JEE Advanced 2013 Paper-1

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 cylinderis

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

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 rectangular blocks, having identical dimensions, can be arranged either in configuration I or in configuration II as shown in the figure. One of the blocks has thermal conductivity K and the other 2K. 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 the configuration I. The time to transport the same amount of heat in the 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 speedoflight is $$3 \times 10^8 ms^{-1}$$. The final momentum of the object is

A particle of mass m is projected from the ground with an initial speed $$u_0$$, 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 $$u_0.$$ The angle that the composite system makes with the horizontal immediately after the collision is

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

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 oflight inside the lens is $$\frac{2}{3}$$ times the wavelength in free space. The radius of the curved surface of the lens 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 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)

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 x-y 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 milliseconds with a velocity $$\overrightarrow{u}_2 = 2\left(\sqrt 3 \widehat{i} + \widehat{j}\right) ms^{-1}$$. The correct statement(s) is (are)

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 2R from the centre of the smaller sphere, along the line joining the centres of the spheres, is zero. The ratio $$\frac{\rho_1}{\rho_2}$$ can be

In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance C. The switch $$S_1$$ is pressedfirst 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 sometime,

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 frequencyplot for Silver to that of Sodium is

A freshly prepared sample of a radioisotope of half-life 1386 s has activity $$10^3$$ disintegrations per second. Given that ln 2 = 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

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 $$ms^{-1})$$ of the particle is zero, the speed (in $$ms^{-1})$$ after 5 s is

A uniform circular disc of mass 50 kg and radius 0.4 m is rotating with an angular velocity of 10 rad $$s^{-1}$$ aboutits 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

A bob of mass m, suspended by string of length $$l_1$$, is given a minimum velocity required to completea full circle in the vertical plane. At the highestpoint,it collides elastically with another bob of mass m suspendedbya string of length $$l_2$$, 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 $$\frac{l_1}{l_2}$$ is

The compound that does NOT liberate $$CO_2$$, on treatment with aqueous sodium bicarbonate solution, is

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

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

Sulfide ores are common for the metals

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 $$H_2S$$, the metal ion that precipitates as a sulfide is

The standard enthalpies of formation of $$CO_2(g), H_2O(l)$$ and glucose(s) at $$25^\circ C$$ are -400 kJ/mol, -300 kJ/mol and -1300 kJ/mol, respectively. The standard enthalpy of combustion per gram of glucoseat $$25^\circ C$$ is

Consider the following complex ions, P, Q and R.
$$P = [FeF_6]^{3-}, Q = [V(H_2O)_6]^{2+}$$ and $$R = [Fe(H_2O)_6]^{2+}$$
The correct order of the complex ions, according to their spin-only magnetic moment values (in B.M.) is

In the reaction,
$$P + Q \rightarrow R + S$$
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

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

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

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

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 $$25^\circ C$$. The $$K_a$$ of HA is

The total number of carboxylic acid groups in the product P 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 $$-NH_2$$ group attached to a chiral center is

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

The total numberof lone-pairs of electrons in melamine is

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 $$-73^\circ C$$ is "M" imes that of the de Broglie wavelength of Ne at $$727^\circ C.$$ M is

Let complex numbers $$\alpha$$ and $$\frac{1}{\alpha}$$ lie on circles $$(x - x_0)^2 + (y - y_0)^2 = r^2$$ and $$(x - x_0)^2 + (y - y_0)^2 = 4r^2$$, respectively. If $$z_0 = x_0 + iy_0$$ Satisfies the equation $$2 |z_0|^2 = r^2 + 2,$$ then $$|\alpha| = $$

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

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

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

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

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

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

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 $$2 \sqrt 2$$. Then

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 feet of perpendiculars lie on the line

Let $$\overrightarrow{PR} = 3\widehat{i} + \widehat{j} - 2\widehat{k}$$ and $$\overrightarrow{SQ} = \widehat{i} - 3 \widehat{j} - 4\widehat{k}$$ determine diagonals of a parallelogram PQRS and $$\overrightarrow{PT} = \widehat{i} + 2\widehat{j} + 3\widehat{k}$$ be another vector. Then the volume of the $$\overrightarrow{PT}, \overrightarrow{PQ}$$ and $$\overrightarrow{PS}$$ is

Let $$S_n = \sum_{k = 1}^{4n}(-1)^{\frac {k (k + 1)}{2}} k^2.$$ Then $$S_n$$ can take value(s)

For $$3 \times 3$$ matrices M and N, whichof the following statement(s) is (are) NOT correct ?

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

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 sides of the rectangular sheet are

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

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

A pack contains n cards numberedfrom 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 =

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 $$(\alpha - 2\beta) p = \alpha \beta$$ and $$(\beta - 3\gamma) p = 2 \beta \gamma$$. All the given probabilities are assumed tolie in the interval (0, 1).
Then $$\frac{Probability of occurrence of E_1}{Probability of occurrence of E_3} = $$

A vertical line passing through the point (h, 0) intersects the ellipse $$\frac{x^2}{4} + {y^2}{3} = 1$$ at the points P and Q. Let the tangents to the ellipse at P and Q meet at the point R. If $$\triangle (h)$$ = area of the triangle PQR,

, then $$\frac{8}{\sqrt 5} \triangle_1 - 8 \triangle_2 =$$

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