A commercially sold conc. HCl is 35% HCl by mass. If the density of this commercial acid is 1.46 g/mL, the molarity of this solution is:

(Atomic mass : Cl = 35.5 amu, H = 1 amu)

If the radius of the 3^rd Bohr's orbit of hydrogen atom is r₃ and the radius of 4^th Bohr's orbit is r₄. Then :

Consider the ions/molecule

O$$_2^ + $$, O₂, O$$_2^ - $$, O$$_2^ {2-} $$

For increasing bond order the correct option is :

The $${\left( {{{\partial E} \over {\partial T}}} \right)_P}$$ of different types of half cells are as follows:

(Where E is the electromotive force)

Which of the above half cells would be preferred to be used as reference electrode?

Choose the correct stability order of group 13 elements in their + 1 oxidation state :

Given below are two statements :

Statement I : In 'Lassaigne's Test', when both nitrogen and sulphur are present in an organic compound, sodium thiocyanate is formed.

Statement II : If both nitrogen and sulphur are present in an organic compound, then the excess of sodium used in sodium fusion will decompose the sodium thiocyanate formed to give NaCN and Na₂S.

In the light of the above statements, choose the most appropriate answer from the options given below :

(C₇H₅O₂)₂ $$\buildrel {hv} \over \longrightarrow $$ [X] $$\to$$ 2C₆H₅ + 2CO₂

Consider the above reaction and identify the intermediate 'X'

Consider the above reaction sequence and identify the product B.

Which will have the highest enol content?

Among the following structures, which will show the most stable enamine formation? (Where Me is $$-$$CH₃)

Which statement is not true with respect to nitrate ion test?

For complete combustion of methanol

CH₃OH(I) + $${3 \over 2}$$O₂(g) $$\to$$ CO₂(g) + 2H₂O(I)

the amount of heat produced as measured by bomb calorimeter is 726 kJ mol^$$-$$1 at 27$$^\circ$$C. The enthalpy of combustion for the reaction is $$-$$x kJ mol^$$-$$1, where x is ___________. (Nearest integer)

(Given : R = 8.3 JK^$$-$$1 mol^$$-$$1)

A 0.5 percent solution of potassium chloride was found to freeze at $$-$$0.24$$^\circ$$C. The percentage dissociation of potassium chloride is ______________. (Nearest integer)

(Molal depression constant for water is 1.80 K kg mol^$$-$$1 and molar mass of KCl is 74.6 g mol^$$-$$1)

50 mL of 0.1 M CH₃COOH is being titrated against 0.1 M NaOH. When 25 mL of NaOH has been added, the pH of the solution will be _____________ $$\times$$ 10^$$-$$2. (Nearest integer)

(Given : pK_a (CH₃COOH) = 4.76)

log 2 = 0.30

log 3 = 0.48

log 5 = 0.69

log 7 = 0.84

log 11 = 1.04

A flask is filled with equal moles of A and B. The half lives of A and B are 100 s and 50 s respectively and are independent of the initial concentration. The time required for the concentration of A to be four times that of B is ___________ s.

(Given : ln 2 = 0.693)

The spin-only magnetic moment value of the most basic oxide of vanadium among V₂O₃, V₂O₄ and V₂O₅ is _____________ B.M. (Nearest integer)

The spin-only magnetic moment value of an octahedral complex among CoCl₃.4NH₃, NiCl₂.6H₂O and PtCl₄.2HCl, which upon reaction with excess of AgNO₃ gives 2 moles of AgCl is ___________ B.M. (Nearest integer)

On complete combustion 0.30 g of an organic compound gave 0.20 g of carbon dioxide and 0.10 g of water. The percentage of carbon in the given organic compound is _____________. (Nearest integer)

Compound 'P' on nitration with dil. HNO₃ yields two isomers (A) and (B). These isomers can be separated by steam distillation. Isomers (A) and (B) show the intramolecular and intermolecular hydrogen bonding respectively. Compound (P) on reaction with conc. HNO₃ yields a yellow compound 'C', a strong acid. The number of oxygen atoms is present in compound 'C' _____________.

The number of oxygens present in a nucleotide formed from a base, that is present only in RNA is ___________.

Let $$f(x) = {{x - 1} \over {x + 1}},\,x \in R - \{ 0, - 1,1\} $$. If $${f^{n + 1}}(x) = f({f^n}(x))$$ for all n $$\in$$ N, then $${f^6}(6) + {f^7}(7)$$ is equal to :

Let $$A = \left\{ {z \in C:\left| {{{z + 1} \over {z - 1}}} \right| < 1} \right\}$$ and $$B = \left\{ {z \in C:\arg \left( {{{z - 1} \over {z + 1}}} \right) = {{2\pi } \over 3}} \right\}$$. Then A $$\cap$$ B is :

The ordered pair (a, b), for which the system of linear equations

3x $$-$$ 2y + z = b

5x $$-$$ 8y + 9z = 3

2x + y + az = $$-$$1

has no solution, is :

The remainder when (2021)²⁰²³ is divided by 7 is :

$$\mathop {\lim }\limits_{x \to {1 \over {\sqrt 2 }}} {{\sin ({{\cos }^{ - 1}}x) - x} \over {1 - \tan ({{\cos }^{ - 1}}x)}}$$ is equal to :

Let f, g : R $$\to$$ R be two real valued functions defined as $$f(x) = \left\{ {\matrix{ { - |x + 3|} & , & {x < 0} \cr {{e^x}} & , & {x \ge 0} \cr } } \right.$$ and $$g(x) = \left\{ {\matrix{ {{x^2} + {k_1}x} & , & {x < 0} \cr {4x + {k_2}} & , & {x \ge 0} \cr } } \right.$$, where k₁ and k₂ are real constants. If (gof) is differentiable at x = 0, then (gof) ($$-$$ 4) + (gof) (4) is equal to :

The sum of the absolute minimum and the absolute maximum values of the

function f(x) = |3x $$-$$ x² + 2| $$-$$ x in the interval [$$-$$1, 2] is :

The area bounded by the curve y = |x² $$-$$ 9| and the line y = 3 is :

Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle. Then the area of $$\Delta$$PQR is :

If the two lines $${l_1}:{{x - 2} \over 3} = {{y + 1} \over {-2}},\,z = 2$$ and $${l_2}:{{x - 1} \over 1} = {{2y + 3} \over \alpha } = {{z + 5} \over 2}$$ are perpendicular, then an angle between the lines l₂ and $${l_3}:{{1 - x} \over 3} = {{2y - 1} \over { - 4}} = {z \over 4}$$ is :

The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6.8. If M is the mean deviation of the numbers about the mean, then 25 M is equal to :

Let $$f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10$$, $$x \in [ - 1,1]$$. If [a, b] is the range of the function f, then 4a $$-$$ b is equal to :

The sum of the cubes of all the roots of the equation

$${x^4} - 3{x^3} - 2{x^2} + 3x + 1 = 0$$ is _________.

There are ten boys B₁, B₂, ......., B₁₀ and five girls G₁, G₂, ........, G₅ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both B₁ and B₂ together should not be the members of a group, is ___________.

Let f(x) = max {|x + 1|, |x + 2|, ....., |x + 5|}. Then $$\int\limits_{ - 6}^0 {f(x)dx} $$ is equal to __________.

Let the solution curve y = y(x) of the differential equation

$$(4 + {x^2})dy - 2x({x^2} + 3y + 4)dx = 0$$ pass through the origin. Then y(2) is equal to _____________.

If $${\sin ^2}(10^\circ )\sin (20^\circ )\sin (40^\circ )\sin (50^\circ )\sin (70^\circ ) = \alpha - {1 \over {16}}\sin (10^\circ )$$, then $$16 + {\alpha ^{ - 1}}$$ is equal to __________.

Let A = {n $$\in$$ N : H.C.F. (n, 45) = 1} and

Let B = {2k : k $$\in$$ {1, 2, ......., 100}}. Then the sum of all the elements of A $$\cap$$ B is ____________.

The value of the integral

$${{48} \over {{\pi ^4}}}\int\limits_0^\pi {\left( {{{3\pi {x^2}} \over 2} - {x^3}} \right){{\sin x} \over {1 + {{\cos }^2}x}}dx} $$ is equal to __________.

Let $$A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } } $$ and $$B = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\max \,\{ i,j\} } } $$. Then A + B is equal to _____________.

Let $$S = (0,2\pi ) - \left\{ {{\pi \over 2},{{3\pi } \over 4},{{3\pi } \over 2},{{7\pi } \over 4}} \right\}$$. Let $$y = y(x)$$, x $$\in$$ S, be the solution curve of the differential equation $${{dy} \over {dx}} = {1 \over {1 + \sin 2x}},\,y\left( {{\pi \over 4}} \right) = {1 \over 2}$$. If the sum of abscissas of all the points of intersection of the curve y = y(x) with the curve $$y = \sqrt 2 \sin x$$ is $${{k\pi } \over {12}}$$, then k is equal to _____________.

An expression for a dimensionless quantity P is given by $$P = {\alpha \over \beta }{\log _e}\left( {{{kt} \over {\beta x}}} \right)$$; where $$\alpha$$ and $$\beta$$ are constants, x is distance; k is Boltzmann constant and t is the temperature. Then the dimensions of $$\alpha$$ will be :

A person is standing in an elevator. In which situation, he experiences weight loss?

An object is thrown vertically upwards. At its maximum height, which of the following quantity becomes zero?

A ball is released from rest from point P of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point Q is A while angular position of point Q is $$\alpha$$ with respect to point P. Which of the following graphs represent the correct relation between A and $$\alpha$$ when ball goes from Q to R?

A thin circular ring of mass M and radius R is rotating with a constant angular velocity 2 rads^$$-$$1 in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass m be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in rads^$$-$$1).

The variation of acceleration due to gravity (g) with distance (r) from the center of the earth is correctly represented by :

(Given R = radius of earth)

Time period of a simple pendulum in a stationary lift is 'T'. If the lift accelerates with $${g \over 6}$$ vertically upwards then the time period will be :

(Where g = acceleration due to gravity)

A thermally insulated vessel contains an ideal gas of molecular mass M and ratio of specific heats 1.4. Vessel is moving with speed v and is suddenly brought to rest. Assuming no heat is lost to the surrounding and vessel temperature of the gas increases by :

(R = universal gas constant)

Two capacitors having capacitance C₁ and C₂ respectively are connected as shown in figure. Initially, capacitor C₁ is charged to a potential difference V volt by a battery. The battery is then removed and the charged capacitor C₁ is now connected to uncharged capacitor C₂ by closing the switch S. The amount of charge on the capacitor C₂, after equilibrium, is :

Given below two statements : One is labelled as Assertion (A) and other is labelled as Reason (R).

Assertion (A) : Non-polar materials do not have any permanent dipole moment.

Reason (R) : When a non-polar material is placed in an electric field, the centre of the positive charge distribution of it's individual atom or molecule coincides with the centre of the negative charge distribution.

In the light of above statements, choose the most appropriate answer from the options given below.

The magnetic flux through a coil perpendicular to its plane is varying according to the relation $$\phi = (5{t^3} + 4{t^2} + 2t - 5)$$ Weber. If the resistance of the coil is 5 ohm, then the induced current through the coil at t = 2 s will be,

An aluminium wire is stretched to make its length, 0.4% larger. The percentage change in resistance is :

A proton and an alpha particle of the same velocity enter in a uniform magnetic field which is acting perpendicular to their direction of motion. The ratio of the radii of the circular paths described by the alpha particle and proton is :

If Electric field intensity of a uniform plane electromagnetic wave is given as $$E = - 301.6\sin (kz - \omega t){\widehat a_x} + 452.4\sin (kz - \omega t){\widehat a_y}{V \over m}$$. Then magnetic intensity 'H' of this wave in Am^$$-$$1 will be :

[Given : Speed of light in vacuum $$c = 3 \times {10^8}$$ ms^$$-$$1, Permeability of vacuum $${\mu _0} = 4\pi \times {10^{ - 7}}$$ NA^$$-$$2]

In free space, an electromagnetic wave of 3 GHz frequency strikes over the edge of an object of size $${\lambda \over {100}}$$, where $$\lambda$$ is the wavelength of the wave in free space. The phenomenon, which happens there will be :

An electron with speed v and a photon with speed c have the same de-Broglie wavelength. If the kinetic energy and momentum of electron are E_e and p_e and that of photon are E_ph and p_ph respectively. Which of the following is correct?

The I-V characteristics of a p-n junction diode in forward bias is shown in the figure. The ratio of dynamic resistance, corresponding to forward bias voltage of 2 V and 4 V respectively, is :

A fighter jet is flying horizontally at a certain altitude with a speed of 200 ms^$$-$$1. When it passes directly overhead an anti-aircraft gun, a bullet is fired from the gun, at an angle $$\theta$$ with the horizontal, to hit the jet. If the bullet speed is 400 m/s, the value of $$\theta$$ will be ___________$$^\circ$$.

A ball of mass 0.5 kg is dropped from the height of 10 m. The height, at which the magnitude of velocity becomes equal to the magnitude of acceleration due to gravity, is ________ m. [Use g = 10 m/s²]

The elastic behaviour of material for linear stress and linear strain, is shown in the figure. The energy density for a linear strain of 5 $$\times$$ 10^$$-$$4 is __________ kJ/m³. Assume that material is elastic upto the linear strain of 5 $$\times$$ 10^$$-$$4.

The elongation of a wire on the surface of the earth is 10^$$-$$4 m. The same wire of same dimensions is elongated by 6 $$\times$$ 10^$$-$$5 m on another planet. The acceleration due to gravity on the planet will be ____________ ms^$$-$$2. (Take acceleration due to gravity on the surface of earth = 10 ms^$$-$$2)

A 10 $$\Omega$$, 20 mH coil carrying constant current is connected to a battery of 20 V through a switch. Now after switch is opened current becomes zero in 100 $$\mu$$s. The average e.m.f. induced in the coil is ____________ V.

A light ray is incident, at an incident angle $$\theta$$₁, on the system of tow plane mirrors M₁ and M₂ having an inclination angle 75$$^\circ$$ between them (as shown in figure). After reflecting from mirror M₁ it gets reflected back by the mirror M₂ with an angle of reflection 30$$^\circ$$. The total deviation of the ray will be _____________ degree.

In a vernier callipers, each cm on the main scale is divided into 20 equal parts. If tenth vernier scale division coincides with nineth main scale division. Then the value of vernier constant will be _________ $$\times$$ 10^$$-$$2 mm.

As per the given circuit, the value of current through the battery will be ____________ A.

A 110 V, 50 Hz, AC source is connected in the circuit (as shown in figure). The current through the resistance 55 $$\Omega$$, at resonance in the circuit, will be __________ A.

An ideal fluid of density 800 kgm^$$-$$3, flows smoothly through a bent pipe (as shown in figure) that tapers in cross-sectional area from a to $${a \over 2}$$. The pressure difference between the wide and narrow sections of pipe is 4100 Pa. At wider section, the velocity of fluid is $${{\sqrt x } \over 6}$$ ms^$$-$$1 for x = ___________. (Given g = 10 ms^$$-$$2)