JEE Advanced 2015 · 24 May 2015 · Paper 1

Q1.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 th …

Q2.Consider a hydrogen atom with its electron in the $n^{\text{th}}$ 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 ( $hc=1242\mathrm{eV}\mathrm{nm}$ )

Q3.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 $1/4^{\text{th}}$ of its value at the surface of the planet. If the escape velocity from the planet i …

Q4.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 same linear spee …

Q5.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 …

Q6.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 a …

Q7.A Young's double slit interference arrangement with slits $S_1$ and $S_2$ 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 $x^2=p^2{m}^2{\lambda }^2-d^2$, where $\lambda$ is the waveleng …

Q8.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 combi …

Q9.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 …

Q10.Planck's constant h , speed of light c and gravitational constant G are used to from a unit of length L and a unit of mass M . Then the correct options(s) is (are)

Q11.Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies ${\omega }_1$ and ${\omega }_2$ and have total energies ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$, respectively. The variations of their momenta p with positions x are shown in …

Q12.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 $\frac{\mathrm{M}}{8}$ at rest at O . These masses can move radially outwards along two massless rods fixed on the …

Q13.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 c …

Q14.Two idenctical glass rods ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ (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 …

Q15.A conductor (shown in the figure) carrying contant current I is kept in the x-y plane in a uniform magnetic field $\vec{B}$. IF F is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is (are):

Q16.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)

Q17.In an aluminum (AI) 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 AI and Fe are $2.7\times 10^{-8}\Omega \mathrm{m}$ and $1.0\times 10^{-7}\Omega \mathrm{m}$, respectively. The elec …

Q18.For photo-electric effect with incident photon wavelength $\lambda$, the stopping potential is ${\mathrm{V}}_0$. Identify the correct variation(s) of ${\mathrm{V}}_0$ with $\lambda$ and $1/\lambda$.

Q19.Match the nuclear processes given in Column I with the appropriate option(s) in Column II.
| (B) | Fission in a nuclear reactor | …

Q20.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 ${\mathrm{U}}_0$ are constants). Match the potential energies in column I …

Q1.The total number of stereoisomers that can exist for M is

Q2.The number of resonance structures for N is

Q3.The total number of lone pairs of electrons in ${\mathrm{N}}_2{\mathrm{O}}_3$ is

Q4.For the octahedral complexes of ${\mathrm{Fe}}^{3+}$ in ${\mathrm{SCN}}^{-}$(thiocyanato-S) and in ${\mathrm{CN}}^{-}$ligand environments, the difference between the spin-only magnetic moments in Bohr magnetons (when approximated to the nearest integer) is
[Atomic number of …

Q5.Among the triatomic molecules/ions, ${\mathrm{BeCl}}_2,{\mathrm{N}}_3^{-},{\mathrm{N}}_2\mathrm{O},{\mathrm{NO}}_2^{+},{\mathrm{O}}_3,{\mathrm{SCl}}_2,{\mathrm{ICl}}_2^{-},{\mathrm{I}}_3^{-}$and ${\mathrm{XeF}}_2$, the total number of linear molecule(s)/ion(s) where t …

Q6.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 ${\mathrm{H}}^{-}$is

Q7.All the energy released from the reaction $\mathbf{X}\to \mathbf{Y},{\Delta }_r{G}^0=-193\mathrm{kJ}{\mathrm{mol}}^{-1}$ is used for oxidizing ${\mathbf{M}}^{+}$as ${\mathbf{M}}^{+}\to {\mathbf{M}}^{3+}+2{\mathrm{e}}^{-},E^0=-0.25V$.
Under standard conditi …

Q8.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 °C, the number of chloride(s) in the coordination sphere of the complex is
$[{\mathrm{K}}_{\mathrm{f}}$ of water …

Q9.Compound(s) that on hydrogenation produce(s) optically inactive compound(s) is(are)

Q11.In the following reaction, the major product is

Q12.The structure of D-(+)-glucose is
[C@H](O)[C@H](O)[C@H](O)CO</smiles>](cdn-74c395fe671eeb76258f9253)
The structure of L-(-)-glucose is

Q13.The major product of the reaction is

Q14.The correct statement(s) about ${\mathrm{Cr}}^{2+}$ and ${\mathrm{Mn}}^{3+}$ is(are)
[Atomic numbers of $\mathrm{Cr}=24$ and $\mathrm{Mn}=25$]

Q15.Copper is purified by electrolytic refining of blister copper. The correct statement(s) about this process is (are)

Q16.${\mathrm{Fe}}^{3+}$ is reduced to ${\mathrm{Fe}}^{2+}$ by using

Q17.The %yield of ammonia as a function of time in the reaction
${\mathrm{N}}_2(g)+3{\mathrm{H}}_2(g)\rightleftharpoons 2{\mathrm{NH}}_3(g),\Delta \mathrm{H}<0$
at $\left(P,T_1\right)$ is given below.
If this reaction is conducted at $\left(P,T_2\right)$, with …

Q18.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

Q19.Match the anionic species given in Column I that are present in the ore(s) given in Column II.
| | (T) Argentite …

Q20.Match the thermodynamic processes given under Column I with the expressions given under Column II.
| (B) | Expansion of 1 mol of an ideal gas into a vacuum und …

Q1.The number of distinct solutions of the equation

in the interval $[0,2\pi ]$ is

Q2.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

Q3.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

Q4.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 consecutive …

Q5.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

Q6.Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)=\{\begin{array}{ll}[x],& x\le 2\\ 0,& x>2\end{array}$,
where $[x]$ is the greatest integer less than or equal to $x$. If $I={\int }_{-1}^2\frac{xf\left(x^2\right)}{2+f(x+1)}dx$, then …

Q7.A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $V{\mathrm{mm}}^3$, 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 …

Q8.Let $F(x)={\int }_x^{x^2+\frac{\pi }{6}}2{\cos }^{2}tdt$ for all $x\in \mathbb{R}$ and $f:\left[0,\frac{1}{2}\right]\to [0,\infty )$ be a continuous function. For $a\in \left[0,\frac{1}{2}\right]$, if $F^{\prime }(a)+2$ is the area of the region bounded by …

Q9.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?

Q10.Which of the following values of $\alpha$ satisfy the equation …

Q11.In ${\mathbb{R}}^3$, 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 …

Q12.In ${\mathbb{R}}^3$, 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 point …

Q13.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 $△OPQ$ is $3\sqrt{2}$, then which of the following is …

Q14.Let $y(x)$ be a solution of the differential equation $\left(1+e^x\right)y^{\prime }+y{e}^x=1$. If $y(0)=2$, then which of the following statements is (are) true?

Q15.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 $P{y}^{\prime \prime }+Q{y}^{\prime }+1=0$, where $P,Q$ are functions of $x,y$ and $y^{\prime }$ (here …

Q16.Let $g:\mathbb{R}\to \mathbb{R}$ be a differentiable function with $g(0)=0,g^{\prime }(0)=0$ and $g^{\prime }(1)\ne 0$. Let $f(x)=\{\begin{array}{cc}\frac{x}{|x|}g(x),& x\ne 0\\ 0,& x=0\end{array}$
and $h(x)=e^{|x|}$ for all $x\in \mathbb{R}$. Let …

Q17.Let $f(x)=\sin \left(\frac{\pi }{6}\sin \left(\frac{\pi }{2}\sin x\right)\right)$ for all $x\in \mathbb{R}$ and $g(x)=\frac{\pi }{2}\sin x$ for all $x\in \mathbb{R}$. Let $(f\circ g)(x)$ denote $f(g(x))$ and $(g\circ f)(x)$ denote $g(f(x))$. Then which of the following is (ar …

Q18.Let $△PQR$ be a triangle. Let $\vec{a}=\overrightarrow{QR},\vec{b}=\overrightarrow{RP}$ and $\vec{c}=\overrightarrow{PQ}$. If $|\vec{a}|=12,|\vec{b}|=4\sqrt{3}$ and $\vec{b}\cdot \vec{c}=24$, then which of the following is (are) true?

Q19.Match the entries in Column I with the entries in Column II.

Column I

(A) In ${\mathbb{R}}^2$, if the magnitude of the projection vector of the vector $\alpha \hat{i}+\beta \hat{j}$ on $\sqrt{3}\hat{i}+\hat{j}$ is $\sqrt{3}$ and if $\alpha =2+\sqrt{3}\beta$, then possible valu …

Q20.Match the entries in Column I with the entries in Column II.

Column I

(A) In a triangle $△XYZ$, let $a,b$ and $c$ be the lengths of the sides opposite to the angles $X,Y$ and $Z$, respectively. If $2\left(a^2-b^2\right)=c^2$ and …