JEE Advanced 2012 Paper-1

Athin uniform rod, pivoted at O,is rotating in the horizontal plane with constant angular speed $$\omega$$, as shownin the figure. At time t = 0, a small insect starts from O and moves with constant speedv with respect to the rod towardsthe other end. It reaches the end of the rod at t = T and stops. The angular speed of the system remains $$\omega$$ throughout. The magnitude of the torque $$(\mid \overrightarrow{\tau} \mid)$$ on the system about O, as a function of time is best represented by which plot ?

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. Thefirst and
third plates are maintained at temperatures 27 and 37 respectively. The temperature of the middle (i.e. second) plate under steady state condition is

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 $$\mid \overrightarrow{E}(r) \mid$$ and the electric potential V(r) with the distance r from the centre, is best represented by which graph?

In the determination of Young’s modulus $$\left(Y = \frac{4MLg}{\pi l d^2}\right)$$ by using Searle’s method,a wire of length L=2 mand diameter d=0.5 mmis used. For a load M= 2.5 kg, an extension l = 0.25 mm in the length of the wire is observed. Quantities d and/ are measured using a screw gauge and a micrometer, respectively. They have the same pitch of 0.5 mm. The number ofdivisions on their circular scale is 100. The contributions to the maximum probable error of the Y measurement

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 thefigure)is fixed. The system lies on a horizontal frictionless surface. The blockis stretched by 0.2 m and released from rest at t= 0. It then executes simple harmonic motion with angular frequency $$\omega = \frac{\pi}{3}$$ rad/s. Simultaneously at t = 0, a small pebble is projected with speed v from point P at an angle of $$45^\circ$$ as shownin thefigure. Point P is at a horizontal distance of 10 m from O. If the pebble hits the block at + = 1s, the value of v is (take g = 10 $$m/s^2$$)

Young's double slit experimentis carried out by using green, red and bluelight, one color at a time. The fringe widths recorded are $$\beta_G, \beta_R, \beta_B$$, respectively. Then,

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 about O and P are denoted by $$\overrightarrow{L_O}$$ and $$\overrightarrow{L_P}$$ respectively, then

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

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 protonis released at rest midway between the two plates. It is found to move at $$45^\circ$$ to the vertical JUSTafter release. Then X is nearly

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 cm. For this bi-convex lens, for an object distance of 40 cm, the image distance will be

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

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

A small block of mass of 0.1 kg lies on 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 g = 10 $$m/s^2$$)

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $$\overrightarrow{E} = E_0 \hat{j}$$ and $$\overrightarrow{B} = B_0 \hat{j}$$. At time t = 0, this charge has velocity $$\overrightarrow{v}$$ in the x-y plane, making an angle $$\theta$$ with the x-axis. Which of the following option(s) is (are) correct for time t > 0?

A person blowsinto open-end of a long pipe. As a result, a high-pressure pulseof air travels down the pipe. Whenthis pulse reachesthe other endof the pipe,

An infinitely long solid cylinder of radius R has a uniform volume charge density p. 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 distance 2R from the axis of the cylinder, is given by the expression $$\frac{23 \rho R}{16k ε_0}$$. The value of k is

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/ flows along the length. If the magnitude of the magnetic field at the point P is given by $$\frac{N}{12}\mu_0 aJ$$, then the value of N is

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 is $$I_O$$ and $$I_P$$, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $$\frac{I_P}{I_O}$$ to the nearest integer is

A circular wire loop of radius R is placed in the x-y plane centeredat the origin O. A square loop of side a (a << R) having twoturnsis placed with its center at $$z = \sqrt{3}R$$ along the axis of the circular wire loop, as shownin figure. The plane of the square loop makes an angle of $$45^\circ$$ with respectto the z-axis. If the mutual inductance between the loops is given by $$\frac{\mu_0 a^2}{2^{\frac{p}{2}}R}$$, then the value of p is

A proton is fired from very far away towards a nucleus with charge Q = 120 e, 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 protonatits start is: (take the proton mass, $$m_p = \left(\frac{5}{3}\right) \times 10^{-27}$$ kg; $$h/e = 4.2 \times 10^{-15}$$ J.s/C; $$\frac{1}{4 \pi ε_0} = 9 \times 10^9$$ m/f; 1 fm = $$10^{-15}$$ m)

Inallene $$(C_3H_4)$$, the type(s) of hybridisation of the carbon atomsis (are)

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

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

A compound $$M_pX_q$$ has cubic close packing (ccp) arrangement of X. Its unit cell structure is shown below. The empirical formula of the compound is

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

The colour oflight absorbed by an aqueous solution of $$CuSO_4$$ is

The carboxyl functional group (-COOH) is present in

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

Which ordering of compoundsis according to the decreasing order of the oxidation state of nitrogen?

As per IUPAC nomenclature, the nameof the complex $$[CO(H_2O)_4(NH_3)_2]Cl_3$$ is

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

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

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

Which of the following hydrogen halides react(s) with $$AgNO_3(aq)$$ to give a precipitate that dissolves in $$Na_2S_2O_3(aq)$$?

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 $$\triangle S$$ as change in entropy and w as work done]

The substituents $$R_1$$ and $$R_2$$ for nine peptides are listed in the table given below. How many of these peptides are positively charged at pH = 7.0 ?

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

$$_{29}^{63}Cu + _{1}^{1}H \rightarrow 6_{0}^{1}n + \alpha + 2 _{1}^{1}H + X$$

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

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

An organic compound undergoes first-order decomposition. The time taken for its decomposition to $$\frac{1}{8}$$ and $$\frac{1}{10}$$ of its initial concentration are $$t_{\frac{1}{8}}$$ and $$t_{\frac{1}{10}}$$ respectively. What is the value of $$\frac{[t_{\frac{1}{8}}]}{[t_{\frac{1}{10}}]} \times 10$$? (take $$\log_{10} 2 = 0.3$$)

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

Let $$f(x) = \begin{cases}x^2 \mid \cos \frac{\pi}{x} \mid & x \neq 0\\0, & x = 0\end{cases}, x \in R$$, then f is

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

If $$\lim_{x \rightarrow \infty}\left(\frac{x^2 + x + 1}{x + 1} - ax - b\right) = 4$$, then

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

The ellipse $$E_1 : \frac{x^2}{9} + \frac{y^2}{4} = 1$$ is inscribed in a rectangle R whosesides 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

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

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

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 7(2, 1, 4) to QR, then the length of the line segment PS is

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

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

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

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. LetX denote the eventthat the ship is operational and let $$X_1, X_2$$, and $$X_3$$ denote respectively the events that the engines $$E_1, E_2$$, and $$E_3$$ are functioning. Which of the following is (are) true ?

Tangents are drawnto 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

If y (x) satisfies the differential equation y'- ytan x = 2x sec x and y(0) = 0, then

Let $$f : IR \rightarrow IR$$ be defined as $$f(x) = \mid x \mid + \mid x^2 - 1 \mid$$. The total numberof points at which fattains either a local maximum or a local minimum is

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

Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and local minimum at x = 3. If p(1) = 6 and p(3) = 2, then p'(0) is

If $$\overrightarrow{a}, \overrightarrow{b}$$ and $$\overrightarrow{c}$$ are unit vectors satisfying $$\mid \overrightarrow{a} - \overrightarrow{b} \mid^2 + \mid \overrightarrow{b} - \overrightarrow{c} \mid^2 + \mid \overrightarrow{c} - \overrightarrow{a} \mid^2 = 9$$, then $$\mid 2 \overrightarrow{a} + 5 \overrightarrow{b} + 5 \overrightarrow{c}\mid$$ is

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 ofthe triangle PQS is