JEE Advanced 2012 · 08 Apr 2012 · Paper 1

Q1.A thin uniform rod, pivoted at $O$, is rotating in the horizontal plane with constant angular speed $\omega$, as shown in the figure. At time $t=0$, a small insect starts from $O$ and moves with constant speed $v$ with respect to the rod towards the other end. It reache …

Q2.Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures $2T$ and $3T$ respectively. The temperature of the …

Q3.Consider a thin spherical shell of radius $R$ with its centre at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field $|\vec{E}(r)|$ and the electric potential $V(r)$ with the distance $r$ from the centre, is b …

Q4.In the determination of Young's modulus $\left(Y=\frac{4ML\mathrm{g}}{\pi l{d}^2}\right)$ by using Searle's method, a wire of length $L=2\mathrm{m}$ and diameter $d=0.5\mathrm{mm}$ is used. For a load $M=2.5\mathrm{kg}$, an extension $l=0.25\mathrm{mm}$ in …

Q5.A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at $t=0$. It then execute …

Q6.Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are ${\beta }_G,{\beta }_R$ and ${\beta }_B$, respectively. Then,

Q7.A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion in the $x-y$ plane with centre at $O$ and constant angular speed $\omega$. If the angular momentum of the system, calculated …

Q8.A mixture of 2 moles of helium gas (atomic mass = 4 amu) and 1 mole of argon gas (atomic mass $=40\mathrm{amu}$ ) is kept at 300 K in a container. The ratio of the rms speeds $\left(\frac{v_{rms}(\text{ helium })}{v_{rms}(\text{ argon })}\right)$ is

Q9.Two large vertical and parallel metal plates having a separation of 1 cm are connected to a DC voltage source of potential difference $X$. A proton is released at rest midway between the two plates. It is found to move at $45^{\circ }$ to the vertical JUST after release. Then …

Q10.A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive index $n$ of the first lens is 1.5 and that of the second lens is 1.2. Both the curved surfaces are of the same radius of curvature $R=14\mathrm{cm}$. For this bi-convex lens, fo …

Q11.A cubical region of side $a$ has its centre at the origin. It encloses three fixed point charges, $-q$ at $(0,-a/4,0),+3q$ at $(0,0,0)$ and $-q$ at $(0,+a/4,0)$. Choose the correct option(s).

Q12.For the resistance network shown in the figure, choose the correct option(s).

Q13.A small block of mass of 0.1 kg lies on a fixed inclined plane $PQ$ which makes an angle $\theta$ with the horizontal. A horizontal force of 1 N acts on the block through its center of mass as shown in the figure. The block remains stationary if ( take …

Q14.Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $\vec{E}=E_0\hat{j}$ and $\vec{B}=B_0\hat{j}$. At time $t=0$, this charge has velocity $\vec{v}$ in the $x-y$ plane, making an angle …

Q15.A person blows into open-end of a long pipe. As a result, a high-pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe,

Q16.An infinitely long solid cylinder of radius $R$ has a uniform volume charge density $\rho$. It has a spherical cavity of radius $R/2$ with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point $P$, which is at a …

Q17.A cylindrical cavity of diameter $a$ exists inside a cylinder of diameter $2a$ as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density $J$ flows along the length. If the magnitude of the magnetic field at the point $P$ is g …

Q18.A lamina is made by removing a small disc of diameter $2R$ from a bigger disc of uniform mass density and radius $2R$, as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I_O$ and $I_P$, respectively. Both thes …

Q19.A circular wire loop of radius $R$ is placed in the $x-y$ plane centered at the origin $O$. A square loop of side $a(a\ll R)$ having two turns is placed with its center at $z=\sqrt{3}R$ along the axis of the circular wire loop, as shown in figure. The plane of the squ …

Q20.A proton is fired from very far away towards a nucleus with charge $Q=120e$, where $e$ is the electronic charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wavelength (in units of fm) of the proton at its start is: (take the proton mass, …

Q1.In allene (${\mathrm{C}}_3{\mathrm{H}}_4$), the type(s) of hybridisation of the carbon atoms is (are)

Q2.For one mole of a van der Waals gas when $b=0$ and $T=300\mathrm{K}$, the $PV$ vs. $1/V$ plot is shown below. The value of the van der Waals constant $a$ (atm.liter${}^2{\mathrm{mol}}^{-2}$) is

Q3.The number of optically active products obtained from the complete ozonolysis of the given compound is

Q4.A compound ${\mathrm{M}}_{\mathrm{p}}{\mathrm{X}}_{\mathrm{q}}$ has cubic close packing (ccp) arrangement of X. Its unit cell structure is shown below. The empirical formula of the compound is

Q5.The number of aldol reaction(s) that occurs in the given transformation is

Q6.The colour of light absorbed by an aqueous solution of ${\mathrm{CuSO}}_4$ is

Q7.The carboxyl functional group $(-\mathrm{COOH})$ is present in

Q8.The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [$a_0$ is Bohr radius]

Q9.Which ordering of compounds is according to the decreasing order of the oxidation state of nitrogen?

Q10.As per IUPAC nomenclature, the name of the complex $[\mathrm{Co}({\mathrm{H}}_2\mathrm{O}{)}_4({\mathrm{NH}}_3{)}_2]{\mathrm{Cl}}_3$ is

Q11.Identify the binary mixture(s) that can be separated into individual compounds, by differential extraction, as shown in the given scheme.

Q12.Choose the correct reason(s) for the stability of the lyophobic colloidal particles.

Q13.Which of the following molecules, in pure form, is (are) unstable at room temperature ?

(A) (see structure)
(B) (see structure)
(C) (see structure)
(D) (see structure)

Q14.Which of the following hydrogen halides react(s) with ${\mathrm{AgNO}}_3(\mathrm{aq})$ to give a precipitate that dissolves in ${\mathrm{Na}}_2{\mathrm{S}}_2{\mathrm{O}}_3(\mathrm{aq})$ ?

Q15.For an ideal gas, consider only $P-V$ work in going from an initial state $X$ to the final state $Z$. The final state $Z$ can be reached by either of the two paths shown in the figure. Which of the following choice(s) is (are) correct ? [take $\Delta S$ as change in ent …

Q16.The substituents ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$ for nine peptides are listed in the table given below. How many of these peptides are positively charged at $\mathrm{pH}=7.0$ ?
| Peptide | ${\mathrm{R}}_1$ | ${\mathrm{R}}_2$ || --- | --- | --- …

Q17.The periodic table consists of 18 groups. An isotope of copper, on bombardment with protons, undergoes a nuclear reaction yielding element $\mathbf{X}$ as shown below. To which group, element X belongs in the periodic table ? …

Q18.When the following aldohexose exists in its D-configuration, the total number of stereoisomers in its pyranose form is

Q19.$29.2\%(\mathrm{w}/\mathrm{w})\mathrm{HCl}$ stock solution has a density of $1.25\mathrm{g}{\mathrm{mL}}^{-1}$. The molecular weight of HCl is $36.5\mathrm{g}{\mathrm{mol}}^{-1}$. The volume (mL) of stock solution required to prepare a 200 mL solution of 0.4 M HCl i …

Q20.An organic compound undergoes first-order decomposition. The time taken for its decomposition to 1/8 and 1/10 of its initial concentration are ${\mathrm{t}}_{1/8}$ and ${\mathrm{t}}_{1/10}$ respectively. What is the value of …

Q1.The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

Q2.Let $f(x)=\{\begin{array}{rl}{x}^2\left|\cos \frac{\pi }{x}\right|,& x\ne 0\\ 0,& x=0\end{array},x\in \mathbb{R}$, then $f$ is

Q3.The function $f:[0,3]\to [1,29]$, defined by $f(x)=2x^3-15x^2+36x+1$, is

Q4.If ${\lim }_{x\to \infty }\left(\frac{x^2+x+1}{x+1}-ax-b\right)=4$, then

Q5.Let $z$ be a complex number such that the imaginary part of $z$ is nonzero and $a=z^2+z+1$ is real. Then $a$ cannot take the value

Q6.The ellipse $E_1:\frac{x^2}{9}+\frac{y^2}{4}=1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_2$ passing through the point (0, 4) circumscribes the rectangle $R$. The eccentricity of the ellipse $E_2$ is

Q7.Let $P=\left[a_{ij}\right]$ be a $3\times 3$ matrix and let $Q=\left[b_{ij}\right]$, where $b_{ij}=2^{i+j}{a}_{ij}$ for $1\le i,j\le 3$. If the determinant of $P$ is 2 , then the determinant of the matrix $Q$ is

Q8.The integral $\int \frac{{\sec }^{2}x}{(\sec x+\tan x{)}^{9/2}}dx$ equals (for some arbitrary constant $K$ )

Q9.The point $P$ is the intersection of the straight line joining the points $Q(2,3,5)$ and $R(1,-1,4)$ with the plane $5x-4y-z=1$. If $S$ is the foot of the perpendicular drawn from the point $T(2,1,4)$ to $QR$, then the length of the line segment $PS$ is

Q10.The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x-5y=20$ to the circle $x^2+y^2=9$ is

Q11.Let $\theta ,\phi \in [0,2\pi ]$ be such that
$2\cos \theta (1-\sin \phi )={\sin }^2\theta \left(\tan \frac{\theta }{2}+\cot \frac{\theta }{2}\right)\cos \phi -1$,
$\tan (2\pi -\theta )>0$ and $-1<\sin \theta <-\frac{\sqrt{3}}{2}$.
Then $\phi$ cannot satisfy

Q12.Let $S$ be the area of the region enclosed by $y=e^{-x^2},y=0,x=0$, and $x=1$. Then

Q13.A ship is fitted with three engines $E_1,E_2$ and $E_3$. The engines function independently of each other with respective probabilities $\frac{1}{2},\frac{1}{4}$ and $\frac{1}{4}$. For the ship to be operational at least two of its engines must function. Let $X$ denote th …

Q14.Tangents are drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$, parallel to the straight line $2x-y=1$. The points of contact of the tangents on the hyperbola are

Q15.If $y(x)$ satisfies the differential equation $y^{\prime }-y\tan x=2x\sec x$ and $y(0)=0$, then

Q16.Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $f(x)=|x|+\left|x^2-1\right|$. The total number of points at which $f$ attains either a local maximum or a local minimum is

Q17.The value of $6+{\log }_{\frac{3}{2}}\left(\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\dots }}}\right)$ is

Q18.Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p^{\prime }(0)$ is

Q19.If $\vec{a},\vec{b}$ and $\vec{c}$ are unit vectors satisfying $|\vec{a}-\vec{b}{|}^2+|\vec{b}-\vec{c}{|}^2+|\vec{c}-\vec{a}{|}^2=9$, then $|2\vec{a}+5\vec{b}+5\vec{c}|$ is

Q20.Let $S$ be the focus of the parabola $y^2=8x$ and let $PQ$ be the common chord of the circle $x^2+y^2-2x-4y=0$ and the given parabola. The area of the triangle $PQS$ is