Phenol on reaction with dilute nitric acid, gives two products. Which method will be most efficient for large scale separation?

The major product in the reaction

The standard entropy change for the reaction

4Fe(s) + 3O₂(g) $\to$ 2Fe₂O₃(s) is $-$550 J K^$-$1 at 298 K.

[Given : The standard enthalpy change for the reaction is $-$165 kJ mol^$-$1]. The temperature in K at which the reaction attains equilibrium is _____________. (Nearest Integer)

If [Cu(H₂O)₄]²⁺ absorbs a light of wavelength 600 nm for d-d transition, then the value of octahedral crystal field splitting energy for [Cu(H₂O)₆]²⁺ will be ____________ $\times$ 10^$-$21 J. [Nearest Integer]

(Given : h = 6.63 $\times$ 10^$-$34 Js and c = 3.08 $\times$ 10⁸ ms^$-$1)

Number of grams of bromine that will completely react with 5.0 g of pent-1-ene is ___________ $\times$ 10^$-$2 g. (Atomic mass of Br = 80 g/mol) [Nearest Integer]

Let A be a 3 $\times$ 3 real matrix such that

A\left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right) = \left( {\matrix{ 1 \cr 1 \cr 1 \cr } } \right);A\left( {\matrix{ 1 \cr 0 \cr 1 \cr } } \right) = \left( {\matrix{ { - 1} \cr 0 \cr 1 \cr } } \right)\( and \)A\left( {\matrix{ 0 \cr 0 \cr 1 \cr } } \right) = \left( {\matrix{ 1 \cr 1 \cr 2 \cr } } \right).

If $X = {({x_1},{x_2},{x_3})^T}$ and I is an identity matrix of order 3, then the system (A - 2I)X = \left( {\matrix{ 4 \cr 1 \cr 1 \cr } } \right) has :

Let f(x) be a polynomial function such that $f(x) + f'(x) + f''(x) = {x^5} + 64$. Then, the value of $\mathop {\lim }\limits_{x \to 1} {{f(x)} \over {x - 1}}$ is equal to:

Let $f:R \to R$ and $g:R \to R$ be two functions defined by $f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$ and $g(x) = {{1 - 2{e^{2x}}} \over {{e^x}}}$. Then, for which of the following range of $\alpha$, the inequality $f\left( {g\left( {{{{{(\alpha - 1)}^2}} \over 3}} \right)} \right) > f\left( {g\left( {\alpha -{5 \over 3}} \right)} \right)$ holds ?

Let A be a 3 $\times$ 3 matrix having entries from the set {$-$1, 0, 1}. The number of all such matrices A having sum of all the entries equal to 5, is ___________.

If $Z = {{{A^2}{B^3}} \over {{C^4}}}$, then the relative error in Z will be :

Which of the following relations is true for two unit vector $\widehat A$ and $\widehat B$ making an angle $\theta$ to each other?

If wattless current flows in the AC circuit, then the circuit is :

The equivalent capacitance between points A and B in below shown figure will be __________ $\mu$F.

A resistor develops 300 J of thermal energy in 15 s, when a current of 2 A is passed through it. If the current increases to 3 A, the energy developed in 10 s is ____________ J.

White precipitate of AgCl dissolves in aqueous ammonia solution due to formation of :

1 L aqueous solution of H₂SO₄ contains 0.02 m mol H₂SO₄. 50% of this solution is diluted with deionized water to give 1 L solution (A). In solution (A), 0.01 m mol of H₂SO₄ are added. Total m mols of H₂SO₄ in the final solution is ___________ $\times$ 10³ m mols.

The value of $\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx}$ is equal to:

Let f : N $\to$ R be a function such that $f(x + y) = 2f(x)f(y)$ for natural numbers x and y. If f(1) = 2, then the value of $\alpha$ for which

$\sum\limits_{k = 1}^{10} {f(\alpha + k) = {{512} \over 3}({2^{20}} - 1)}$

holds, is :

Let f : R $\to$ R be defined as $f(x) = {x^3} + x - 5$. If g(x) is a function such that $f(g(x)) = x,\forall 'x' \in R$, then g'(63) is equal to ________________.

Let A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]\(. If M and N are two matrices given by \)M = \sum\limits_{k = 1}^{10} {{A^{2k}}} \( and \)N = \sum\limits_{k = 1}^{10} {{A^{2k - 1}}} then MN² is :

Let $x = 2t$, $y = {{{t^2}} \over 3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA \bot BA$, where A is any point on the conic. If k is the ordinate of the centroid of the $\Delta$SAB, then $\mathop {\lim }\limits_{t \to 1} k$ is equal to :

Let a circle C in complex plane pass through the points ${z_1} = 3 + 4i$, ${z_2} = 4 + 3i$ and ${z_3} = 5i$. If $z( \ne {z_1})$ is a point on C such that the line through z and z₁ is perpendicular to the line through z₂ and z₃, then $arg(z)$ is equal to :

Let $\theta$ be the angle between the vectors $\overrightarrow a$ and $\overrightarrow b$, where $|\overrightarrow a | = 4,$ $|\overrightarrow b | = 3$ and $\theta \in \left( {{\pi \over 4},{\pi \over 3}} \right)$. Then ${\left| {\left( {\overrightarrow a - \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow b } \right)} \right|^2} + 4{\left( {\overrightarrow a \,.\,\overrightarrow b } \right)^2}$ is equal to __________.

Let $f:R \to R$ be a function defined by

$f(x) = {\left( {2\left( {1 - {{{x^{25}}} \over 2}} \right)(2 + {x^{25}})} \right)^{{1 \over {50}}}}$. If the function $g(x) = f(f(f(x))) + f(f(x))$, then the greatest integer less than or equal to g(1) is ____________.

In the figure, a very large plane sheet of positive charge is shown. P₁ and P₂ are two points at distance l and 2l from the charge distribution. If $\sigma$ is the surface charge density, then the magnitude of electric fields E₁ and E₂ at P₁ and P₂ respectively are :

A long straight wire with a circular cross-section having radius R, is carrying a steady current I. The current I is uniformly distributed across this cross-section. Then the variation of magnetic field due to current I with distance r (r < R) from its centre will be :

The electric field in an electromagnetic wave is given by E = 56.5 sin $\omega$(t $-$ x/c) NC^$-$1. Find the intensity of the wave if it is propagating along x-axis in the free space.

(Given : $\varepsilon$₀ = 8.85 $\times$ 10^$-$12C²N^$-$1m^$-$2)

A light wave travelling linearly in a medium of dielectric constant 4, incidents on the horizontal interface separating medium with air. The angle of incidence for which the total intensity of incident wave will be reflected back into the same medium will be :

(Given : relative permeability of medium $\mu$_r = 1)

The velocity of upper layer of water in a river is 36 kmh^$-$1. Shearing stress between horizontal layers of water is 10^$-$3 Nm^$-$2. Depth of the river is __________ m. (Co-efficient of viscosity of water is 10^$-$2 Pa.s)

The pair, in which ions are isoelectronic with AI³⁺ is :

Number of electron deficient molecules among the following

PH₃, B₂H₆, CCl₄, NH₃, LiH and BCl₃ is

Among the following, which is the strongest oxidizing agent?

The IUPAC name of ethylidene chloride is :

The reaction of with bromine and KOH gives RNH₂ as the end product. Which one of the following is the intermediate product formed in this reation?

The longest wavelength of light that can be used for the ionisation of lithium atom (Li) in its ground state is x $\times$ 10^$-$8 m. The value of x is ___________. (Nearest Integer).

(Given : Energy of the electron in the first shell of the hydrogen atom is $-$2.2 $\times$ 10^$-$18 J ; h = 6.63 $\times$ 10^$-$34 Js and c = 3 $\times$ 10⁸ ms^$-$1)

The standard free energy change ($\Delta$G$^\circ$) for 50% dissociation of N₂O₄ into NO₂ at 27$^\circ$C and 1 atm pressure is $-$ x J mol^$-$1. The value of x is ___________. (Nearest Integer)

[Given : R = 8.31 J K^$-$1 mol^$-$1, log 1.33 = 0.1239 ln 10 = 2.3]

For a given chemical reaction

$\gamma$₁A + $\gamma$₂B $\to$ $\gamma$₃C + $\gamma$₄D

Concentration of C changes from 10 mmol dm^$-$3 to 20 mmol dm^$-$3 in 10 seconds. Rate of appearance of D is 1.5 times the rate of disappearance of B which is twice the rate of disappearance A. The rate of appearance of D has been experimentally determined to be 9 mmol dm^$-$3 s^$-$1. Therefore, the rate of reaction is _____________ mmol dm^$-$3 s^$-$1. (Nearest Integer)

Let E₁ and E₂ be two events such that the conditional probabilities $P({E_1}|{E_2}) = {1 \over 2}$, $P({E_2}|{E_1}) = {3 \over 4}$ and $P({E_1} \cap {E_2}) = {1 \over 8}$. Then :

Let $g:(0,\infty ) \to R$ be a differentiable function such that

$\int {\left( {{{x(\cos x - \sin x)} \over {{e^x} + 1}} + {{g(x)\left( {{e^x} + 1 - x{e^x}} \right)} \over {{{({e^x} + 1)}^2}}}} \right)dx = {{x\,g(x)} \over {{e^x} + 1}} + c}$, for all x > 0, where c is an arbitrary constant. Then :

Let $\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$ ${a_i} > 0$, $i = 1,2,3$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of $\overrightarrow a$ on the vector $3\widehat i + 4\widehat j$ be 7. Let $\overrightarrow b$ be a vector obtained by rotating $\overrightarrow a$ with 90$^\circ$. If $\overrightarrow a$, $\overrightarrow b$ and x-axis are coplanar, then projection of a vector $\overrightarrow b$ on $3\widehat i + 4\widehat j$ is equal to:

The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is _____________.

The relation between root mean square speed (v_rms) and most probable sped (v_p) for the molar mass M of oxygen gas molecule at the temperature of 300 K will be :

Match List-I with List-II.

Choose the correct answer from the options given below :

The two light beams having intensities I and 9I interfere to produce a fringe pattern on a screen. The phase difference between the beams is $\pi$/2 at point P and $\pi$ at point Q. Then the difference between the resultant intensities at P and Q will be :

The ratio for the speed of the electron in the 3^rd orbit of He⁺ to the speed of the electron in the 3^rd orbit of hydrogen atom will be :

The first overtone frequency of an open organ pipe is equal to the fundamental frequency of a closed organ pipe. If the length of the closed organ pipe is 20 cm. The length of the open organ pipe is _____________ cm.

A force on an object of mass 100 g is $\left( {10\widehat i + 5\widehat j} \right)$ N. The position of that object at t = 2 s is $\left( {a\widehat i + b\widehat j} \right)$ m after starting from rest. The value of ${a \over b}$ will be ___________.

Bonding in which of the following diatomic molecule(s) become(s) stronger, on the basis of MO Theory, by removal of an electron?

(A) NO

(B) N₂

(C) O₂

(D) C₂

(E) B₂

Choose the most appropriate answer from the options given below :

Cerium (IV) has a noble gas configuration. Which of the following is correct statement about it?

In the following structures, which on is having staggered conformation with maximum dihedral angle?

The product formed in the following reaction.

The intermediate X, in the reaction :

In the following reaction :

The compounds A and B respectively are :

The number of N atoms in 681 g of C₇H₅N₃O₆ is x $\times$ 10²¹. The value of x is (N_A = 6.02 $\times$ 10²³ mol^$-$1) (Nearest Integer)

In a cell, the following reactions take place

\matrix{ {F{e^{2 + }} \to F{e^{3 + }} + {e^ - }} & {E_{F{e^{3 + }}/F{e^{2 + }}}^o = 0.77\,V} \cr {2{I^ - } \to {I_2} + 2{e^ - }} & {E_{{I_2}/{I^ - }}^o = 0.54\,V} \cr }

The standard electrode potential for the spontaneous reaction in the cell is x $\times$ 10^$-$2 V 298 K. The value of x is ____________. (Nearest Integer)

If ${1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}$, then the remainder when K is divided by 6 is :

Let $y = y(x)$ be the solution of the differential equation $(x + 1)y' - y = {e^{3x}}{(x + 1)^2}$, with $y(0) = {1 \over 3}$. Then, the point $x = - {4 \over 3}$ for the curve $y = y(x)$ is :

If the solution curve $y = y(x)$ of the differential equation ${y^2}dx + ({x^2} - xy + {y^2})dy = 0$, which passes through the point (1, 1) and intersects the line $y = \sqrt 3 x$ at the point $(\alpha ,\sqrt 3 \alpha )$, then value of ${\log _e}(\sqrt 3 \alpha )$ is equal to :

Let the abscissae of the two points P and Q be the roots of $2{x^2} - rx + p = 0$ and the ordinates of P and Q be the roots of ${x^2} - sx - q = 0$. If the equation of the circle described on PQ as diameter is $2({x^2} + {y^2}) - 11x - 14y - 22 = 0$, then $2r + s - 2q + p$ is equal to __________.

$\overrightarrow A$ is a vector quantity such that $|\overrightarrow A |$ = non-zero constant. Which of the following expression is true for $\overrightarrow A$ ?

If force $\overrightarrow F = 3\widehat i + 4\widehat j - 2\widehat k$ acts on a particle position vector $2\widehat i + \widehat j + 2\widehat k$ then, the torque about the origin will be :

The height of any point P above the surface of earth is equal to diameter of earth. The value of acceleration due to gravity at point P will be : (Given g = acceleration due to gravity at the surface of earth).

The terminal velocity (v_t) of the spherical rain drop depends on the radius (r) of the spherical rain drop as :

The photodiode is used to detect the optical signals. These diodes are preferably operated in reverse biased mode because :

The difference of speed of light in the two media A and B (v_A $-$ v_B) is 2.6 $\times$ 10⁷ m/s. If the refractive index of medium B is 1.47, then the ratio of refractive index of medium B to medium A is : (Given : speed of light in vacuum c = 3 $\times$ 10⁸ ms^$-$1)

A teacher in his physics laboratory allotted an experiment to determine the resistance (G) of a galvanometer. Students took the observations for ${1 \over 3}$ deflection in the galvanometer. Which of the below is true for measuring value of G?

A uniform chain of 6 m length is placed on a table such that a part of its length is hanging over the edge of the table. The system is at rest. The co-efficient of static friction between the chain and the surface of the table is 0.5, the maximum length of the chain hanging from the table is ___________ m.

A 0.5 kg block moving at a speed of 12 ms^$-$1 compresses a spring through a distance 30 cm when its speed is halved. The spring constant of the spring will be _______________ Nm^$-$1.

The total current supplied to the circuit as shown in figure by the 5 V battery is ____________ A.

The current in a coil of self inductance 2.0 H is increasing according to I = 2 sin(t²) A. The amount of energy spent during the period when current changes from 0 to 2 A is ____________ J.