JEE Advanced 2015 Paper 1 Offline

Paper was held on Sat, May 23, 2015 9:00 PM

Chemistry

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

Copper is purified by electrolytic refining of blister copper. The correct statement(s) about this process is(are)

All the energy released from the reaction
$$X \to Y, \Delta _tG^o $$ = -193 kJ mol^-1 is used for oxidizing M⁺ as M⁺ $$\to$$ M³⁺ + 2e^-, E^o = -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^–1]

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^oC, the number of chloride(s) in the coordination sphere of the complex is [K_f of water = 1.86 K kg mol^–1 ]

If the unit cell of a mineral has cubic close packed (ccp) array of oxygen atoms with m fraction of octahedral holes occupied by aluminium ions and n fraction of tetrahedral holes occupied by magnesium ions, m and n, respectively, are

Match the thermodynamics processes given under column I with expression given under column II

Column I
(A) Freezing water at 273 K and 1 atm
(B) Expansion of 1 mol of an ideal gas into a vacuum under isolated conditions.
(C) Mixing of equal volumes of two ideal gases at constant temperature and pressure in an isolated container.
(D) Reversible heating of H₂(g) at 1 atm from 300K to 600K, followed by reversible cooling to 300K at 1 atm

Column II
(p) q = 0
(q) w = 0
(r) $$\Delta S_{sys}$$ < 0
(s) $$\Delta U$$ = 0
(t) $$\Delta G$$ = 0

Among the triatomic molecules/ions, BeCl₂, $$N_3^-$$, N₂O, $$NO_2^+$$, O₃, SCl₂, $$ICl_2^-$$, $$I_3^-$$ 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]

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 ___________.

The total number of stereoisomers that can exist for M is ___________.

The number of resonance structure for N is _________.

The number of lone pairs of electrons in N₂O₃ is ___________.

For the octahedral complexes of Fe³⁺ in SCN^$$-$$ (thiocyana-to-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 Fe = 26]

Compound(s) that on hydrogenation produce(s) optically inactive compound(s) is(are)

The major product of the following reaction is

In the following reaction, the major product is

The structure of D-(+)-glucose is

The structure of L-($$-$$)-glucose is

The major product of the reaction is

The correct statements about Cr²⁺ and Mn³⁺ is(are) (Atomic numbers of Cr = 24 and Mn = 25)

Fe³⁺ is reduced to Fe²⁺ by using

The % yield of ammonia as a function of time in the reaction

N₂(g) + 3H₂(g) $$\rightleftharpoons$$ 2NH₃(g), $$\Delta$$H < 0 at (P, T₁) is given below:

If this reactions is conducted at (P, T₂), with T₂ > T₁, the % yield of ammonia as a function of time is represented by

Mathematics

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 $${m \over n}$$ is

The number of distinct solutions of the equation

$${5 \over 4}{\cos ^2}\,2x + {\cos ^4}\,x + {\sin ^4}\,x + {\cos ^6}\,x + {\sin ^6}\,x\, = \,2$$

in the interval $$\left[ {0,\,2\pi } \right]$$ is

Match the following :

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column $$I$$
(A)$$\,\,\,\,$$ In $${R^2},$$ If the magnitude of the projection vector of the vector $$\alpha \widehat i + \beta \widehat j$$ on $$\sqrt 3 \widehat i + \widehat j$$ and If $$\alpha = 2 + \sqrt 3 \beta ,$$ then possible value of $$\left| \alpha \right|$$ is/are
(B)$$\,\,\,\,$$ Let $$a$$ and $$b$$ be real numbers such that the function $$f\left( x \right) = \left\{ {\matrix{ { - 3a{x^2} - 2,} & {x < 1} \cr {bx + {a^2},} & {x \ge 1} \cr } } \right.$$ if differentiable for all $$x \in R$$. Then possible value of $$a$$ is (are)
(C)$$\,\,\,\,$$ Let $$\omega \ne 1$$ be a complex cube root of unity. If $${\left( {3 - 3\omega + 2{\omega ^2}} \right)^{4n + 3}} + {\left( {2 + 3\omega - 3{\omega ^2}} \right)^{4n + 3}} + {\left( { - 3 + 2\omega + 3{\omega ^2}} \right)^{4n + 3}} = 0,$$ then possible value (s) of $$n$$ is (are)
(D)$$\,\,\,\,$$ Let the harmonic mean of two positive real numbers $$a$$ and $$b$$ be $$4.$$ If $$q$$ is a positive real nimber such that $$a, 5, q, b$$ is an arithmetic progression, then the value(s) of $$\left| {q - a} \right|$$ is (are)

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column $$II$$
(p)$$\,\,\,\,$$ $$1$$
(q)$$\,\,\,\,$$ $$2$$
(r)$$\,\,\,\,$$ $$3$$
(s)$$\,\,\,\,$$ $$4$$
(t)$$\,\,\,\,$$ $$5$$

Let $$P$$ and $$Q$$ be distinct points on the parabola $${y^2} = 2x$$ 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 $$\Delta OPQ$$ is $${3\sqrt 2 ,}$$ then which of the following is (are) the coordinates of $$P$$?

If the normals of the parabola $${y^2} = 4x$$ drawn at the end points of its latus rectum are tangents to the circle $${\left( {x - 3} \right)^2} + {\left( {y + 2} \right)^2} = {r^2}$$, then the value of $${r^2}$$ is

Let the curve $$C$$ be the mirror image of the parabola $${y^2} = 4x$$ 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

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $$V$$ $$m{m^3}$$, 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 $${V \over {250\pi }}$$ is

Let $$y(x)$$ be a solution of the differential equation
$$\left( {1 + {e^x}} \right)y' + y{e^x} = 1.$$
If $$y(0)=2$$, then which of the following statement is (are) true?

Consider the family of all circles whose centres lie on the straight line $$y=x,$$ If this family of circle is represented by the differential equation $$Py'' + Qy' + 1 = 0,$$ where $$P, Q$$ are functions of $$x,y$$ and $$y'$$ $$\left( {here\,\,\,y' = {{dy} \over {dx}},y'' = {{{d^2}y} \over {d{x^2}}}} \right)$$ then which of the following statements is (are) true?

Let $$f:R \to R$$ be a function defined by
$$f\left( x \right) = \left\{ {\matrix{ {\left[ x \right],} & {x \le 2} \cr {0,} & {x > 2} \cr } } \right.$$ where $$\left[ x \right]$$ is the greatest integer less than or equal to $$x$$, if $$I = \int\limits_{ - 1}^2 {{{xf\left( {{x^2}} \right)} \over {2 + f\left( {x + 1} \right)}}dx,} $$ then the value of $$(4I-1)$$ is

Let $$F\left( x \right) = \int\limits_x^{{x^2} + {\pi \over 6}} {2{{\cos }^2}t\left( {dt} \right)} $$ for all $$x \in R$$ and $$f:\left[ {0,{1 \over 2}} \right] \to \left[ {0,\infty } \right]$$ be a continuous function. For $$a \in \left[ {0,{1 \over 2}} \right],\,$$ $$F'(a)+2$$ is the area of the region bounded by $$x=0, y=0, y=f(x)$$ and $$x=a,$$ then $$f(0)$$ is

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

In $${R^3},$$ consider the planes $$\,{P_1}:y = 0$$ and $${P_2}:x + z = 1.$$ Let $${P_3}$$ be the plane, different from $${P_1}$$ and $${P_2}$$, which passes through the intersection of $${P_1}$$ and $${P_2}.$$ If the distance of the point $$(0,1, 0)$$ from $${P_3}$$ is $$1$$ and the distance of a point $$\left( {\alpha ,\beta ,\gamma } \right)$$ from $${P_3}$$ is $$2,$$ then which of the following relations is (are) true?

In $${R^3},$$ let $$L$$ be a straight lines passing through the origin. Suppose that all the points on $$L$$ are at a constant distance from the two planes $${P_1}:x + 2y - z + 1 = 0$$ and $${P_2}:2x - y + z - 1 = 0.$$ Let $$M$$ be the locus of the feet of the perpendiculars drawn from the points on $$L$$ to the plane $${P_1}.$$ Which of the following points lie (s) on $$M$$?

Let $$\Delta PQR$$ be a triangle. Let $$\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP} $$ and $$\overrightarrow c = \overrightarrow {PQ} .$$ If $$\left| {\overrightarrow a } \right| = 12,\,\,\left| {\overrightarrow b } \right| = 4\sqrt 3 ,\,\,\,\overrightarrow b .\overrightarrow c = 24,$$ then which of the following is (are) true?

Match the following :

	Column I		Column I
(A)	$\begin{array}{l}\text { In a triangle } \Delta X Y Z \text {, let } a, b \text { and } c \text { be the lengths of the sides } \\\text { opposite to the angles } X, Y \text { and } Z \text {, respectively. If } 2\left(a^2-b^2\right)=c^2 \\\text { and } \lambda=\frac{\sin (X-Y)}{\sin Z} \text {, then possible values of } n \text { for which } \cos (n \lambda) \\=0 \text { is (are) }\end{array}$	(P)	1
(B)	$\begin{array}{l}\text { In a triangle } \triangle X Y Z \text {, let } a, b \text { and } c \text { be the lengths of the sides } \\\text { opposite to the angles } X, Y \text { and } Z \text {, respectively. If } 1+\cos 2 X-2 \\\cos 2 Y=2 \sin X \sin Y \text {, then possible value(s) of } \frac{a}{b} \text { is (are) }\end{array}$	(Q)	2
(C)	$\begin{array}{l}\text { In } \mathbb{R}^2 \text {, let } \sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j} \text { and } \beta \hat{i}+(1-\beta) \hat{j} \text { be the position } \\\text { vectors of } X, Y \text { and } Z \text { with respect of the origin } \mathrm{O} \text {, respectively. If } \\\text { the distance of } \mathrm{Z} \text { from the bisector of the acute angle of } \overrightarrow{\mathrm{OX}} \text { with } \\\overrightarrow{\mathrm{OY}} \text { is } \frac{3}{\sqrt{2}} \text {, then possible value(s) of }\|\beta\| \text { is (are) }\end{array}$	(R)	3
(D)	$\begin{array}{l}\text { Suppose that } F(\alpha) \text { denotes the area of the region bounded by } \\x=0, x=2, y^2=4 x \text { and } y=\|\alpha x-1\|+\|\alpha x-2\|+\alpha x \text {, } \\\text { where, } \alpha \in\{0,1\} \text {. Then the value(s) of } F(\alpha)+\frac{8}{2} \sqrt{2} \text {, when } \alpha=0 \\\text { and } \alpha=1 \text {, is (are) }\end{array}$	(S)	5
		(T)	6

Let X and Y be two arbitrary, 3 $$\times$$ 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 $$\times$$ 3, non-zero, symmetric matrix. Then which of the following matrices is(are) skew symmetric?

Which of the following values of $$\alpha$$ satisfy the equation

$$\left| {\matrix{ {{{(1 - \alpha )}^2}} & {{{(1 + 2\alpha )}^2}} & {{{(1 + 3\alpha )}^2}} \cr {{{(2 + \alpha )}^2}} & {{{(2 + 2\alpha )}^2}} & {{{(2 + 3\alpha )}^2}} \cr {{{(3 + \alpha )}^2}} & {{{(3 + 2\alpha )}^2}} & {{{(3 + 3\alpha )}^2}} \cr } } \right| = - 648\alpha $$ ?

Let $$g:R \to R$$ be a differentiable function with $$g(0) = 0$$, $$g'(0) = 0$$ and $$g'(1) \ne 0$$. Let

$$f(x) = \left\{ {\matrix{ {{x \over {|x|}}g(x),} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$$

and $$h(x) = {e^{|x|}}$$ for all $$x \in R$$. Let $$(f\, \circ \,h)(x)$$ denote $$f(h(x))$$ and $$(h\, \circ \,f)(x)$$ denote $$f(f(x))$$. Then which of the following is (are) true?

Let $$f(x) = \sin \left( {{\pi \over 6}\sin \left( {{\pi \over 2}\sin x} \right)} \right)$$ for all $$x \in R$$ and g(x) = $${{\pi \over 2}\sin x}$$ for all x$$\in$$R. Let $$(f \circ g)(x)$$ denote f(g(x)) and $$(g \circ f)(x)$$ denote g(f(x)). Then which of the following is/are true?

Physics

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $$\lambda $$ 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 positions, then the correct statement(s) is (are)

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 $$\left( {{{{\lambda _A}} \over {{\lambda _B}}}} \right)$$ of their wavelengths $${{\lambda _A}}$$ and $${{\lambda _B}}$$ at which the peaks occur in their respective radiation curves is

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 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 $${\left( {{1 \over 4}} \right)^{th}}$$ of its value at the surface of the planet. If the escape velocity from the planet is $${v_{esc}} = v\sqrt N $$, then the value of N is (ignore energy loss due to atmosphere)

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 complete rotation of the circular scale moves it by two divisions on the linear scale. Then:

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 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 = p²m²$$\lambda$$² $$-$$ d², where $$\lambda$$ 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

JEE Advanced 2015 Paper 1 Offline Physics - Wave Optics Question 12 English

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 $${7 \over 6}$$, the magnification becomes M₂. The magnitude $$\left| {{{{M_2}} \over {{M_1}}}} \right|$$ is

JEE Advanced 2015 Paper 1 Offline Physics - Geometrical Optics Question 39 English

An infinitely long uniform line charge distribution of charge per unit length $$\lambda$$ lies parallel to the y-axis in the y-z plane at $$z = {{\sqrt 3 } \over 2}$$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 $${{\lambda L} \over {n{\varepsilon _0}}}$$ ($${{\varepsilon _0}}$$ = permittivity of free space), then the value of n is

JEE Advanced 2015 Paper 1 Offline Physics - Electrostatics Question 27 English

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 of n is (hc = 1242 eV nm)

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)

JEE Advanced 2015 Paper 1 Offline Physics - Rotational Motion Question 37 English

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

A ring of mass M and radius R is rotating with angular speed $$\omega$$ about a fixed vertical axis passing through its centre O with two point masses each of mass $${M \over 8}$$ 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 $${8 \over 9}$$$$\omega$$ and one of the masses is at a distance of $${3 \over 5}$$R from O. At this instant, the distance of the other mass from O is

JEE Advanced 2015 Paper 1 Offline Physics - Rotational Motion Question 36 English

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

JEE Advanced 2015 Paper 1 Offline Physics - Geometrical Optics Question 38 English

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 statements is/are

JEE Advanced 2015 Paper 1 Offline Physics - Magnetism Question 35 English

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 $$\times$$ 10^$$-$$8$$\Omega$$ m and 1.0 $$\times$$ 10^$$-$$7 $$\Omega$$ m, respectively. The electrical resistance between the two faces P and Q of the composite bar is

JEE Advanced 2015 Paper 1 Offline Physics - Current Electricity Question 20 English

For photo-electric effect with incident photon wavelength $$\lambda$$, the stopping potential is V₀. Identify the correct variation(s) of V₀ with $$\lambda$$ and $${1 \over \lambda }$$.

Two independent harmonic oscillators of equal masses are oscillating about the origin with angular frequencies $$\omega$$₁ and $$\omega$$₂ and have total energies E₁ and E₂, respectively. The variations of their momenta p with positions x are shown in the figures. If $${a \over b} = {n^2}$$ and $${a \over R} = n$$, then the correct equations is/are

JEE Advanced 2015 Paper 1 Offline Physics - Simple Harmonic Motion Question 17 English

Match the nuclear processes given in Column I with the appropriate option(s) in Column II:

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: