 fm

Q. 2 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 maximum number of days after which the laboratory can be considered safe for use?

A. 64

B. 90

C. 108

D. 120

Q. 3 A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure Pᵢ = 10⁵ Pa and volume Vⁱ = 10⁻³ m³ changes to a final state at P = (1/32) x 10⁵ Pa and V = 8 x 10⁻³ m³ in an adiabatic quasi-static process, such that P³.V⁵ = 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 Pᵢ followed by an isochoric (isovolumetric) process at volume V. The amount of heat supplied to the system in the two-step process is approximately

A. 112 J

B. 294 J

C. 588 J

D. 813 J

Q. 4 The ends Q and R of two thin wires, PQ and RS, are soldered (joined) together. Initially each of the wires has a lenght of 1 m at 10 ⁰C. Now the end P is maintained at 10 ⁰C, while the end S is heated and maintained at 400 ⁰C. 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 x 10⁻⁵K⁻¹, the change in lenght of the wire PQ is

A. 0.78 mm

B. 0.90 mm

C. 1.56 mm

D. 2.34 mm

Q. 5 A small object is placed 50 cm to the left of a thin convex lens of focal lenght 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 of θ = 30⁰ 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

A. (0, 0)

B. (50 – 25√3, 25)

C. (25, 25√3)

D. (125/3, 25/√3)

Q. 6 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 (C₁) has 10 equal divisions that correspond to 9 main scale divisions. The Vernier scale of the other caliper (C₂) 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 C₁ and C₂, respectively, are

A. 2.85 and 2.82

B. 2.87 and 2.83

C. 2.87 and 2.86

D. 2.87 and 2.87

Q. 7 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 lenght l = √24a through their centres. 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 ω . The angular momentum of the entire assembly about the point ‘O’ is L (see the figure). Which of the following statements(s) is (are) true?

A. The center of mass of the assembly rotates about the z – axis with an angular speed of ω/5

B. The magnitude of an angular momentum of center of mass of the assembly about the point O is 81ma²ω

C. The magnitude of angular momentum of the assembly about its center of mass is

17ma²ω/2

D. The magnitude of the z – component of L is 55 ma²ω

Q. 8 Light of wavelenght λₚₕ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is Φ 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 wavelenght of the electrons passing through the anode is λₑ, which of the following statements(s) is (are) true?

A. λₑ, decreases with increase in Φ and λₚₕ

B. λₑ is approximately halved, if d is doubled

C. For large potential difference (V≪ Φ/e), λₑ is approximately halved if V is made four times

D. λₑ increases at the same rate as λₚₕ for λₚₕ <he/Φ

Q. 9 In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a period motion is T = (2π) √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?

A. The error in the measurement of r is 10%

B. The error in the measurement of T is 3.57%

C. The error in the measurement of T is 2%

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

Q. 10 Consider two identical galvanometers and two identical resistors with resistance R. If the internal resistance of the galvanometers Rc < R/2, which of the following statement(s) about any one of the galvanometers is (are) true?

A. The maximum voltage range is obtained when all the components are connected in

Series

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.

C. The maximum current range is obtained when all the components in parallel

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.

Q. 11 In the circuit shown below, the key is pressed at time t = 0. Which of the following statement(s) is(are) true?

A. The volunteer displays -5 V as soon as the key is pressed, and displays +5 V after a

long time.

B. The voltmeter will display 0 V at time t = In 2 seconds

C. The current in the ammeter becomes 1/e of the initial value after 1 second

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

Q. 12 A block with mass M is connected by a massless spring with stiffness constant ƙ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position x₀. Consider two cases : (i) when the block is at x₀ ; and (ii) when then block is at x = x₀ + A. In both the cases, a particle with mass m (

A. The amplitude of oscillation in the first case changes by a factor of √[(M/(m+M)],

whereas in the second case it remains unchanged

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

C. The total energy decreases in both the cases

D. The instantaneous speed at x₀ of the combined masses decreases in both the cases

Q. 13 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 (S₁,S₂) 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 S₁S₂ , 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 S₁S₂. Which of the following is(are) true of the intensity pattern on the screen?

A. Straight bright and dark bands parallel to the x – axis

B. The region very close to the point O will be dark

C. Hyberbolic bright and dark bands with foci symmetrically placed about O in the x –

direction

D. Semi circular bright and dark bands centered at point 0

Q. 14 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 v₀ 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 B₀ into the plane of the paper, as shown in the question figure. For sufficiently large v₀, 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 given schematic plot(s) is/are correct? (Ignore gravity)

A. A

B. B

C. C

D. D

Questions: 15 – 16

Q. 15 The distance r of the block at time t is

A. (R/4) [ eʷᵗ+ e⁻ʷᵗ ]

B. (R/2) cos ωt

C. (R/4) [ e²ʷᵗ + e⁻²ʷᵗ ]

D. (R/2) cos 2ωt

Q. 16 The net reaction of the disc on the block is

A. (1/2) mω²R ( e²ʷᵗ – e⁻²ʷᵗ ) ĵ + mgk̂

B. (1/2) mω²R ( eʷᵗ – e²ʷᵗ ) ĵ + mgk̂

C. -mω²R cos ωt ĵ – mgk̂

D. mω²R sin ωt ĵ – mgk̂

Questions: 17 – 18

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 lightweight 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 +V₀ and the top plate is at -V₀. 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)

Q. 17 Which one of the following statements is correct?

A. The balls will stick to the top plate and remain there

B. The balls will bounce back to the bottom plate carrying the same charge they went up with

C. The balls will bounce back to the bottom plate carrying the opposite charge they went up with

D. The balls will execute simple harmonic motion between the two plates

Q. 18 The average current in the steady state registered by the ammeter in the circuit will be

A. zero

B. proportional to the potential V₀

C. proportional to V

D. proportional to V

Q. 19 For the following electrochemical cell at 298 K ,

Pt (s) | H₂ (g,1 bar) | H⁺ (aq, 1 M) || M⁴⁺ (aq), M²⁺ (aq) | Pt (s)

E_cell = 0.092 V when [M²⁺ (aq)] / [M⁴⁺ (aq)] = 10ˣ

Given: E⁰ [M⁴⁺ / M²⁺)] = 0.151 V; 2.303 (RT/V) = 0.059 V.

The value of x is

A. -2

B. -1

C. 1

D. 2

Q. 20 The given qualitative sketches I, II and III shows the variation of surface tension with a molar concentration of three different aqueous solutions of KC₁, CH₃OH and CH₃(CH₂)₁₁OSO₃⁻Na⁺ at room temperature. The correct assignment of the sketches is

A. A

B. B

C. C

D. D

Q. 21 In the given reaction sequence in aqueous solution, the species X Y and Z, respectively, are

A. A

B. B

C. C

D. D

Q. 22 The geometries of the ammonia complexes of Ni²⁺ , Pt²⁺ and Zn²⁺, respectively, are

A. octahedral, square planar and tetrahedral

B. square planar, octahedral and tetrahedral

C. tetrahedral, square planar and octahedral

D. octahedral, tetrahedral and square planar

Q. 23 The correct order of acidity for the given compounds is

A. I > II >I II > IV

B. III > I > II >I V

C. III > IV > II > I

D. I >I II > IV > II

Q. 24 The major product of the given reaction sequence is

A. A

B. B

C. C

D. D

Q. 25 According to Molecular Orbital Theory,

A. C₂²⁻ is expected to be diamagnetic

B. O₂²⁺ is expected to have a longer bond length than O₂

C. N₂⁺ and N₂⁻ have the same bond order

D. He₂⁺ has the same energy as two isolated He atoms

Q. 26 Mixture(s) showing positive deviation from Raoult’s law at 35 ᵒC is (are)

A. carbon tetrachloride + methanol

B. carbon disulphide + acetone

C. benzene + toluene

D. phenol + aniline

Question 27 ONE OR MORE THAN ONE of the four given options is (are) correct.

Q. 27 The CORRECT statement(s) for cubic close packed (ccp) three dimensional structure is(are)

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

B. The efficiency of atom packing is 74%

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

D. The unit cell edge length is 2√2 times the radius of the atom

Q. 28 Extraction of copper from copper pyrite (CuFeS₂) involves

A. crushing followed by concentration of the ore by froth-flotation

B. removal of iron as slag

C. self-reduction step to produce ‘blister copper’ following evolution of SO₂

D. refining of ‘blister copper’ by carbon reduction

Q. 29 The nitrogen containing compound produced in the reaction of HNO₃ with P₄O₁₀

A. can also be prepared by reaction of P₄ and HNO₃

B. is diamagnetic

C. contains one N-N bond

D. reacts with Na metal producing a brown gas

Q. 30 For “invert sugar”, the correct statement(s) is(are)

(Given : specific rotations of (+) -sucrose, (+)-maltose, L-(-)glucose and L-(+)-fructose in aqueous solution are +66⁰ , +140⁰ , -52⁰ and +92⁰ , respectively)

A. ‘ínvert sugar’ is prepared by acid catalyzed hydrolysis of maltose

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

C. specific rotation of ‘invert sugar’ is -20⁰

D. on reaction with Br₂ water, ‘invert sugar’ forms saccharic acids as one of the products

Q. 31 Reagent(s) which can be used to bring about the given transformation is (are)

A. LiAlH₄ in (C₂H₅)₂O

B. BH₃ in THF

C. NaBH₄ in C₂H₅OH

D. Raney Ni/H₂ in THF

Q. 32 Among the given, reaction(s) which give(s) tert-butyl benzene as the major product is(are)

A. A

B. B

C. C

D. D

Questions: 33 – 34

•Read the paragraph and answer the following questions

•Each question has ONE correct option

Thermal decomposition of gaseous X₂ to gaseous X at 298K takes place according to the following equation:

X₂ (g) ⇄ 2X (g)

The standard reaction Gibbs energy, ΔᵣG°, of this reaction is positive. At the start of the reaction, there is one mole of X₂ and no X. As the reaction proceeds the number of moles of X formed is given by β. Thus, β(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 gasses to behave ideally. (Given: R = 0.083L bar K⁻¹mol⁻¹)

Q. 33 The equilibrium constant K_ρ for this reaction at 298 K, in terms of β_equilibrium, is

A. 8(β_equilibrium)² / (2 – β_equilibrium)

B. 8(β_equilibrium)² / (4 – β_equilibrium)

C. 4(β_equilibrium)² / (2 – β_equilibrium)

D. 4(β_equilibrium)² / (4 – β_equilibrium)

Q. 34 The INCORRECT statement among the following, for this reaction, is

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

B. At the start of the reaction , dissociation of gaseous X₂ takes place spontaneusly

C. β_equilibrium = 0.7

D. K_C < 1

Question 35

Treatment of compound O with KMnO₄/H+ gave P, which on heating with ammonia gave P. The compound Q on treatment with Br₂/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.

Q. 35 The compound R is

A. A

B. B

C. C

D. D

Question 36

Treatment of compound O with KMnO₄/H+ gave P, which on heating with ammonia gave P. The compound Q on treatment with Br₂/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.

Q. 36 The compound T is

A. glycine

B. alanine

C. valine

D. serine

Q. 37 Let P be the given lower triangular matrix and I be the identity matrix of order 3. If Q = [qᵥ ] is a matrix such that P⁵⁰ – Q = I, then (q₃₁ + q₃₂) / q₂₁ equals

A. 52

B. 103

C. 201

D. 205

Q. 38 Let bᵢ > 1 for i = 1, 2, …, 101. Suppose loge b₁, loge b₂, …, loge b₁₀₁ are in Arithmetic Progression (A. P.) with the common difference logₑ2. Suppose a₁, a₂, …, a₁₀₁ are in A. P. such that a₁ = b₁ and a₅₁ = b₅₁. If t = b₁ + b₂ + … + b₅₁ and s = a₁ + a₂ + … + a₅₁, then

A. s > t and a₁₀₁ > b₁₀₁

B. s > t and a₁₀₁ < b₁₀₁

C. s < t and a₁₀₁ > b₁₀₁

D. s < t and a₁₀₁ < b₁₀₁

Q. 39 The value of the given summation is equal to

K=1121sin 4+(k – 1) 6 sin 4 + k6

A. 3 – √3

B. 2(3 – √3)

C. 2(√3 – 1)

D. 2(2 + √3)

Q. 40 The value of the given integral is

A. Π²/4 – 2

B. Π²/4 + 2

C. Π² – e^(Π/2)

D. Π² + e^(Π/2)

Q. 41 Area of the region { (x, y) ∈ R² : y ≥ √ (|x + 3|), 5y ≤ x + 9 ≤ 15 } is equal to

A. 1/6

B. 4/3

C. 3/2

D. 5/3

Q. 42 Let P be the image of the point (3, 1, 7) with respect to the plane x – y + z = 3. The the equation of the plane passing through P and containing the straight line x/1 = y/2 = z/1 is

A. x + y – 3z = 0

B. 3x + z = 0

C. x – 4y + 7z = 0

D. 2x – y = 0

Q. 43 Let the given equality be true for all x > 0. Then

A. f(1/2) ≥ f(1)

B. f(1/3) ≤ f(2/3)

C. f'(2) ≤ 0

D. f'(3)/f(3) ≥ f'(2)/f(2)

Q. 44 Let a, b ∈ R and f : R → R be defined by f(x) = a cos (|x³ – x|) + b sin (|x³ + x|). Then f is

A. differentiable at x = 0 if a = 0 and b = 1

B. differentiable at x = 1 if a = 1 and b = 0

C. NOT differentiable at x = 0 if a = 1 and b = 0

D. NOT differentiable at x = 1 if a = 1 and b = 1

Q. 45 Let f : R → (0, ∞) and g : R → R be twice differentiable functions such that f” and g” are continuous functions on R. Suppose f'(2) = g(2) = 0, f”(2) ≠ 0 and g'(2) ≠ 0.

If limit_(x → 2) [f(x) g(x)] / [f'(x) g'(x)] = 1, then

A. f has a local minimum at x = 2

B. f has a local maximum at x = 2

C. f”(2) > f(2)

D. f(x) – f”(x) = 0 for at least one x ∈ R

Q. 46 Let f : [-1/2, 2] → R and g : [-1/2, 2] → R be functions defined by f(x) = [x² – 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 ∈ R. Then

A. f is discontinuous exactly at three points in [-1/2, 2]

B. f is discontinuous exactly at four points in [-1/2, 2]

C. g is NOT differentiable exactly at four points in (-1/2, 2)

D. g is NOT differentiable exactly at five points in (-1/2, 2)

Q. 47 Let a, b ∈ R and a² + b² ≠ 0. Suppose S = { z ∈ C : z = 1/(a + ibt), t ∈ R, t ≠ 0 }, where i = √-1. If z = x + iy and z ∈ S, then (x, y) lies on

A. the circle with radius 1/2a and centre (1/2a, 0) for a > 0, b ≠ 0

B. the circle with radius -1/2a and centre (-1/2a, 0) for a < 0, b ≠ 0

C. the x-axis for a ≠ 0, b = 0

D. the y-axis for a = 0, b ≠ 0

Q. 48 Let P be the point on the parabola y² = 4x which is at the shortest distance from the center S of the circle x² + y² – 4x – 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Then

A. SP = 2√5

B. SQ : QP = (√5 + 1) : 2

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

D. the slope of the tangent to the circle at Q is 1/2

Q. 49 Let a, λ, μ ∈ R. Consider the system of liner equations

ax + 2y = λ

3x – 2y = μ

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

A. If a = -3, then the system has infinitely many solutions for all values of λ and μ

B. If a ≠ -3, then the system has a unique solution for all values of λ and μ

C. If λ + μ = 0, then the system has infinitely many solutions for a = -3

D. If λ + μ ≠ 0, then the system has no solution for a = -3

Q. 50 Let û = u_1 î + u_2 ĵ + u_3 k̂ be a unit vector in R³ and ŵ = (1/√6)(î + ĵ + 2k̂). Given that there exists a vector v in R³ such that |û x v̂| = 1 and ŵ . (û x v̂) = 1. Which of the following statement(s) is(are correct?

A. There is exactly one choice for such vector v

B. There are indefinitely many choices for such vector v

C. If û lies in the xy-plane then |u_1| = |u_2|

D. If û lies in the xz-plane then 2 |u_1| = |u_3|

Questions: 51 – 52

Football teams T1 and T2 have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of T1 winning, drawing and losing a game against T2 are 1/2, 1/6 and 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 T1 and T2, respectively, after two games.

Q. 51 P(X > Y) is

A. 1/4

B. 5/12

C. 1/2

D. 7/12

Q. 52 P(X = Y) is

A. 11/36

B. 1/3

C. 13/36

D. 1/2

Questions: 53 – 54

Let F₁(x₁, 0) and F₂(x₂, 0), for x₁ < 0 and x₂ > 0, be the foci of the ellipse x²/9 + y²/8 = 1. Suppose a parabola having vertex at the origin and focus at F₂ intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant.

Q. 53 The orthocentre of the triangle F₁MN is

A. (-9/10, 0)

B. (2/3, 0)

C. (9/10, 0)

D. (2/3, √6)

Q. 54 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 MF₁NF₂ is

A. 3 : 4

B. 4 : 5

C. 5 : 8

D. 2 : 3

Answer Sheet |
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Question |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |

Answer | C | C | C | A | C | B | AC | C | ABD | BC |

Question |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |

Answer | ABCD | ABD | BD | AB | A | B | C | D | D | D |

Question |
21 |
22 |
23 |
24 |
25 |
26 |
27 |
28 |
29 |
30 |

Answer | A | A | A | A | AC | AB | BCD | ABC | BD | BC |

Question |
31 |
32 |
33 |
34 |
35 |
36 |
37 |
38 |
39 |
40 |

Answer | CD | BCD | B | C | A | B | B | B | C | A |

Question |
41 |
42 |
43 |
44 |
45 |
46 |
47 |
48 |
49 |
50 |

Answer | C | C | BC | AB | AD | BC | ACD | ACD | BCD | BC |

Question |
51 |
52 |
53 |
54 |
||||||

Answer | B | C | A | C |