JEE Advanced 2015 Paper-1

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 $$z = \frac{\sqrt{3}}{2} a$$ (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 $$\frac{\lambda L}{n ε_0}$$ ($$ε_0$$ = permittivity of free space), then the value of n is

Consider a hydrogen atom with its electron in the $$n^{th}$$ 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)

A bullet is fired vertically upwards with velocity v from the surface of a spherical planet. Whenit reaches its maximum height, its acceleration due to the planet’s gravity is $$\frac{1}{4^{th}}$$ 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)

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds $$v_1$$ and $$v_2$$, respectively, and always remain in contact with the surfaces. If they reach B and D with the samelinear speed and $$v_1 = 3 m/s$$, the $$v_2$$ in m/s is ($$g = 10 m/s^2$$)

Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits $$10^4$$ times the power emitted from B. The ratio $$\left(\frac{\lambda_A}{\lambda_B}\right)$$ of their wavelengths $$\lambda_A$$ and $$\lambda_B$$ at which the peaks occurin their respective radiation curves is

A nuclear power plant supplying electrical power to a village uses a radioactive material of half life 7’ years as the fuel. The amountof fuel at the beginningis 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 needsof the village for a maximum period of nT years, then the valueof n is

A Young’s double slit interference arrangement with slits $$S_1$$ and $$S_2$$ is immersed in water (refractive index = $$\frac{4}{3}$$) as shown in the figure. The positions of maxima on the surface of water are given by $$x^2 = p^2m^2 \lambda^2 - d^2$$, where $$\lambda$$ is the wave length of light in air (refractive index = 1), 2d is the separation between theslits and m is an integer. The valueofp is

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 $$M_1$$. When the set-up is kept in a medium of refractive index $$\frac{7}{6}$$, the magnification becomes $$M_2$$. The magnitude $$\mid \frac{M_2}{M_1} \mid$$ is

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 movesit 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. Thenthe correct option(s) is(are)

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies $$\omega_1$$ and $$\omega_2$$ and have total energies $$E_1$$ and $$E_2$$, respectively. The variations of their momenta p with positions x are shown in the figures. If $$\frac{a}{b} = n^2$$ and $$\frac{a}{R} = n$$, then the correct equation(s) is(are)

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 $$\frac{M}{8}$$ 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 $$\frac{8}{9}\omega$$ and one of the masses is at a distance of $$\frac{3}{5}R$$ from O. At this instant the distance of the other mass from O is

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 identical glass rods $$S_1$$ and $$S_2$$ (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 $$S_1$$ 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 $$S_2$$. The distance d is

A conductor (shown in the figure) carrying constant current I is kept in the x-y plane in a uniform magnetic field $$\overrightarrow{B}$$. If F is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is(are)

A containerof fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature T. Assuming The average energy per mole of the gas mixture is 2RT.the gases are ideal, the correct statement(s) is(are)

In an aluminum (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 $$2.7 \times 10^{-8} Ω m$$ and $$1.0 \times 10^{-7} Ω m,$$ respectively. The electrical resistance between the two faces P and Q of the composite bar is

For photo-electric effect with incident photon wavelength $$\lambda$$, the stopping potential is $$V_0$$. Identify the correct variation(s) of $$V_0$$ with $$\lambda$$ and $$\frac{1}{\lambda}$$.

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 $$U_0$$ are constants). Match the potential energies in column I to the corresponding statement(s) in column II.

The total number of stereoisomers that can exist for M is

The number of resonance structures for N is

The total number of lone pairs of electrons in $$N_2O_3$$ is

For the octahedral complexes of $$Fe^{3+}$$ 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]

Among the triatomic molecules/ions, $$BeCl_5, N_3^-, N_2O, NO_2^+, O_3, SCl_2,ICl_2^-, I_3^-$$ and $$XeF_2$$, 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, 1=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

All the energy released from the reaction $$X \rightarrow Y, \triangle_r G^0 = - 193 kJ mol^{-1}$$ is used for oxidizing $$M^+$$ as $$M^+ \rightarrow M^{3+} + 2e^-, E^0 = -0.25 V$$ 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 $$-0.0558 ^\circ_C$$ the numberof chloride(s) in the coordination sphere of the complex is $$[K_f$$ of water = 1.86 K kg $$mol^{-1}$$]

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 statement(s) about $$Cr^{2+}$$ and $$Mn^{3+}$$ is(are)
[Atomic numbers of Cr = 24 and Mn = 25]

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

$$Fe^{3+}$$ is reduced to $$Fe^{2+}$$ by using

The % yield of ammonia as a function of time in the reaction
$$N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g), \triangle H < 0$$ at $$(P, T_1)$$ is given below.

If this reaction is conducted at $$(P, T_2)$$, with $$T_2 > T_1$$, the %yield of ammonia as a function of time is represented by

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 anionic species given in Column I that are present in the ore(s) given in Column II.

Match the thermodynamic processes given under Column I with the expressions given under Column II.

The number of distinct solutions of the equation
$$\frac{5}{4} \cos^2 2x + \cos^4 x + \sin^4x + \cos^6 x + \sin^6 x = 2$$
in the interval $$[0, 2\pi]$$ 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

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

Let n be the numberof 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 $$\frac{m}{n}$$ is

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

Let $$f : R \rightarrow R$$ be a function defined by $$f(x) = \begin{cases}[x], & x \leq 2\\0 & x > 2\end{cases}$$,
where [x] is the greatest integer less than or equal to x. If $$I = \int_{-1}^{2} \frac {xf(x^2)}{2 + f(x + 1)} dx,$$ then the value of (4I - 1) is

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 $$\frac{V}{250 \pi}$$ is

Let $$F(x) = \int_{x}^{x^2 + \frac{\pi}{6}} 2 \cos^2t dt$$ for all $$x \in R$$ and $$f : \left[0, \frac{1}{2}\right] \rightarrow [0, \infty)$$ be a continuous function. For $$a \in \left[0, \frac{1}{2}\right]$$, 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

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

In $$R^3$$, consider the planes $$P_1 : y = 0$$ and $$P_2 : x + z = 1$$. Let $$P_3$$ be a 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 $$(\alpha, \beta, \gamma)$$ from $$P_3$$ is 2, then which of the following relationsis (are) true?

In $$R^3$$ 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 $$P_1 : x + 2y - z + 1 = 0$$ 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 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 $$\triangle OPQ$$ is $$3 \sqrt 2$$, then which of the following is (are) the coordinates of P ?

Let y(x) be a solution of the differential equation $$(1 + e^x)y' + ye^x = 1$$. If y(0) = 2, then which of the following statements is (are) true?

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 $$y' = \frac{dy}{dx}, y'' = \frac{d^2y}{dx^2}$$), then which of the following statements is (are) true?

Let g: $$R \rightarrow R$$ be a differentiable function with g(0) = 0, g'(0) = 0 and $$g’(1) \neq 0$$. Let
$$f(x) = \begin{cases}\frac{x}{|x|}g(x) & x \neq 0\\0 & x = 0\end{cases}$$
and $$h(x) = e^{|x|}$$ for all $$x \in R$$. Let (f o h) (x) denote f(h(x)) and (h o f) (x) denote h(f(x)). Then which of the following is (are) true?

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

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