JEE Advanced 2013 Paper 1 Offline

Paper was held on Sat, Jun 1, 2013 9:00 PM

Chemistry

The atomic masses of He and Ne are 4 and 20 a.m.u., respectively. The value of the de Broglie wavelength of He gas at –73^oC is “M” times that of the de Broglie wavelength of Ne at 727^oC. M is

The total number of lone-pairs of electrons in melamine is

The standard enthalpies of formation of CO₂(g), H₂O(l) and glucose(s) at 25^oC are –400 kJ/mol, –300 kJ/mol and –1300 kJ/mol, respectively. The standard enthalpy of combustion per gram of glucose at 25^oC is

The initial rate of hydrolysis of methyl acetate (1M) by a weak acid (HA, 1M) is 1/100^th of that of a strong acid (HX, 1M), at 25^oC. The K_a of HA is

The hyperconjugative stabilities of tert-butyl cation and 2-butene, respectively, are due to

Methylene blue, from its aqueous solution, is adsorbed on activated charcoal at 25^oC. For this process, the correct statement is

Benzene and naphthalene form an ideal solution at room temperature. For this process, the true statement(s) is(are)

Sulphide ores are common for the metals

KI in acetone, undergoes S_N2 reaction with each of P, Q, R and S. The rates of the reaction vary as

The compound that does NOT liberate CO₂, on treatment with aqueous sodium bicarbonate solution, is

Consider the following complex ions : P, Q and R.

$$P = {[Fe{F_6}]^{3 - }}$$, $$Q = {[V{({H_2}O)_6}]^{2 + }}$$ and $$R = {[Fe{({H_2}O)_6}]^{2 + }}$$

The correct order of the complex ions, according to their spin-only magnetic moment values (in B.M.) is

In the reaction, P + Q $$\to$$ R + S, the time taken for 75% reaction of P is twice the time taken for 50% reaction of P. The concentration of Q varies with reaction time as shown in the figure. The overall order of the reaction is

Concentrated nitric acid, upon long standing, turns yellow-brown due to the formation of

The arrangement of X^$$-$$ ions around A⁺ ion in solid AX is given in the figure (not drawn to scale). If the radius of X^$$-$$ is 250 pm, the radius of A⁺ is

Upon treatment with ammoniacal H₂S, the metal ion that precipitates as a sulphide is

The pair(s) of coordination complexes/ions exhibiting the same kind of isomerism is(are)

Among P, Q, R and S, the aromatic compound(s) is(are)

The total number of carboxylic acid groups is the product P is _________.

EDTA^4$$-$$ is ethylenediaminetetraacetate ion. The total number of N$$-$$Co$$-$$O bond angles in [Co(EDTA)]^1$$-$$ complex ion is ________.

A tetrapeptide has $$-$$COOH group on alanine. This produces glycine (Gly), valine (Val), phenyl alanine (Phe) and alanine (Ala), on complete hydrolysis. For this tetrapeptide, the number of possible sequences (primary structures) with $$-$$NH₂ group attached to a chiral center is _______.

Mathematics

Let $$f\left( x \right) = x\sin \,\pi x,\,x > 0.$$ Then for all natural numbers $$n,\,f'\left( x \right)$$ vanishes at

A pack contains $$n$$ cards numbered from $$1$$ to $$n.$$ Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is $$1224.$$ If the smaller of the numbers on the removed cards is $$k,$$ then $$k-20=$$

Let complex numbers $$\alpha \,and\,{1 \over {\overline \alpha }}\,$$ lie on circles $${\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \right)^2} = {r^2}$$ and $$\,{\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \right)^2} = 4{r^2}$$ respextively. If $${z_0} = {x_0} + i{y_0}$$ satisfies the equation $$2{\left| {{z_0}} \right|^2}\, = {r^2} + 2,\,then\,\left| a \right| = $$

The number of points in $$\left( { - \infty \,\infty } \right),$$ for which $${x^2} - x\sin x - \cos x = 0,$$ is

The coefficient of three consecutive terms of $${\left( {1 + x} \right)^{n + 5}}$$ are in the ratio $$5:10:14.$$ Then $$n$$ =

Consider the set of eight vectors $$V = \left\{ {a\,\hat i + b\,\hat j + c\hat k:a,\,b,\,c\, \in \left\{ { - 1,\,1} \right\}} \right\}$$. Three non-coplanar vectors can be chosen from v in $${2^p}$$ ways. Then p is

Let $${S_n} = {\sum\limits_{k = 1}^{4n} {\left( { - 1} \right)} ^{{{k\left( {k + 1} \right)} \over 2}}}{k^2}.$$ Then $${S_n}$$can take value(s)

Perpendiculars are drawn from points on the line $\frac{x+2}{2}=\frac{y+1}{-1}=\frac{z}{3}$ to the plane $x+y+$ $z=3$. The foot of perpendiculars lie on the line

For $$a > b > c > 0,$$ the distance between $$(1, 1)$$ and the point of intersection of the lines $$ax + by + c = 0$$ and $$bx + ay + c = 0$$ is less than $$\left( {2\sqrt 2 } \right)$$. Then

A vertical line passing through the point $$(h,0)$$ intersects the ellipse $${{{x^2}} \over 4} + {{{y^2}} \over 3} = 1$$ at the points $$P$$ and $$Q$$. Let the tangents to the ellipse at $$P$$ and $$Q$$ meet at the point $$R$$. If $$\Delta \left( h \right)$$$$=$$ area of the triangle $$PQR$$, $${{\Delta _1}}$$ $$ = \mathop {\max }\limits_{1/2 \le h \le 1} \Delta \left( h \right)$$ and $${{\Delta _2}}$$ $$ = \mathop {\min }\limits_{1/2 \le h \le 1} \Delta \left( h \right)$$, then $${8 \over {\sqrt 5 }}{\Delta _1} - 8{\Delta _2} = $$

The value of $$\cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}} \left( {1 + \sum\limits_{k = 1}^n {2k} } \right)} \right)$$ is

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $$8:15$$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $$100$$, the resulting box has maximum volume. Then the lengths of the vsides of the rectangular sheet are

For 3 × 3 matrices M and N, which of the following statement(s) is(are) NOT correct?

The area enclosed by the curves $$y = \sin x + {\mathop{\rm cosx}\nolimits} $$ and $$y = \left| {\cos x - \sin x} \right|$$ over the interval $$\left[ {0,{\pi \over 2}} \right]$$ is

Let $$f$$ $$:\,\,\left[ {{1 \over 2},1} \right] \to R$$ (the set of all real number) be a positive,
non-constant and differentiable function such that
$$f'\left( x \right) < 2f\left( x \right)$$ and $$f\left( {{1 \over 2}} \right) = 1.$$ Then the value of $$\int\limits_{1/2}^1 {f\left( x \right)} \,dx$$ lies in the interval

A curve passes through the point $$\left( {1,{\pi \over 6}} \right)$$. Let the slope of
the curve at each point $$(x,y)$$ be $${y \over x} + \sec \left( {{y \over x}} \right),x > 0.$$
Then the equation of the curve is

Four persons independently solve a certain problem correctly with probabilities $${1 \over 2},{3 \over 4},{1 \over 4},{1 \over 8}.$$ Then the probability that the problem is solved correctly by at least one of them is

Of the three independent events $${E_1},{E_2}$$ and $${E_3},$$ the probability that only $${E_1}$$ occurs is $$\alpha ,$$ only $${E_2}$$ occurs is $$\beta $$ and only $${E_3}$$ occurs is $$\gamma .$$ Let the probability $$p$$ that none of events $${E_1},{E_2}$$ or $${E_3}$$ occurs satisfy the equations $$\left( {\alpha -2\beta } \right)p = \alpha \beta $$ and $$\left( {\beta - 3\gamma } \right)p = 2\beta \gamma .$$ All the given probabilities are assumed to lie in the interval $$(0, 1)$$.

Then $${{\Pr obability\,\,of\,\,occurrence\,\,of\,\,{E_1}} \over {\Pr obability\,\,of\,\,occurrence\,\,of\,\,{E_3}}}$$

A line $$l$$ passing through the origin is perpendicular to the lines $$$\,{l_1}:\left( {3 + t} \right)\widehat i + \left( { - 1 + 2t} \right)\widehat j + \left( {4 + 2t} \right)\widehat k,\,\,\,\,\, - \infty < t < \infty $$$ $$${l_2}:\left( {3 + 2s} \right)\widehat i + \left( {3 + 2s} \right)\widehat j + \left( {2 + s} \right)\widehat k,\,\,\,\,\, - \infty < s < \infty $$$
Then, the coordinate(s) of the points(s) on $${l_2}$$ at a distance of $$\sqrt {17} $$ from the point of intersection of $$l$$ and $${l_1}$$ is (are)

Let $\overrightarrow{\mathrm{PR}}=3 \hat{i}+\hat{j}-2 \hat{k}$ and $ \overrightarrow{\mathrm{SQ}}=\hat{i}-3 \hat{j}-4 \hat{k}$ determine diagonals of a parallelogram $P Q R S$ and $\overrightarrow{\mathrm{PT}}=\hat{i}+2 \hat{j}+3 \hat{k}$ be another vector. Then the volume of the parallelopiped determined by the vectors $\overrightarrow{\mathrm{PT}}, \overrightarrow{\mathrm{PQ}}$ and $\overrightarrow{\mathrm{PS}}$ is :

Physics

A uniform circular disc of mass 50 kg and radius 0.4 m is rotating with an angular velocity of 10 rad s^-1 about its own axis, which is vertical. Two uniform circular rings, each of mass 6.25 kg and radius 0.2 m, are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity (in rad s^-1 ) of the system is

The image of an object, formed by a plano-convex lens at a distance of 8 m behind the lens, is real and is one-third the size of the object. The wavelength of light inside the lens is $${2 \over 3}$$ times the wavelength in free space. The radius of the curved surface of the lens is

The work done on a particle of mass m by a force $$$K\left[ {{x \over {{{\left( {{x^2} + {y^2}} \right)}^{3/2}}}}\widehat i + {y \over {{{\left( {{x^2} + {y^2}} \right)}^{3/2}}}}\widehat j} \right]$$$ (K being a constant of appropriate dimensions), when the particle is taken from the point $$\left( {a,0} \right)$$ to the point $$\left( {0,a} \right)$$ along a circular path of radius a about the origin in the x-y plane is

A particle of mass 0.2 kg is moving in one dimension under a force that delivers a constant power 0.5 W to the particle. If the initial speed (in m/s) of the particle is zero, the speed (in m/s) after 5 s is

A particle of mass m is projected from the ground with an initial speed u₀ at an angle $$\alpha $$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed u₀. The angle that the composite system makes with the horizontal immediately after the collision is

A bob of mass m, suspended by a string of length l₁ is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length l₂, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio $${{{l_1}} \over {{l_2}}}$$ is

One end of a horizontal thick copper wire of length 2L and radius 2R is welded to an end of another horizontal thin copper wire of length L and radius R. When the arrangement is stretched by applying forces at two ends, the ratio of the elongation in the thin wire to that in the thick wire is

A solid sphere of radius R and density $$\rho $$ is attached to one end of a mass-less spring of force constant k. The other end of the spring is connected to another solid sphere of radius R and density 3$$\rho $$. The complete arrangement is placed in a liquid of density 2$$\rho $$ and is allowed to reach equilibrium. The correct statement(s) is (are)

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2 : 3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3. The ratio of their densities is

Two non-conducting solid spheres of radii $$R$$ and $$2R,$$ having uniform volume charge densities $${\rho _1}$$ and $${\rho _2}$$ respectively, touch each other. The net electric field at a distance $$2$$ $$R$$ from the center of the smaller sphere, along the line joining the centers of the spheres, is zero. The ratio $${{{\rho _1}} \over {{\rho _2}}}$$ can be

In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance $$C.$$ The switch $${S_1}$$ is pressed first to fully charge the capacitor $${C_1}$$ and then released. The switch $${S_2}$$ is then pressed to charge the capacitor $${C_2}.$$ After some time, $${S_2}$$ is released and then $${S_3}$$ is pressed. After some time

JEE Advanced 2013 Paper 1 Offline Physics - Capacitor Question 15 English

The diameter of a cylinder is measured using a Vernier callipers with no zero error. It is found that the zero of the Vernier scale lies between 5.10 cm and 5.15 cm of the main scale. The Vernier scale has 50 divisions equivalent to 2.45 cm. The 24^th division of the Vernier scale exactly coincides with one of the main scale divisions. The diameter of the cylinder is

A ray of light travelling in the direction $${1 \over 2}\left( {\widehat i + \sqrt 3 \widehat j} \right)$$ is incident on a plane mirror. After reflection, it travels along the direction $${1 \over 2}\left( {\widehat i - \sqrt 3 \widehat j} \right)$$. The angle of incidence is

Two rectangular blocks, having identical dimensions, can be arranged in either configuration-I or configuration-II as shown in the figure. One of the blocks has thermal conductivity $$\kappa $$ and the other 2$$\kappa $$. The temperature difference between the ends along the x-axis is the same in both the configurations. It takes 9 s to transport a certain amount of heat from the hot end to the cold end in configuration-I. The time to transport the same amount of heat in configuration-II is

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 mW and the speed of light is 3 $$\times$$ 10⁸ ms^$$-$$1. The final momentum of the object is

In the Young's double-slit experiment using a monochromatic light of wavelength $$\lambda$$, the path difference (in terms of an integer n) corresponding to any point having half the peak intensity is

A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation

y(x, t) = (0.01 m) sin[(62.8 m^$$-$$1)x] cos[(628 s^$$-$$1)t]

Assuming $$\pi$$ = 3.14, the correct statement(s) is(are)

A particle of mass M and positive charge Q, moving with a constant velocity $${\overrightarrow u _1} = 4\widehat i$$ ms^$$-$$1 enters a region of uniform static magnetic field, normal to the xy plane. The region of the magnetic field extends from x = 0 to x = L for all values of y. After passing through this region, the particle emerges on the other side after 10 ms with a velocity $${\overrightarrow u _2} = 2\left( {\sqrt 3 \widehat i + \widehat j} \right)$$ ms^$$-$$1. The correct statement(s) is(are)

The work functions of silver and sodium are 4.6 and 2.3 eV, respectively. The ratio of the slope of the stopping potential versus frequency plot for silver to that of sodium is ___________.

A freshly prepared sample of a radioisotope of half-life 1386 s has activity 10³ disintegrations per second. Given that ln2 = 0.693, the fraction of the initial number of nuclei (expressed in nearest integer percentage) that will decay in the first 80 s after preparation of the sample is __________.