JEE Advanced 2013 · 02 Jun 2013 · Paper 2

Q1.Using the expression $2d\sin \theta =\lambda$, one calculates the values of $d$ by measuring the corresponding angles $\theta$ in the range 0 to 90°. The wavelength $\lambda$ is exactly known and the error in $\theta$ is constant for all values of $\theta$. As …

Q2.Two non-conducting spheres of radii $R_1$ and $R_2$ and carrying uniform volume charge densities $+\rho$ and $-\rho$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,

Q3.The figure below shows the variation of specific heat capacity $(C)$ of a solid as a function of temperature $(T)$. The temperature is increased continuously from 0 to 500 K at a constant rate. Ignoring any volume change, the following statement(s) is (are) correct to a reaso …

Q4.The radius of the orbit of an electron in a Hydrogen-like atom is $4.5a_0$, where $a_0$ is the Bohr radius. Its orbital angular momentum is $\frac{3h}{2\pi }$. It is given that $h$ is Planck constant and $R$ is Rydberg constant. The possible wavelength(s), when th …

Q5.Two bodies, each of mass $M$, are kept fixed with a separation $2L$. A particle of mass $m$ is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is $G$. The correct statement(s) is (are)

Q6.A particle of mass $m$ is attached to one end of a mass-less spring of force constant $k$, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time $t=0$ with an initial vel …

Q7.A steady current $I$ flows along an infinitely long hollow cylindrical conductor of radius $R$. This cylinder is placed coaxially inside an infinite solenoid of radius $2R$. The solenoid has $n$ turns per unit length and carries a steady current $I$. Consider a point …

Q8.Two vehicles, each moving with speed $u$ on the same horizontal straight road, are approaching each other. Wind blows along the road with velocity $w$. One of these vehicles blows a whistle of frequency $f_1$. An observer in the other vehicle hears the frequency of the wh …

Q9.Paragraph for Questions 9 and 10

A point charge $Q$ is moving in a circular orbit of radius $R$ in the $x-y$ plane with an angular velocity $\omega$. This can be considered as equivalent to a loop carrying a steady current $\frac{Q\omega }{2\pi }$. A uniform magnetic f …

Q10.Paragraph for Questions 9 and 10

A point charge $Q$ is moving in a circular orbit of radius $R$ in the $x-y$ plane with an angular velocity $\omega$. This can be considered as equivalent to a loop carrying a steady current $\frac{Q\omega }{2\pi }$. A uniform magnetic f …

Q11.Paragraph for Questions 11 and 12

The mass of a nucleus ${}_Z^{A}X$ is less than the sum of the masses of ( $A-Z$ ) number of neutrons and $Z$ number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of …

Q12.Paragraph for Questions 11 and 12

The mass of a nucleus ${}_Z^{A}X$ is less than the sum of the masses of ( $A-Z$ ) number of neutrons and $Z$ number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of …

Q13.Paragraph for Questions 13 and 14

A small block of mass $1kg$ is released from rest at the top of a rough track. The track is a circular arc of radius $40m$. The block slides along the track without toppling and a frictional force acts on it in the direction opposite to th …

Q14.Paragraph for Questions 13 and 14

A small block of mass $1kg$ is released from rest at the top of a rough track. The track is a circular arc of radius $40m$. The block slides along the track without toppling and a frictional force acts on it in the direction opposite to th …

Q15.Paragraph for Questions 15 and 16

A thermal power plant produces electric power of $600kW$ at $4000V$, which is to be transported to a place 20 km away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying c …

Q16.Paragraph for Questions 15 and 16

A thermal power plant produces electric power of $600kW$ at $4000V$, which is to be transported to a place 20 km away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying c …

Q17.Match List I with List II and select the correct answer using the codes given below the lists :
| 1. | …

Q18.A right angled prism of refractive index ${\mu }_1$ is placed in a rectangular block of refractive index ${\mu }_2$, which is surrounded by a medium of refractive index ${\mu }_3$, as shown in the figure. A ray of light 'e' enters the rectangular block at normal incidence. Dep …

Q19.Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists :
| R. | …

Q20.One mole of a monatomic ideal gas is taken along two cyclic processes $\mathrm{E}\to \mathrm{F}\to \mathrm{G}\to \mathrm{E}$ and $\mathrm{E}\to \mathrm{F}\to \mathrm{H}\to \mathrm{E}$ as shown in the PV diagram. The proces …

Q1.The correct statement(s) about $O_3$ is(are)

Q2.In the nuclear transmutation

(X, Y) is(are)

Q3.The carbon-based reduction method is NOT used for the extraction of

Q4.The thermal dissociation equilibrium of ${\mathrm{CaCO}}_3(s)$ is studied under different conditions.
${\mathrm{CaCO}}_3(\mathrm{s})\rightleftharpoons \mathrm{CaO}(\mathrm{s})+{\mathrm{CO}}_2(\mathrm{g})$
For this equilibrium, the correct statement(s) is(are)

Q5.The $K_{sp}$ of ${\mathrm{Ag}}_2{\mathrm{CrO}}_4$ is $1.1\times 10^{-12}$ at 298 K. The solubility (in $\mathrm{mol}/\mathrm{L}$) of ${\mathrm{Ag}}_2{\mathrm{CrO}}_4$ in a $0.1\mathrm{M}{\mathrm{AgNO}}_3$ solution is

Q6.In the following reaction, the product(s) formed is(are)

Q7.The major product(s) of the following reaction is(are)

Q8.After completion of the reactions (I and II), the organic compound(s) in the reaction mixtures is(are)

Reaction I:

Reaction II:

Q9.A fixed mass 'm' of a gas is subjected to transformation of states from K to L to M to N and back to K as shown in the figure

The succeeding operations that enable this transformation of states are

Q10.A fixed mass 'm' of a gas is subjected to transformation of states from K to L to M to N and back to K as shown in the figure

The pair of isochoric processes among the transformation of states is

Q11.The reactions of ${\mathrm{Cl}}_2$ gas with cold-dilute and hot-concentrated NaOH in water give sodium salts of two (different) oxoacids of chlorine, $\mathbf{P}$ and $\mathbf{Q}$, respectively. The ${\mathrm{Cl}}_2$ gas reacts with $S{O}_2$ gas, in presence of …

Q12.The reactions of ${\mathrm{Cl}}_2$ gas with cold-dilute and hot-concentrated NaOH in water give sodium salts of two (different) oxoacids of chlorine, $\mathbf{P}$ and $\mathbf{Q}$, respectively. The ${\mathrm{Cl}}_2$ gas reacts with $S{O}_2$ gas, in presence of …

Q13.An aqueous solution of a mixture of two inorganic salts, when treated with dilute $HCl$, gave a precipitate $(\mathbf{P})$ and a filtrate $(\mathbf{Q})$. The precipitate $\mathbf{P}$ was found to dissolve in hot water. The filtrate $(\mathbf{Q})$ remaine …

Q14.An aqueous solution of a mixture of two inorganic salts, when treated with dilute $HCl$, gave a precipitate $(\mathbf{P})$ and a filtrate $(\mathbf{Q})$. The precipitate $\mathbf{P}$ was found to dissolve in hot water. The filtrate $(\mathbf{Q})$ remaine …

Q15.$P$ and $Q$ are isomers of dicarboxylic acid ${\mathrm{C}}_4{\mathrm{H}}_4{\mathrm{O}}_4$. Both decolorize ${\mathrm{Br}}_2/{\mathrm{H}}_2\mathrm{O}$. On heating, $\mathbf{P}$ forms the cyclic anhydride. Upon treatment with dilute alkaline ${\mathrm{KMnO}}_4$, …

Q16.$P$ and $Q$ are isomers of dicarboxylic acid ${\mathrm{C}}_4{\mathrm{H}}_4{\mathrm{O}}_4$. Both decolorize ${\mathrm{Br}}_2/{\mathrm{H}}_2\mathrm{O}$. On heating, $\mathbf{P}$ forms the cyclic anhydride. Upon treatment with dilute alkaline ${\mathrm{KMnO}}_4$, …

Q17.Match the chemical conversions in List I with the appropriate reagents in List II and select the correct answer using the code given below the lists:
| 3. | Et-B …

Q18.The unbalanced chemical reactions given in List I show missing reagent or condition (?) which are provided in List II. Match List I with List II and select the correct answer using the code given below the lists:
| P. | …

Q19.The standard reduction potential data at 25 °C is given below. …

Q20.An aqueous solution of $\mathbf{X}$ is added slowly to an aqueous solution of $\mathbf{Y}$ as shown in List I. The variation in conductivity of these reactions is given in List II. Match List I with List II and select the correct answer using the code given below the lists: …

Q1.Let $w=\frac{\sqrt{3}+\mathrm{i}}{2}$ and $P=\left\{w^n:n=1,2,3,\dots \right\}$. Further $H_1=\left\{\mathrm{z}\in \mathbb{C}:\mathrm{Re}z>\frac{1}{2}\right\}$ and $H_2=\left\{\mathrm{z}\in \mathbb{C}:\mathrm{Re}z<\frac{-1}{2}\right\}$, where …

Q2.If $3^x=4^{x-1}$, then $x=$

Q3.Let $\omega$ be a complex cube root of unity with $\omega \ne 1$ and $P=\left[p_{ij}\right]$ be a $n\times n$ matrix with $p_{ij}={\omega }^{i+j}$. Then $P^2\ne 0$, when $n=$

Q4.The function $f(x)=2|x|+|x+2|-||x+2|-2|x||$ has a local minimum or a local maximum at $x=$

Q5.For $a\in \mathbb{R}$ (the set of all real numbers), $a\ne -1$,
${\lim }_{\mathrm{n}\to \infty }\frac{\left(1^a+2^a+\dots +{\mathrm{n}}^a\right)}{(n+1{)}^{a-1}[(na+1)+(na+2)+\dots +(na+n)]}=\frac{1}{60}$
Then $a=$

Q6.Circle(s) touching $x$-axis at a distance 3 from the origin and having an intercept of length $2\sqrt{7}$ on $y$-axis is (are)

Q7.Two lines $L_1:x=5,\frac{y}{3-\alpha }=\frac{z}{-2}$ and $L_2:x=\alpha ,\frac{y}{-1}=\frac{z}{2-\alpha }$ are coplanar. Then $\alpha$ can take value(s)

Q8.In a triangle $PQR,P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the incircle of the triangle touches the sides $PQ,QR$ and $RP$ at $N,L$ and $M$ respectively, such that the lengths of $PN,QL$ and $RM$ are consecutive even integers. Then possible length(s) …

Q9.Let $S=S_1\cap S_2\cap S_3$, where
$S_1=\{z\in \mathbb{C}:|z|<4\}$, $S_2=\left\{z\in \mathbb{C}:\mathrm{Im}\left[\frac{z-1+\sqrt{3}i}{1-\sqrt{3}i}\right]>0\right\}$ and
$S_3=\{z\in \mathbb{C}:\mathrm{Re}z>0\}$.

Area of $S=$

Q10.Let $S=S_1\cap S_2\cap S_3$, where
$S_1=\{z\in \mathbb{C}:|z|<4\}$, $S_2=\left\{z\in \mathbb{C}:\mathrm{Im}\left[\frac{z-1+\sqrt{3}i}{1-\sqrt{3}i}\right]>0\right\}$ and
$S_3=\{z\in \mathbb{C}:\mathrm{Re}z>0\}$.

${\min }_{z\in S}|1-3i-z|=$

Q11.A box $B_1$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B_2$ contains 2 white balls, 3 red balls and 4 black balls. A third box $B_3$ contains 3 white balls, 4 red balls and 5 black balls.

If 1 ball is drawn from each of the boxes $B_1,B_2$ and …

Q12.A box $B_1$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B_2$ contains 2 white balls, 3 red balls and 4 black balls. A third box $B_3$ contains 3 white balls, 4 red balls and 5 black balls.

If 2 balls are drawn (without replacement) from a randomly se …

Q13.Let $f:[0,1]\to \mathbb{R}$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f(0)=f(1)=0$ and satisfies $f^{\prime \prime }(x)-2f^{\prime }(x)+f(x)\ge e^x,x\in [0,1]$.

Which of the following is true for $0<x<1$ ?

Q14.Let $f:[0,1]\to \mathbb{R}$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f(0)=f(1)=0$ and satisfies $f^{\prime \prime }(x)-2f^{\prime }(x)+f(x)\ge e^x,x\in [0,1]$.

If the function ${\mathrm{e}}^{-x}f(x)$ assumes its …

Q15.Let $PQ$ be a focal chord of the parabola $y^2=4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a,a>0$.

Length of chord $PQ$ is

Q16.Let $PQ$ be a focal chord of the parabola $y^2=4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a,a>0$.

If chord $PQ$ subtends an angle $\theta$ at the vertex of $y^2=4ax$, then $\tan \theta =$

Q17.A line $L:y=mx+3$ meets $y$-axis at $E(0,3)$ and the arc of the parabola $y^2=16x$, $0\le y\le 6$ at the point $F\left(x_0,y_0\right)$. The tangent to the parabola at $F\left(x_0,y_0\right)$ intersects the $y$-axis at $G\left(0,y_1\right)$. The slope $m$ of …

Q18.Match List-I with List-II and select the correct answer using the code given below the lists :
| P. | …

Q19.Consider the lines $L_1:\frac{x-1}{2}=\frac{y}{-1}=\frac{z+3}{1},L_2:\frac{x-4}{1}=\frac{y+3}{1}=\frac{z+3}{2}$ and the planes $P_1:7x+y+2z=3,P_2:3x+5y-6z=4$. Let $ax+by+cz=d$ be the equation of the plane passing through the point of intersection of lines …

Q20.Match List-I with List-II and select the correct answer using the code given below the lists :
| P. | Volume of parallelepiped determined by vectors $\vec{a},\vec{b}$ and $\vec{c}$ is 2. Then the volume of the parallelepiped de …