JEE Advanced 2012 · 08 Apr 2012 · Paper 2

Q1.A loop carrying current $I$ lies in the $x-y$ plane as shown in the figure. The unit vector $\hat{k}$ is coming out of the plane of the paper. The magnetic moment of the current loop is

Q2.An infinitely long hollow conducting cylinder with inner radius $R/2$ and outer radius $R$ carries a uniform current density along its length. The magnitude of the magnetic field, $|\vec{B}|$ as a function of the radial distance $r$ from the axis is best represented by

Q3.A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If ${\rho }_c$ is the relative density of the material of the shell with respect to water, then the correct statement is that the sh …

Q4.In the given circuit, a charge of $+80\mu \mathrm{C}$ is given to the upper plate of the $4\mu \mathrm{F}$ capacitor. Then in the steady state, the charge on the upper plate of the $3\mu \mathrm{F}$ capacitor is

Q5.Two moles of ideal helium gas are in a rubber balloon at 30°C. The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to 35°C. The amount of heat required in raising the temperature is …

Q6.Consider a disc rotating in the horizontal plane with a constant angular speed $\omega$ about its centre $O$. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown …

Q7.A student is performing the experiment of Resonance Column. The diameter of the column tube is 4 cm . The frequency of the tuning fork is 512 Hz . The air temperature is $38^{\circ }\mathrm{C}$ in which the speed of sound is 336 m/s. The zero of the meter scale coincides with t …

Q8.Two identical discs of same radius $R$ are rotating about their axes in opposite directions with the same constant angular speed $\omega$. The discs are in the same horizontal plane. At time $t=0$, the points $P$ and $Q$ are facing each other as shown in the figure. The …

Q9.Most materials have the refractive index, $n>1$. So, when a light ray from air enters a naturally occurring material, then by Snell's law, $\frac{\sin {\theta }_1}{\sin {\theta }_2}=\frac{n_2}{n_1}$, it is understood that the refracted ray bends towards the normal. But it n …

Q10.Most materials have the refractive index, $n>1$. So, when a light ray from air enters a naturally occurring material, then by Snell's law, $\frac{\sin {\theta }_1}{\sin {\theta }_2}=\frac{n_2}{n_1}$, it is understood that the refracted ray bends towards the normal. But it n …

Q11.The $\beta$-decay process, discovered around 1900, is basically the decay of a neutron $(n)$. In the laboratory, a proton $(p)$ and an electron $\left(e^{-}\right)$are observed as the decay products of the neutron. Therefore, considering the decay of a neutron as a two-bo …

Q12.The $\beta$-decay process, discovered around 1900, is basically the decay of a neutron $(n)$. In the laboratory, a proton $(p)$ and an electron $\left(e^{-}\right)$are observed as the decay products of the neutron. Therefore, considering the decay of a neutron as a two-bo …

Q13.The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin unifo …

Q14.The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin unifo …

Q15.In the given circuit, the AC source has $\omega =100\mathrm{rad}/\mathrm{s}$. Considering the inductor and capacitor to be ideal, the correct choice(s) is(are)

Q16.A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it. The correct statement(s) is(are)

Q17.Six point charges are kept at the vertices of a regular hexagon of side $L$ and centre $O$, as shown in the figure. Given that $K=\frac{1}{4\pi {\epsilon }_0}\frac{q}{L^2}$, which of the following statement(s) is(are) correct ?

Q18.Two solid cylinders $P$ and $Q$ of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder $P$ has most of its mass concentrated near its surface, while $Q$ has most of its mass concentrated near the axis. Which …

Q19.Two spherical planets $P$ and $Q$ have the same uniform density $\rho$, masses $M_P$ and $M_Q$, and surface areas $A$ and $4A$, respectively. A spherical planet $R$ also has uniform density $\rho$ and its mass is $\left(M_P+M_Q\right)$. The escape vel …

Q20.The figure shows a system consisting of (i) a ring of outer radius $3R$ rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and (ii) an inner disc of radius $2R$ rotating anti-clockwise with angular speed $\omega /2$. The ring and dis …

Q1.${\mathrm{NiCl}}_2{\left\{\mathrm{P}{\left({\mathrm{C}}_2{\mathrm{H}}_5\right)}_2\left({\mathrm{C}}_6{\mathrm{H}}_5\right)\right\}}_2$ exhibits temperature dependent magnetic behaviour (paramagnetic / diamagnetic). The coordination geometries of ${\mathrm{Ni}}^{2+}$ in the pa …

Q2.In the cyanide extraction process of silver from argentite ore, the oxidizing and reducing agents used are

Q4.The shape of ${\mathrm{XeO}}_2{\mathrm{F}}_2$ molecule is

Q5.For a dilute solution containing 2.5 g of a non-volatile non-electrolyte solute in 100 g of water, the elevation in boiling point at 1 atm pressure is 2°C. Assuming concentration of solute is much lower than the concentration of solvent, the vapour pressure (mm of Hg) of the solu …

Q6.The compound that undergoes decarboxylation most readily under mild condition is

Q7.Using the data provided, calculate the multiple bond energy $\left(\mathrm{kJ}{\mathrm{mol}}^{-1}\right)$ of a $\mathrm{C}\equiv \mathrm{C}$ bond in ${\mathrm{C}}_2{\mathrm{H}}_2$. That energy is (take the bond energy of a C-H bond as …

Q8.The major product $\mathbf{H}$ of the given reaction sequence is

Q9.Bleaching powder and bleach solution are produced on a large scale and used in several household products. The effectiveness of bleach solution is often measured by iodometry.

Bleaching powder contains a salt of an oxoacid as one of its components. The anhydride of that oxoacid …

Q10.Bleaching powder and bleach solution are produced on a large scale and used in several household products. The effectiveness of bleach solution is often measured by iodometry.

25 mL of household bleach solution was mixed with 30 mL of 0.50 M KI and 10 mL of 4 N acetic acid. In t …

Q11.The electrochemical cell shown below is a concentration cell.
$\mathrm{M}\mid {\mathrm{M}}^{2+}$ (saturated solution of a sparingly soluble salt, ${\mathrm{MX}}_2$ ) \| ${\mathrm{M}}^{2+}\left(0.001\mathrm{mol}{\mathrm{dm}}^{-3}\right)\mid \mathrm{M}$
The emf of the cell dep …

Q12.The electrochemical cell shown below is a concentration cell.
$\mathrm{M}\mid {\mathrm{M}}^{2+}$ (saturated solution of a sparingly soluble salt, ${\mathrm{MX}}_2$ ) \| ${\mathrm{M}}^{2+}\left(0.001\mathrm{mol}{\mathrm{dm}}^{-3}\right)\mid \mathrm{M}$
The emf of the cell dep …

Q13.In the following reaction sequence, the compound J is an intermediate. …

Q14.In the following reaction sequence, the compound J is an intermediate. …

Q15.The reversible expansion of an ideal gas under adiabatic and isothermal conditions is shown in the figure. Which of the following statement(s) is (are) correct?

Q16.The given graphs / data I, II, III and IV represent general trends observed for different physisorption and chemisorption processes under mild conditions of temperature and pressure. Which of the following choice(s) about I, II, III and IV is (are) correct?

Q17.For the given aqueous reactions, which of the statement(s) is (are) true ?

Q18.With respect to graphite and diamond, which of the statement(s) given below is (are) correct ?
(A) Graphite is harder than diamond.
(B) Graphite has higher electrical conductivity than diamond.
(C) Graphite has higher thermal conductivity than diamond.
(D) Graphite has higher C-C …

Q19.With reference to the scheme given, which of the given statment(s) about T, U, V and W is (are) correct ?

Q20.Which of the given statement(s) about $\mathbf{N},\mathbf{O},\mathbf{P}$ and $\mathbf{Q}$ with respect to $\mathbf{M}$ is (are) correct?

(A) $\mathbf{M}$ and $\mathbf{N}$ are non-mirror image stereoisomers
(B) $\mathbf{M}$ and $\mathbf{O}$ are identical
(C) M and …

Q1.The equation of a plane passing through the line of intersection of the planes $x+2y+3z=2$ and $x-y+z=3$ and at a distance $\frac{2}{\sqrt{3}}$ from the point $(3,1,-1)$ is

Q2.Let $PQR$ be a triangle of area $△$ with $a=2,b=\frac{7}{2}$ and $c=\frac{5}{2}$, where $a,b$ and $c$ are the lengths of the sides of the triangle opposite to the angles at $P,Q$ and $R$ respectively. Then $\frac{2\sin P-\sin 2P}{2\sin P+\sin 2P}$ equals

Q3.If $\vec{a}$ and $\vec{b}$ are vectors such that $|\vec{a}+\vec{b}|=\sqrt{29}$ and $\vec{a}\times (2\hat{i}+3\hat{j}+4\hat{k})=(2\hat{i}+3\hat{j}+4\hat{k})\times \vec{b}$, then a possible value of $(\vec{a}+\vec{b})\cdot (-7\hat{i}+2\hat{j}+3\hat{k})$ is

Q4.If $P$ is a $3\times 3$ matrix such that $P^T=2P+I$, where $P^T$ is the transpose of $P$ and $I$ is the $3\times 3$ identity matrix, then there exists a column matrix …

Q5.Let $\alpha (a)$ and $\beta (a)$ be the roots of the equation $(\sqrt[3]{1+a}-1)x^2+(\sqrt{1+a}-1)x+(\sqrt[6]{1+a}-1)=0$ where $a>-1$.
Then ${\lim }_{a\to 0^{+}}\alpha (a)$ and ${\lim }_{a\to 0^{+}}\beta (a)$ are

Q6.Four fair dice $D_1,D_2,D_3$ and $D_4$, each having six faces numbered $1,2,3,4,5$ and 6, are rolled simultaneously. The probability that $D_4$ shows a number appearing on one of $D_1,D_2$ and $D_3$ is

Q7.The value of the integral ${\int }_{-\pi /2}^{\pi /2}\left(x^2+\ln \frac{\pi +x}{\pi -x}\right)\cos x\mathrm{d}x$ is

Q8.Let $a_1,a_2,a_3,\dots$ be in harmonic progression with $a_1=5$ and $a_{20}=25$. The least positive integer $n$ for which $a_n<0$ is

Q9.Let $a_n$ denote the number of all $n$-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0. Let $b_n=$ the number of such $n$-digit integers ending with digit 1 and $c_n=$ the number of such $n$-digit integers ending with d …

Q10.Let $a_n$ denote the number of all $n$-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0. Let $b_n=$ the number of such $n$-digit integers ending with digit 1 and $c_n=$ the number of such $n$-digit integers ending with d …

Q11.Let $f(x)=(1-x{)}^2{\sin }^{2}x+x^2$ for all $x\in \mathbb{R}$, and let $g(x)={\int }_1^x\left(\frac{2(t-1)}{t+1}-\ln t\right)f(t)dt$ for all $x\in (1,\infty )$.

Which of the following is true?

Q12.Let $f(x)=(1-x{)}^2{\sin }^{2}x+x^2$ for all $x\in \mathbb{R}$, and let $g(x)={\int }_1^x\left(\frac{2(t-1)}{t+1}-\ln t\right)f(t)dt$ for all $x\in (1,\infty )$.

Consider the statements :
$\mathbf{P}$ : There exists some $x\in \mathbb{R}$ such that …

Q13.A tangent $PT$ is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3},1)$. A straight line $L$, perpendicular to $PT$ is a tangent to the circle $(x-3{)}^2+y^2=1$.

A possible equation of $L$ is

Q14.A tangent $PT$ is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3},1)$. A straight line $L$, perpendicular to $PT$ is a tangent to the circle $(x-3{)}^2+y^2=1$.

A common tangent of the two circles is

Q15.For every integer $n$, let $a_n$ and $b_n$ be real numbers. Let function $f:\mathbb{R}\to \mathbb{R}$ be given by …

Q16.If $f(x)={\int }_0^x{e}^{t^2}(t-2)(t-3)dt$ for all $x\in (0,\infty )$, then

Q17.If the straight lines $\frac{x-1}{2}=\frac{y+1}{k}=\frac{z}{2}$ and $\frac{x+1}{5}=\frac{y+1}{2}=\frac{z}{k}$ are coplanar, then the plane(s) containing these two lines is(are)

Q18.Let $X$ and $Y$ be two events such that $P(X\mid Y)=\frac{1}{2},P(Y\mid X)=\frac{1}{3}$ and $P(X\cap Y)=\frac{1}{6}$. Which of the following is (are) correct?

Q19.If the adjoint of a $3\times 3$ matrix $P$ is $\left[\begin{array}{lll}1& 4& 4\\ 2& 1& 7\\ 1& 1& 3\end{array}\right]$, then the possible value(s) of the determinant of $P$ is (are)

Q20.Let $f:(-1,1)\to \mathbb{R}$ be such that $f(\cos 4\theta )=\frac{2}{2-{\sec }^2\theta }$ for $\theta \in \left(0,\frac{\pi }{4}\right)\cup \left(\frac{\pi }{4},\frac{\pi }{2}\right)$. Then the value(s) of $f\left(\frac{1}{3}\right)$ is (are)