JEE Advanced 2016 · 22 May 2016 · Paper 1

Q1.In a historical experiment to determine Planck's constant, a metal surface was irradiated with light of different wavelengths. The emitted photoelectron energies were measured by applying a stopping potential. The relevant data for the wavelength $(\lambda )$ of incident light a …

Q2.A uniform wooden stick of mass 1.6 kg and length $l$ rests in an inclined manner on a smooth, vertical wall of height $h(<l)$ such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick m …

Q3.A water cooler of storage capacity 120 litres can cool water at a constant rate of $P$ watts. In a closed circulation system (as shown schematically in the figure), the water from the cooler is used to cool an external device that generates constantly 3 kW of heat (thermal load …

Q4.A parallel beam of light is incident from air at an angle $\alpha$ on the side PQ of a right angled triangular prism of refractive index $n=\sqrt{2}$. Light undergoes total internal reflection in the prism at the face PR when $\alpha$ has a minimum value of 45°. The angle …

Q5.An infinite line charge of uniform electric charge density $\lambda$ lies along the axis of an electrically conducting infinite cylindrical shell of radius $R$. At time $t=0$, the space inside the cylinder is filled with a material of permittivity $\epsilon$ and electr …

Q6.Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge $Ze$ are defined by their principal quantum number $n$, where $n\gg 1$. Which of the following statement(s) is(are) true?
(A) Relative change in the radii of two consecutive orbi …

Q7.Two loudspeakers $M$ and $N$ are located 20 m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A car is initially at a point $P$, 1800 m away from the midpoint $Q$ of the line $MN$ and moves towards $Q$ constantly at 60 km/hr along the perpendicula …

Q8.An incandescent bulb has a thin filament of tungsten that is heated to high temperature by passing an electric current. The hot filament emits black-body radiation. The filament is observed to break up at random locations after a sufficiently long time of operation due to non-uni …

Q9.A plano-convex lens is made of a material of refractive index $n$. When a small object is placed 30 cm away in front of the curved surface of the lens, an image of double the size of the object is produced. Due to reflection from the convex surface of the lens, another faint im …

Q10.A length-scale $(l)$ depends on the permittivity $(\epsilon )$ of a dielectric material, Boltzmann constant $(k_B)$, the absolute temperature $(T)$, the number per unit volume $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particle …

Q11.A conducting loop in the shape of a right angled isosceles triangle of height 10 cm is kept such that the 90° vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel …

Q12.The position vector $\vec{r}$ of a particle of mass $m$ is given by the following equation

where $\alpha =10/3\mathrm{m}{\mathrm{s}}^{-3}$, $\beta =5\mathrm{m}{\mathrm{s}}^{-2}$ and $m=0.1\mathrm{kg}$. At …

Q13.A transparent slab of thickness $d$ has a refractive index $n(z)$ that increases with $z$. Here $z$ is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices $n_1$ and $n_2(>n_1)$, as sh …

Q14.A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated ($P$) by the metal. The sensor has a scale that displays ${\log }_2(P/P_0)$, where $P_0$ is a constant. When the metal surface is at a temperature of 487°C, the senso …

Q15.The isotope ${}_5^{12}\mathrm{B}$ having a mass 12.014 u undergoes $\beta$-decay to ${}_6^{12}\mathrm{C}$. ${}_6^{12}\mathrm{C}$ has an excited state of the nucleus $({}_6^{12}{\mathrm{C}}^{\ast })$ at 4.041 MeV above its ground state. If ${}_5^{12}\mathrm{B}$ dec …

Q16.A hydrogen atom in its ground state is irradiated by light of wavelength 970 Å. Taking $hc/e=1.237\times 10^{-6}$ eV m and the ground state energy of hydrogen atom as -13.6 eV, the number of lines present in the emission spectrum is

Q17.Consider two solid spheres P and Q each of density 8 gm cm${}^{-3}$ and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm cm${}^{-3}$ and viscosity $\eta =3$ poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm cm${}^{-3}$ an …

Q18.Two inductors $L_1$ (inductance 1 mH, internal resistance 3 $\Omega$) and $L_2$ (inductance 2 mH, internal resistance 4 $\Omega$), and a resistor $R$ (resistance 12 $\Omega$) are all connected in parallel across a 5 V battery. The circuit is switched on at time …

Q1.$P$ is the probability of finding the $1s$ electron of hydrogen atom in a spherical shell of infinitesimal thickness, $dr$, at a distance $r$ from the nucleus. The volume of this shell is $4\pi r^{2}dr$. The qualitative sketch of the dependence of $P$ on $r$ is

Q2.One mole of an ideal gas at 300 K in thermal contact with surroundings expands isothermally from 1.0 L to 2.0 L against a constant pressure of 3.0 atm. In this process, the change in entropy of surroundings ($\Delta S_{\text{surr}}$) in ${\mathrm{JK}}^{-1}$ is
(1 L atm = 101.3 J)

Q3.The increasing order of atomic radii of the following Group 13 elements is

Q4.Among $[\mathrm{Ni}(\mathrm{CO}{)}_4]$, $[{\mathrm{NiCl}}_4{]}^{2-}$, $[\mathrm{Co}({\mathrm{NH}}_3{)}_4{\mathrm{Cl}}_2]\mathrm{Cl}$, ${\mathrm{Na}}_3[{\mathrm{CoF}}_6]$, ${\mathrm{Na}}_2{\mathrm{O}}_2$ and ${\mathrm{CsO}}_2$, the total number of paramagnetic compounds is

Q5.On complete hydrogenation, natural rubber produces

Q6.According to the Arrhenius equation,

Q7.A plot of the number of neutrons $(N)$ against the number of protons $(P)$ of stable nuclei exhibits upward deviation from linearity for atomic number, $Z>20$. For an unstable nucleus having $N/P$ ratio less than 1, the possible mode(s) of decay is(are)

Q8.The crystalline form of borax has

Q9.The compound(s) with TWO lone pairs of electrons on the central atom is(are)

Q10.The reagent(s) that can selectively precipitate ${\mathrm{S}}^{2-}$ from a mixture of ${\mathrm{S}}^{2-}$ and ${\mathrm{SO}}_4^{2-}$ in aqueous solution is(are)

Q11.Positive Tollen's test is observed for

Q12.The product(s) of the following reaction sequence is(are)

(see structure)

i) Acetic anhydride/pyridine
ii) ${\mathrm{KBrO}}_3/\mathrm{HBr}$
iii) ${\mathrm{H}}_3{\mathrm{O}}^{+}$, heat
iv) ${\mathrm{NaNO}}_2/\mathrm{HCl}$, 273-278 K
v) $\mathrm{Cu}/\mathrm{HBr}$

Q13.The correct statement(s) about the following reaction sequence is(are) …

Q14.The mole fraction of a solute in a solution is 0.1. At 298 K, molarity of this solution is the same as its molality. Density of this solution at 298 K is $2.0{\mathrm{g}\mathrm{cm}}^{-3}$. The ratio of the molecular weights of the solute and solvent, …

Q15.The diffusion coefficient of an ideal gas is proportional to its mean free path and mean speed. The absolute temperature of an ideal gas is increased 4 times and its pressure is increased 2 times. As a result, the diffusion coefficient of this gas increases $x$ times. The value o …

Q16.In neutral or faintly alkaline solution, 8 moles of permanganate anion quantitatively oxidize thiosulphate anions to produce $\mathbf{X}$ moles of a sulphur containing product. The magnitude of $\mathbf{X}$ is

Q17.The number of geometric isomers possible for the complex $[{\mathrm{CoL}}_2{\mathrm{Cl}}_2{]}^{-}$ ($\mathrm{L}={\mathrm{H}}_2{\mathrm{NCH}}_2{\mathrm{CH}}_2{\mathrm{O}}^{-}$) is

Q18.In the following monobromination reaction, the number of possible chiral products is

(see structure)

(1.0 mole)
(enantiomerically pure)

Q1.Let $-\frac{\pi }{6}<\theta <-\frac{\pi }{12}$. Suppose ${\alpha }_1$ and ${\beta }_1$ are the roots of the equation $x^2-2x\sec \theta +1=0$ and ${\alpha }_2$ and ${\beta }_2$ are the roots of the equation $x^2+2x\tan \theta -1=0$. If ${\alpha }_1>{\beta }_1$ and …

Q2.A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

Q3.Let $S=\left\{x\in (-\pi ,\pi ):x\ne 0,\pm \frac{\pi }{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3}\sec x+\mathrm{cosec}x+2(\tan x-\cot x)=0$ in the set $S$ is equal to

Q4.A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces 20% and plant $T_2$ produces 80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that
$P$ (computer turns out to be defect …

Q5.The least value of $\alpha \in \mathbb{R}$ for which $4\alpha x^2+\frac{1}{x}\ge 1$, for all $x>0$, is

Q6.Consider a pyramid $OPQRS$ located in the first octant $(x\ge 0,y\ge 0,z\ge 0)$ with $O$ as origin, and $OP$ and $OR$ along the $x$-axis and the $y$-axis, respectively. The base $OPQR$ of the pyramid is a square with $OP=3$. The point $S$ is directly above the mid-point …

Q7.Let $f:(0,\infty )\to \mathbb{R}$ be a differentiable function such that $f^{\prime }(x)=2-\frac{f(x)}{x}$ for all $x\in (0,\infty )$ and $f(1)\ne 1$. Then

Q8.Let $P=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \alpha \\ 3& -5& 0\end{array}\right]$, where $\alpha \in \mathbb{R}$. Suppose $Q=\left[q_{ij}\right]$ is a matrix such that $PQ=kI$, where $k\in \mathbb{R},k\ne 0$ and $I$ is the identity matrix of order 3. If …

Q9.In a triangle $XYZ$, let $x,y,z$ be the lengths of sides opposite to the angles $X,Y,Z$, respectively, and $2s=x+y+z$. If $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$ and area of incircle of the triangle $XYZ$ is $\frac{8\pi }{3}$, then

Q10.A solution curve of the differential equation $\left(x^2+xy+4x+2y+4\right)\frac{dy}{dx}-y^2=0,x>0$, passes through the point (1, 3). Then the solution curve

Q11.Let $f:\mathbb{R}\to \mathbb{R},g:\mathbb{R}\to \mathbb{R}$ and $h:\mathbb{R}\to \mathbb{R}$ be differentiable functions such that $f(x)=x^3+3x+2$, $g(f(x))=x$ and $h(g(g(x)))=x$ for all $x\in \mathbb{R}$. Then

Q12.The circle $C_1:x^2+y^2=3$, with centre at $O$, intersects the parabola $x^2=2y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_1$ at $P$ touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$, respectively. Suppose $C_2$ and …

Q13.Let $RS$ be the diameter of the circle $x^2+y^2=1$, where $S$ is the point $(1,0)$. Let $P$ be a variable point (other than $R$ and $S$) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$. The normal to the circle at $P$ intersects a line drawn thro …

Q14.The total number of distinct $x\in \mathbb{R}$ for which $\left|\begin{array}{ccc}x& x^2& 1+x^3\\ 2x& 4x^2& 1+8x^3\\ 3x& 9x^2& 1+27x^3\end{array}\right|=10$ is

Q15.Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x{)}^2+(1+x{)}^3+\cdots +(1+x{)}^{49}+(1+mx{)}^{50}$ is $(3n+1{)}^{51}{C}_3$ for some positive integer $n$. Then the value of $n$ is

Q16.The total number of distinct $x\in [0,1]$ for which ${\int }_0^x\frac{t^2}{1+t^4}dt=2x-1$ is

Q17.Let $\alpha ,\beta \in \mathbb{R}$ be such that ${\lim }_{x\to 0}\frac{x^2\sin (\beta x)}{\alpha x-\sin x}=1$. Then $6(\alpha +\beta )$ equals

Q18.Let $z=\frac{-1+\sqrt{3}i}{2}$, where $i=\sqrt{-1}$, and $r,s\in \{1,2,3\}$. Let $P=\left[\begin{array}{cc}(-z{)}^r& z^{2s}\\ z^{2s}& z^r\end{array}\right]$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r,s)$ for which $P^2=-I$ is