JEE Advanced 2017 · 21 May 2017 · Paper 2

Q1.Consider an expanding sphere of instantaneous radius $R$ whose total mass remains constant. The expansion is such that the instantaneous density $\rho$ remains uniform throughout the volume. The rate of fractional change in density …

Q2.Consider regular polygons with number of sides $n=3,4,5\dots$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum in …

Q3.A photoelectric material having work-function ${\varphi }_0$ is illuminated with light of wavelength $\lambda \left(\lambda <\frac{hc}{{\varphi }_0}\right)$. The fastest photoelectron has a de Broglie wavelength ${\lambda }_d$. A change in wavelength of the incident light by …

Q4.A symmetric star shaped conducting wire loop is carrying a steady state current $I$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $4a$. The magnitude of the magnetic field at the center of the loop is

Q5.Three vectors $\vec{P},\vec{Q}$ and $\vec{R}$ are shown in the figure. Let $S$ be any point on the vector $\vec{R}$. The distance between the points $P$ and $S$ is $b|\vec{R}|$. The general relation among vectors $\vec{P},\vec{Q}$ and $\vec{S}$ is

Q6.A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is $3\times 10^5$ times heavier than the Earth and is at a distance $2.5\times 10^4$ times larger than the radius of the Earth. The escape v …

Q7.A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is $\delta T=0.01$ seconds and he measures the depth of the well to be $L=20$ mete …

Q8.A uniform magnetic field $B$ exists in the region between $x=0$ and $x=\frac{3R}{2}$ (region 2 in the figure) pointing normally into the plane of the paper. A particle with charge $+Q$ and momentum $p$ directed along $x$-axis enters region 2 from region 1 at point …

Q9.The instantaneous voltages at three terminals marked $X,Y$ and $Z$ are given by …

Q10.A point charge $+Q$ is placed just outside an imaginary hemispherical surface of radius $R$ as shown in the figure. Which of the following statements is/are correct?

Q11.Two coherent monochromatic point sources $S_1$ and $S_2$ of wavelength $\lambda =600\mathrm{nm}$ are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance $d=1.8\mathrm{mm}$. This arrangement produces inte …

Q12.A source of constant voltage $V$ is connected to a resistance $R$ and two ideal inductors $L_1$ and $L_2$ through a switch $S$ as shown. There is no mutual inductance between the two inductors. The switch $S$ is initially open. At $t=0$, the switch is closed and …

Q13.A rigid uniform bar AB of length $L$ is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is $\theta$. Which of the following statements about its motion is/are correct?

Q14.A wheel of radius $R$ and mass $M$ is placed at the bottom of a fixed step of height $R$ as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $\tau$ about an axis …

Q15.Consider a simple $RC$ circuit as shown in Figure 1.

Process 1: In the circuit the switch $S$ is closed at $t=0$ and the capacitor is fully charged to voltage $V_0$ (i.e., charging continues for time $T\gg RC$). In the process some dissipation $\left(E_D\right)$ …

Q16.Consider a simple $RC$ circuit as shown in Figure 1.

Process 1: In the circuit the switch $S$ is closed at $t=0$ and the capacitor is fully charged to voltage $V_0$ (i.e., charging continues for time $T\gg RC$). In the process some dissipation $\left(E_D\right)$ …

Q17.One twirls a circular ring (of mass $M$ and radius $R$) near the tip of one's finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the pa …

Q18.One twirls a circular ring (of mass $M$ and radius $R$) near the tip of one's finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the pa …

Q1.Pure water freezes at 273 K and 1 bar. The addition of 34.5 g of ethanol to 500 g of water changes the freezing point of the solution. Use the freezing point depression constant of water as $2\mathrm{K}\mathrm{kg}{\mathrm{mol}}^{-1}$. The figures shown below represent plots o …

Q2.For the following cell,

when the concentration of ${\mathrm{Zn}}^{2+}$ is 10 times the concentration of ${\mathrm{Cu}}^{2+}$, the expression for $\Delta G$ (in …

Q3.The standard state Gibbs free energies of formation of C (graphite) and C (diamond) at $\mathrm{T}=298\mathrm{K}$ are …

Q4.Which of the following combination will produce ${\mathrm{H}}_2$ gas?

Q5.The order of the oxidation state of the phosphorus atom in ${\mathrm{H}}_3{\mathrm{PO}}_2,{\mathrm{H}}_3{\mathrm{PO}}_4,{\mathrm{H}}_3{\mathrm{PO}}_3$, and ${\mathrm{H}}_4{\mathrm{P}}_2{\mathrm{O}}_6$ is

Q6.The major product of the following reaction is

(see structure)

Q7.The order of basicity among the following compounds is

(see structure) I

(see structure) II

(see structure) III

(see structure) IV

Q8.The correct statement(s) about surface properties is(are)

Q9.For a reaction taking place in a container in equilibrium with its surroundings, the effect of temperature on its equilibrium constant $K$ in terms of change in entropy is described by

Q10.In a bimolecular reaction, the steric factor P was experimentally determined to be 4.5. The correct option(s) among the following is(are)

Q11.For the following compounds, the correct statement(s) with respect to nucleophilic substitution reactions is(are)

(see structure) I

(see structure) II

(see structure) III

(see structure) IV

Q12.Among the following, the correct statement(s) is(are)

Q13.The option(s) with only amphoteric oxides is(are)

Q14.Compounds P and R upon ozonolysis produce Q and S, respectively. The molecular formula of $\mathbf{Q}$ and $\mathbf{S}$ is ${\mathrm{C}}_8{\mathrm{H}}_8\mathrm{O}$. Q undergoes Cannizzaro reaction but not haloform reaction, whereas $\mathbf{S}$ undergoes haloform reaction but n …

Q15.Upon heating ${\mathrm{KClO}}_3$ in the presence of catalytic amount of ${\mathrm{MnO}}_2$, a gas $\mathbf{W}$ is formed. Excess amount of $\mathbf{W}$ reacts with white phosphorus to give $\mathbf{X}$. The reaction of $\mathbf{X}$ with pure ${\mathrm{HNO}}_3$ gives Y and Z.

W a …

Q16.Upon heating ${\mathrm{KClO}}_3$ in the presence of catalytic amount of ${\mathrm{MnO}}_2$, a gas $\mathbf{W}$ is formed. Excess amount of $\mathbf{W}$ reacts with white phosphorus to give $\mathbf{X}$. The reaction of $\mathbf{X}$ with pure ${\mathrm{HNO}}_3$ gives Y and Z.

Y a …

Q17.The reaction of compound $\mathbf{P}$ with ${\mathrm{CH}}_3\mathrm{MgBr}$ (excess) in ${\left({\mathrm{C}}_2{\mathrm{H}}_5\right)}_2\mathrm{O}$ followed by addition of ${\mathrm{H}}_2\mathrm{O}$ gives $\mathbf{Q}$. The compound $\mathbf{Q}$ on treatment with …

Q18.The reaction of compound $\mathbf{P}$ with ${\mathrm{CH}}_3\mathrm{MgBr}$ (excess) in ${\left({\mathrm{C}}_2{\mathrm{H}}_5\right)}_2\mathrm{O}$ followed by addition of ${\mathrm{H}}_2\mathrm{O}$ gives $\mathbf{Q}$. The compound $\mathbf{Q}$ on treatment with …

Q1.The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes $2x+y-2z=5$ and $3x-6y-2z=7$, is

Q2.Let $O$ be the origin and let $PQR$ be an arbitrary triangle. The point $S$ is such that …

Q3.If $y=y(x)$ satisfies the differential equation

and $y(0)=\sqrt{7}$, then $y(256)=$

Q4.If $f:\mathbb{R}\to \mathbb{R}$ is a twice differentiable function such that $f^{\prime \prime }(x)>0$ for all $x\in \mathbb{R}$, and $f\left(\frac{1}{2}\right)=\frac{1}{2},f(1)=1$, then

Q5.How many $3\times 3$ matrices $M$ with entries from $\{0,1,2\}$ are there, for which the sum of the diagonal entries of $M^{T}M$ is 5?

Q6.Let $S=\{1,2,3,\dots ,9\}$. For $k=1,2,\dots ,5$, let $N_k$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1+N_2+N_3+N_4+N_5=$

Q7.Three randomly chosen nonnegative integers $x,y$ and $z$ are found to satisfy the equation $x+y+z=10$. Then the probability that $z$ is even, is

Q8.If $g(x)={\int }_{\sin x}^{\sin (2x)}{\sin }^{-1}(t)dt$, then

Q9.Let $\alpha$ and $\beta$ be nonzero real numbers such that $2(\cos \beta -\cos \alpha )+\cos \alpha \cos \beta =1$. Then which of the following is/are true?

Q10.If $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $f^{\prime }(x)>2f(x)$ for all $x\in \mathbb{R}$, and $f(0)=1$, then

Q11.Let $f(x)=\frac{1-x(1+|1-x|)}{|1-x|}\cos \left(\frac{1}{1-x}\right)$ for $x\ne 1$. Then

Q12.If $f(x)=\left|\begin{array}{ccc}\cos (2x)& \cos (2x)& \sin (2x)\\ -\cos x& \cos x& -\sin x\\ \sin x& \sin x& \cos x\end{array}\right|$, then

Q13.If the line $x=\alpha$ divides the area of region $R=\left\{(x,y)\in {\mathbb{R}}^2:x^3\le y\le x,0\le x\le 1\right\}$ into two equal parts, then

Q14.If $I={\sum }_{k=1}^{98}{\int }_k^{k+1}\frac{k+1}{x(x+1)}dx$, then

Q15.Let $O$ be the origin, and $\overrightarrow{OX},\overrightarrow{OY},\overrightarrow{OZ}$ be three unit vectors in the directions of the sides $\overrightarrow{QR},\overrightarrow{RP},\overrightarrow{PQ}$, respectively, of a triangle $PQR$. …

Q16.Let $O$ be the origin, and $\overrightarrow{OX},\overrightarrow{OY},\overrightarrow{OZ}$ be three unit vectors in the directions of the sides $\overrightarrow{QR},\overrightarrow{RP},\overrightarrow{PQ}$, respectively, of a triangle $PQR$.

If the triangle $PQR$ varie …

Q17.Let $p,q$ be integers and let $\alpha ,\beta$ be the roots of the equation $x^2-x-1=0$, where $\alpha \ne \beta$. For $n=0,1,2,\dots$, let $a_n=p{\alpha }^n+q{\beta }^n$.

FACT: If $a$ and $b$ are rational numbers and $a+b\sqrt{5}=0$, then $a=0=b$. …

Q18.Let $p,q$ be integers and let $\alpha ,\beta$ be the roots of the equation $x^2-x-1=0$, where $\alpha \ne \beta$. For $n=0,1,2,\dots$, let $a_n=p{\alpha }^n+q{\beta }^n$.

FACT: If $a$ and $b$ are rational numbers and $a+b\sqrt{5}=0$, then $a=0=b$.

I …