JEE Advanced 2015 Question Paper with Solutions

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:

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)

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 large spherical mass M is fixed at one position and two identical masses m are kept on a line passing through the centre of M (see figure). The point masses are connected by a rigid massless rod of length l and this assembly is free to move along the line connecting them. All three masses interact only through their mutual gravitational interaction. When the point mass nearer to M is at a distance r = 3l from M the tension in the rod is zero for m = k ( M 288 ) . The value of k is

In plotting stress versus strain curves for two materials P and Q, a student by mistake puts strain on the y-axis and stress on the x-axis as shown in the figure. Then, the correct statements is/are

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

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 ___________.

Among the triatomic molecules/ions, BeCl 2 , $N_3^-$ , N 2 O, $NO_2^+$ , 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, I = 53 and Xe = 54]

Fe 3+ is reduced to Fe 2+ by using

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 H 2 (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

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

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 $P$ 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 $P$ ?

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha {x^2} - x + \alpha = 0$ has two distinct real roots ${x_1}$ and ${x_2}$ satisfying the inequality $\left| {{x_1} - {x_2}} \right| < 1.$ Which of the following intervals is (are) $a$ subset(s) os $S$ ?

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 ${1 \over \lambda }$ .

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)

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 ${7 \over 6}$ , the magnification becomes M 2 . The magnitude $\left| {{{{M_2}} \over {{M_1}}}} \right|$ is

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

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

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?

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 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?

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?

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

The major product of the following reaction is

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

The correct statements about Cr 2+ and Mn 3+ is(are) (Atomic numbers of Cr = 24 and Mn = 25)

The number of resonance structure for N is _________.

For the octahedral complexes of Fe 3+ 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]