JEE Advanced 2015 Paper 1
Q. 1 An infinitely long uniform line charge distribution of charge per unit length x lies parallel to the y-axis in the y-z plane at z = (root of 3/2)a (see figure). If the magnitude of the flux of the electric field through the rectangular surface ABCD lying in the x-y plane with its centre at the origin is xL/nE0 (E0 = permittivity of free space), then the value of n is
Q. 2 Consider a hydrogen atom with its electron in the nth orbital. An electromagnetic radiation of wavelength 90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV, then the value on n is (hc = 1242 eV nm)
Q. 3 A bullet is fired vertically upwards with velocity v from the surface of a spherical planet. When it reaches its maximum height, its acceleration due to the planet’s gravity is 1/4th of its value at the surface of the planet. If the escape velocity from the planet is vesc = v(root of N), then the value of N is (ignore energy loss due to atmosphere)
Q. 4 Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds v₁ and v₂, respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and v₁ = 3 m/s, then v₂ in m/s is (g = 10 m/s²)
Q. 5 Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits 10⁴ times the power emitted from B. The ratio (xA/xB) of their wavelengths xA and xB at which the peaks occur in their respective radiation curves is
Q. 6 A nuclear power plant supplying electrical power to a village uses a radioactive material of half life T years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is 12.5% of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of nT years, then the value of n is
Q. 7 A Young’s double slit interference arrangement with slits S₁ and S₂ is immersed in water (refractive index = 4/3) as shown in the figure. The positions of maxima on the surface of water are given by x² = p²m²λ² – d², where λ is the wavelength of light in air (refractive index = 1), 2d is the separation between the slits and m is an integer. The value of p is
Q. 8 Consider a concave mirror and a convex lens (refractive index = 1.5) of focal length 10 cm each, separated by a distance of 50 cm in air (refractive index = 1) as shown in the figure. An object is placed at a distance of 15 cm from the mirror. Its erect image formed by this combination has magnification M₁. When the set-up is kept in a medium of refractive index 7/6, the magnification becomes M₂. The magnitude |M₂/M₁| is
Q. 9 Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier scale coincide with 4 divisions on the main scale and in the screw gauge, one compete rotation of the circular scale moves it by two divisions on the linear scale. Then:
A. If the pitch of the screw gauge is twice the least count of the Vernier callipers, the
least count of the screw gauge is 0.01 mm.
B. If the pitch of the screw gauge is twice the least count of the screw gauge is 0.005 mm.
C. If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.01 mm.
D. If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge isi 0.005 mm.
Q. 10 Planck’s constant h, speed of light c and gravitational constant G are used to form a unit of length L and a unit of mass M. Then the correct option(s) is (are)
A. M ∝ √c
B. M ∝ √G
C. L ∝ √h
D. L ∝ √G
Q. 11 Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies w₁ and w₂ and have total energies E₁ and E₂, respectively. The variations of their momenta p with positions x are shown in the figures. If a/b = n² and a/R = n, then the correct equation(s) is(are)
A. E₁w₁ = E₂w₂
B. w₂/w₁ = n²
C. w₁w₂ = n²
D. E₁/w₁ = E₂/w₂
Q. 12 A ring of mass M and radius R is rotating with angular speed w about a fixed vertical axis passing through its centre O with two point masses each of mass M/8 at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is 8w/9 and one of the masses is at a distance of 3R/5 from O. At this instant the distance of the other mass from O is
Q. 13 The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density are kept parallel to each other. In their resulting electric field, point charges q and -q are kept in equilibrium between them. The point charges are confined to move in the x direction only. If they are given a small displacement about their equilibrium then the correct statement(s) is(are)
A. Both charges execute simple harmonic motion.
B. Both charges will continue moving in the direction of their displacement.
C. Charge +q executes simple harmonic motion while charge -q continues moving in the direction of its displacement.
D. Charge -q executes simple harmonic motion while charge +q continues moving in the direction of its displacement.
Q. 14 Two identical glass rods S₁ and S₂ (refractive index = 1.5) have one convex end of radius of curvature 10 cm. They are placed with the curved surfaces at a distance d as shown in the figure, with their axes (shown by the dashed line) aligned. When a point source of light P is placed inside rod S₁ on its axis at a distance of 50 cm from the curved face, the light rays emanating from it are found to be parallel to the axis inside S₂. The distance d is
A. 60 cm
B. 70 cm
C. 80 cm
D. 90 cm
Q. 15 A conductor (shown in the figure) carrying constant current I is kept in the x-y plane in a uniform magnetic field B. If F is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is(are)
A. If B is along z, F ∝ (L + R)
B. If B is along x, F = 0
C. If B is along y, F ∝ (L + R)
D. If B is along z, F = 0
Q. 16 A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature T. Assuming the gases are ideal, the correct statement(s) is (are)
A. The average energy per mole of the gas mixture is 2RT.
B. The ratio of speed of sound in the gas mixture to that in helium gas is root of 6/5
C. The ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/2
D. The ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/√2
Q. 17 In an aluminium (Al) bar of square cross section, a square hole is drilled and is filled with iron (Fe) as shown in the figure The electrical resistivities of Al and Fe are 2.7 x 10⁻⁸ ohm m and 1.0 x 10⁷ ohm m, respectively. The electrical resistance between the two faces P and Q of the composite bar is
Q. 18 For photo-electric effect with incident photon wavelength lambda, the stopping potential is V₀. Identify the correct variation(s) of V₀ with λ and 1/λ.
Q. 19 Match the nuclear processes given in column I with the appropriate option(s) in column II.
A. A – R,T ; B – P,S ; C – Q,T ; D – R
B. A – P,Q ; B – R,S ; C – T ; D – P
C. A – S ; B – P,T ; C – Q ; D – R
D. A – P ; B – R ; C – S,T ; D – P,Q
Q. 20 A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I (a and U₀ are constants). Match the potential energies in column I to the corresponding statement(s) in column II.
A. A – Q,S ; B – P,R ; C – P,T ; D – P
B. A – P,T ; B – R,S ; C – Q ; D – Q,R
C. A – P,Q,R,T ; B – Q,S ; C – P,Q,R,S ; D – P,R,T
D. A – P,T ; B – P,R ; C – S,Q ; D – T
Q. 21 The total number of sterioisomers that can exist for M is
Q. 22 The number of resonance structures for N is
Q. 23 The total number of lone pairs of electrons in N₂O₃ is
Q. 24 For the octahedral complexes of Fe³+ in SCN (thiocyanato-S) and in CN⁻ ligand environments, the difference between the spin-only magnetic moments in Bohr magnetons (when approximated to the nearest integer) is
[Atomic number of Fe = 26]
Q. 25 Among the triatomic molecules/ions, BeCl₂, N₃⁻, N₂O, NO₂⁺, O₃, SCl₂, ICl₂⁻, I₃⁻ and XeF₂, the total number of linear molecule(s)/ion(s) where the hybridization of the central atom does not have contribution from the d-orbital(s) is
[Atomic number: S = 16, Cl = 17, I = 53 and Xe = 54]
Q. 26 Not considering the electronic spin, the degeneracy of the second excited state (n = 3) of H atom is 9, while the degeneracy of the second excited state of H⁻ is
Q. 27 All the energy released from the reaction X —-> Y, delta, G⁰ = -193 kJ mol⁻¹ is used for oxidizing M⁺ as M⁺ —-> M³⁺ + 2e⁻, E⁰ = -0.25 V. Under standard conditions, the number of moles of M⁺ oxidized when one mole of X is converted to Y is [F = 96500 C mol⁻²]
Q. 28 If the freezing point of a 0.01 molal aqueous solution of a cobalt (III) chloride-ammonia complex (which behaves as a strong electrolyte) is -0.0558 degrees C, the number of chloride(s) in the coordination sphere of the complex is
[Kf of water = 1.86 K kg mol⁻¹]
Q. 29 Compound(s) that on hydrogenation produce(s) optically inactive compound(s) is(are)
Q. 30 The major product of the following reaction is
Q. 31 In the following reaction the major product is
Q. 32 The structure of D-(+)-glucose is given. The stricture of L-(-)-glucose is
Q. 33 The major product of the reaction is
Q. 34 The correct statement(s) about Cr²+ and Mn³+ is(are) [Atomic numbers of Cr = 24 and Mn = 25
A. Cr²+ is a reducing agent
B. Mn³⁺ is an oxidizing agent
C. Both Cr²+ and Mn³+ exhibit d⁴ electronic configuration
D. When Cr²+ is used as a reducing agent, the thromium ion attains d⁵ electronic
Q. 35 Copper is purified by electrolytic refining of blister copper. The correct statement(s) about this process is(are)
A. Impure Cu strip is used as cathode
B. Acidified aqueous CuSO₄ is used as electrolyte
C. Pure Cu deposits at cathode
D. Impurities settle as anode-mud
Q. 36 Fe³⁺ is reduced to Fe²⁺ by using
A. H₂O₂ in presence of NaOH
B. Na₂O₂ in water
C. H₂O₂ in presence of H₂SO₄
D. Na₂O₂ in presence of H₂SO₄
Q. 37 The % yield of ammonia as a function of time in the reaction at (P, T₁) is given. If this reaction is conducted at (P, T₂) with T₂>T₁, the % yield of ammonia as a function of time is represented by
Q. 38 If the unit cell of a mineral has cubic close packed (ccp) array of oxygen atoms wih m fraction of octahedral holes occupied by aluminium ions and n fraction of tetrahedral holes occupied by magnesium ions, m and n, respectively, are
A. 1/2, 1/8
B. 1, 1/4
C. 1/2, 1/2
D. 1/4, 1/8
Q. 39 Match the anionic species given in Column I that are present in the ore(s) given in Column II.
Column – I Column – II
(A) Carbonate (P) Siderite
(B) Sulphide (Q) Malachite
(C) Hydroxide (R) Bauxite
(D) Oxide (S) Calamine (T) Argentite
A. A – P,T ; B – R ; C- T,S ; D – Q
B. A – P,Q,S ; B – T ; C – Q,R ; D – R
C. A – P,Q,S ; B – T ; C – P,R ; D – R
D. A – P,Q,S ; B – P,R ; C – S ; D – T,R
Q. 40 Match the thermodynamic processes given under column I with the expressions given under column II.
A. A – P,T ; B – Q,S ; C – S,T ; D – R
B. A – Q ; B – P,S ; C – R,T ; D – P,Q,S,T
C. A – R,T ; B – P,Q,S ; C – P,Q,S ; D – P,Q,S,T
D. A – P,S ; B-P,Q,S ; C- P,S ; D -P,Q
Q. 41 The number of distinct solutions of the equation 5/4 cos²(2x) + cos⁴x + sin⁴x + cos⁶x + sin⁶x = 2 in the interval [0, 2pi] is
Q. 42 Let the curve C be the mirror image of the parabola y² = 4x with respect to the line x + y + 4 = 0. If A and B are the points of intersection of C with the line y = -5, then the distance between A and B is
Q. 43 The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is
Q. 44 Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of m/n is
Q. 45 If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents to the circle (x-3)² + (y+2)² = r², then the value of r² is
Q. 46 Let f : R —> R be a function defined by
f(x) = [x], x <=2
f(x) = 0, x > 2
where [x] is the greatest integer less than or equal to x. The value of (4I – 1) is
Q. 47 A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of V mm^3, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of V/250pi is
Q. 48 For a is an element of [0, 1/2], if F'(a) + 2 is the area of the region bounded by x = 0, y = 0, y = f(x) and x = a, then f(0) is
Q. 49 Let X and Y be two arbitrary, 3×3, non-zero, skew-symmetric matrices and Z be an arbitrary 3×3, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?
A. Y³Z⁴ – Z⁴Y³
B. X⁴⁴ + Y⁴⁴
C. X⁴Z³ – Z³X⁴
D. X²³ + Y²³
Q. 50 Which of the following values of a satisfy the equation
Q. 51 In R³, consider the planes P₁ : y = 0 and P₂ : x + z = 1. Let P₃ be a plane, different from P₁ and P₂, which passes through the intersection of P₁ and P₂. If the distance of the point (0, 1, 0) from P₃ is 1 and the distance of a point (a, b, c) from P₃ is 2, then which of the following relations is (are) true?
A. 2a + b + 2c + 2 = 0
B. 2a – b + 2c + 4 = 0
C. 2a + b – 2c – 10 = 0
D. 2a – b + 2c – 8 = 0
Q. 52 In R³, let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P₁ : x + 2y – z + 1 = 0 and P₂ : 2x – y + z = 0. Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P₁. Which of the following points lie(s) on M?
A. (0, -5/6, -2/3)
B. (-1/6, -1/3, 1/6)
C. (-5/6, 0, 1/6)
D. (-1/3, 0, 2/3)
Q. 53 Let P and Q be distinct points on the parabola y² = 2x such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle OPQ is 3√2, then which of the following is (are) the coordinates of P?
A. 4 , 2√2
B. 9 , 3√2
C. 1/4 , 1⁄√2
D. 1 , √2
Q. 54 Let y(x) be a solution of the differential equation (1 + eˣ)y’+ yeˣ = 1. If y(0) = 2, then which of the following statements is (are) true?
A. y(-4) = 0
B. y(-2) = 0
C. y(x) has a critical point in the interval (-1, 0)
D. y(x) has no critical point in the interval (-1, 0)
Q. 55 Consider the family of all circles whose centers lie on the straight line y = x. If this family of circles is represented by the differential equation Py” + Qy’ + 1 = 0, where P, Q are functions of x, y and y’ (here y’ = dy/dx, y” = d²y/dx²), then which of the following statements is (are) true?
A. P = y + x
B. P = y – x
C. P + Q = 1 – x + y + y’ + (y’)²
D. P – Q = x + y – y’ – (y’)²
Q. 56 Let g : R—>R be a differentiable function with g(0) = 0, g'(0) = 0 and g'(0) = 0 and g'(1) is not = 0. Let
f(x) = xg(x)/|x|, x is not = 0
f(x) = 0, x = 0
and h(x) = e^|x| for all x is element of R. Let (f . h)(x) denote f(h(x)) and (h . f)(x) denote h(f(x)). Then which of the following is (are) true?
A. f is differentiable at x = 0
B. h is differentiable at x = 0
C. f . h is differentiable at x = 0
D. h . f is differentiable at x = 0
Q. 57 Let f(x) = sin(pi/6 sin(pi/2 sinx)) for all x is element of R and g(x) = pi/2 sinx for all x element of R. Let (f . g)(x) denote f(g(x)) and (g . f)(x) denote g(f(x)). Then which of the following is (are) true?
A. Range of f is [-1/2, 1/2]
B. Range of f . g is [-1/2, 1/2]
C. lim(x—>0) f(x)/g(x) = pi/6
D. There is an x element of R such that (g . f)(x) = 1
Q. 58 Let PQR be a triangle. Let a = QR, b = RP and c = PQ. If |a| = 12, |b| = 4√3 and b.c = 24, then which of the following is (are) true?
A. |c|²/2 – |a| = 12
B. |c|²/2 + |a| = 30
C. |a x b + c x a| = 48√3
D. a.b = -72
Q. 59 Match the column
A. A – P,Q ; B – P,Q ; C – P,Q,S,T ; D – Q,T
B. A – P,S ; B – R,T ; C – S,T ; D – Q,T
C. A – R,T ; B – P,S ; C – S,T ; D – P,T
D. A – P,Q ; B – R,S ; C – S,T ; D – Q,S
Q. 60 Match the column
A. A – P,Q,T ; B – S,R ; C – P,T ; D – R
B. A – P,R,S ; B – P ; C – P,Q; D – S,T
C. A – P,T ; B – Q,R ; C – S,R ; D – R
D. A – Q,S ; B – S,T ; C – P,T D – S