'A' in the given reaction is

When the hydrogen ion concentration [H$$^+$$] changes by a factor of 1000, the value of pH of the solution __________

Match List I with List II

Choose the correct answer from the options given below :

What is the mass ratio of ethylene glycol ($$\mathrm{C_2H_6O_2}$$, molar mass = 62 g/mol) required for making 500 g of 0.25 molal aqueous solution and 250 mL of 0.25 molar aqueous solution?

Find out the major product from the following reaction.

The isomeric deuterated bromide with molecular formula $$\mathrm{C_4H_8DBr}$$ having two chiral carbon atoms is

Match List I with List II

Choose the correct answer from the options given below :

A. Ammonium salts produce haze in atmosphere.

B. Ozone gets produced when atmospheric oxygen reacts with chlorine radicals.

C. Polychlorinated biphenyls act as cleansing solvents.

D. 'Blue baby' syndrome occurs due to the presence of excess of sulphate ions in water.

Choose the correct answer from the options given below :

Statement I : Dipole moment is a vector quantity and by convention it is depicted by a small arrow with tail on the negative centre and head pointing towards the positive centre.

Statement II : The crossed arrow of the dipole moment symbolizes the direction of the shift of charges in the molecules.

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

Match List I with List II

Choose the correct answer from the options given below :

A chloride salt solution acidified with dil.HNO$$_3$$ gives a curdy white precipitate, [A], on addition of AgNO$$_3$$. [A] on treatment with NH$$_4$$OH gives a clear solution B. A and B are respectively :

Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R

Assertion A : Butylated hydroxy anisole when added to butter increases its shelf life.

Reason R : Butylated hydroxy anisole is more reactive towards oxygen than food.

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

Potassium dichromate acts as a strong oxidizing agent in acidic solution. During this process, the oxidation state changes from :

28.0 L of CO$$_2$$ is produced on complete combustion of 16.8 L gaseous mixture of ethene and methane at 25$$^\circ$$C and 1 atm. Heat evolved during the combustion process is ___________ kJ.

Given : $$\mathrm{\Delta H_c~(CH_4)=-900~kJ~mol^{-1}}$$

$$\mathrm{\Delta H_c~(C_2H_4)=-1400~kJ~mol^{-1}}$$

The number of pairs of the solutions having the same value of the osmotic pressure from the following is _________.

(Assume 100% ionization)

A. 0.500 $$\mathrm{M~C_2H_5OH~(aq)}$$ and 0.25 $$\mathrm{M~KBr~(aq)}$$

B. 0.100 $$\mathrm{M~K_4[Fe(CN)_6]~(aq)}$$ and 0.100 $$\mathrm{M~FeSO_4(NH_4)_2SO_4~(aq)}$$

C. 0.05 $$\mathrm{M~K_4[Fe(CN)_6]~(aq)}$$ and 0.25 $$\mathrm{M~NaCl~(aq)}$$

D. 0.15 $$\mathrm{M~NaCl~(aq)}$$ and 0.1 $$\mathrm{M~BaCl_2~(aq)}$$

E. 0.02 $$\mathrm{M~KCl.MgCl_2.6H_2O~(aq)}$$ and 0.05 $$\mathrm{M~KCl~(aq)}$$

A first order reaction has the rate constant, $$\mathrm{k=4.6\times10^{-3}~s^{-1}}$$. The number of correct statement/s from the following is/are __________

Given : $$\mathrm{\log3=0.48}$$

A. Reaction completes in 1000 s.

B. The reaction has a half-life of 500 s.

C. The time required for 10% completion is 25 times the time required for 90% completion.

D. The degree of dissociation is equal to ($$\mathrm{1-e^{-kt}}$$)

E. The rate and the rate constant have the same unit.

$$Pt(s)|{H_2}(g)(1\,bar)|{H^ + }(aq)(1\,M)||{M^{3 + }}(aq),{M^ + }(aq)|Pt(s)$$

The $$\mathrm{E_{cell}}$$ for the given cell is 0.1115 V at 298 K when $${{\left[ {{M^ + }(aq)} \right]} \over {\left[ {{M^{3 + }}(aq)} \right]}} = {10^a}$$

The value of $$a$$ is ____________

Given : $$\mathrm{E_{{M^{3 + }}/{M^ + }}^\theta = 0.2}$$ V

$${{2.303RT} \over F} = 0.059V$$

The number of given orbitals which have electron density along the axis is _________

$$\mathrm{p_x,p_y,p_z,d_{xy},d_{yz},d_{xz},d_{z^2},d_{x^2-y^2}}$$

Total number of moles of AgCl precipitated on addition of excess of AgNO$$_3$$ to one mole each of the following complexes $$\mathrm{[Co(NH_3)_4Cl_2]Cl,[Ni(H_2O)_6]Cl_2,[Pt(NH_3)_2Cl_2]}$$ and $$\mathrm{[Pd(NH_3)_4]Cl_2}$$ is ___________.

Number of compounds giving (i) red colouration with ceric ammonium nitrate and also (ii) positive iodoform test from the following is ___________

Number of hydrogen atoms per molecule of a hydrocarbon A having 85.8% carbon is __________

(Given : Molar mass of A = 84 g mol$$^{-1}$$)

The equations of two sides of a variable triangle are $$x=0$$ and $$y=3$$, and its third side is a tangent to the parabola $$y^2=6x$$. The locus of its circumcentre is :

The foot of perpendicular of the point (2, 0, 5) on the line $${{x + 1} \over 2} = {{y - 1} \over 5} = {{z + 1} \over { - 1}}$$ is ($$\alpha,\beta,\gamma$$). Then, which of the following is NOT correct?

The number of functions

$$f:\{ 1,2,3,4\} \to \{ a \in Z|a| \le 8\} $$

satisfying $$f(n) + {1 \over n}f(n + 1) = 1,\forall n \in \{ 1,2,3\} $$ is

Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that $$N-2,\sqrt{3N},N+2$$ are in geometric progression be $$\frac{k}{48}$$. Then the value of k is :

If the function $$f(x) = \left\{ {\matrix{ {(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 < x < {\pi \over 2}} \cr \mu & , & {x = {\pi \over 2}} \cr e^{{{\cot 6x} \over {{}\cot 4x}}} & , & {{\pi \over 2} < x < \pi } \cr } } \right.$$

is continuous at $$x = {\pi \over 2}$$, then $$9\lambda + 6{\log _e}\mu + {\mu ^6} - {e^{6\lambda }}$$ is equal to

Let A, B, C be 3 $$\times$$ 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements

(S1) A$$^{13}$$ B$$^{26}$$ $$-$$ B$$^{26}$$ A$$^{13}$$ is symmetric

(S2) A$$^{26}$$ C$$^{13}$$ $$-$$ C$$^{13}$$ A$$^{26}$$ is symmetric

Then,

The shortest distance between the lines $$x+1=2y=-12z$$ and $$x=y+2=6z-6$$ is :

Let T and C respectively be the transverse and conjugate axes of the hyperbola $$16{x^2} - {y^2} + 64x + 4y + 44 = 0$$. Then the area of the region above the parabola $${x^2} = y + 4$$, below the transverse axis T and on the right of the conjugate axis C is :

Let the function $$f(x) = 2{x^3} + (2p - 7){x^2} + 3(2p - 9)x - 6$$ have a maxima for some value of $$x < 0$$ and a minima for some value of $$x > 0$$. Then, the set of all values of p is

Let $$z$$ be a complex number such that $$\left| {{{z - 2i} \over {z + i}}} \right| = 2,z \ne - i$$. Then $$z$$ lies on the circle of radius 2 and centre :

The integral $$16\int\limits_1^2 {{{dx} \over {{x^3}{{\left( {{x^2} + 2} \right)}^2}}}} $$ is equal to

Let $$f:\mathbb{R}\to\mathbb{R}$$ be a function defined by $$f(x) = {\log _{\sqrt m }}\{ \sqrt 2 (\sin x - \cos x) + m - 2\} $$, for some $$m$$, such that the range of $$f$$ is [0, 2]. Then the value of $$m$$ is _________

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is :

Let $$y=y(t)$$ be a solution of the differential equation $${{dy} \over {dt}} + \alpha y = \gamma {e^{ - \beta t}}$$ where, $$\alpha > 0,\beta > 0$$ and $$\gamma > 0$$. Then $$\mathop {\lim }\limits_{t \to \infty } y(t)$$

Let $$A = \left[ {\matrix{ {{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr {{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}} \cr } } \right]$$ and $$B = \left[ {\matrix{ 1 & { - i} \cr 0 & 1 \cr } } \right]$$, where $$i = \sqrt { - 1} $$. If $$\mathrm{M=A^T B A}$$, then the inverse of the matrix $$\mathrm{AM^{2023}A^T}$$ is

$$\sum\limits_{k = 0}^6 {{}^{51 - k}{C_3}} $$ is equal to :

Let $$f(x) = 2{x^n} + \lambda ,\lambda \in R,n \in N$$, and $$f(4) = 133,f(5) = 255$$. Then the sum of all the positive integer divisors of $$(f(3) - f(2))$$ is

25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $$\frac{k}{10}%$$. Then the value of k is __________.

A triangle is formed by X-axis, Y-axis and the line $$3x+4y=60$$. Then the number of points P(a, b) which lie strictly inside the triangle, where a is an integer and b is a multiple of a, is ____________.

The remainder when (2023)$$^{2023}$$ is divided by 35 is __________.

If $$\int\limits_{{1 \over 3}}^3 {|{{\log }_e}x|dx = {m \over n}{{\log }_e}\left( {{{{n^2}} \over e}} \right)} $$, where m and n are coprime natural numbers, then $${m^2} + {n^2} - 5$$ is equal to _____________.

Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is ____________

For the two positive numbers $$a,b,$$ if $$a,b$$ and $$\frac{1}{18}$$ are in a geometric progression, while $$\frac{1}{a},10$$ and $$\frac{1}{b}$$ are in an arithmetic progression, then $$16a+12b$$ is equal to _________.

If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line $${{x - 1} \over 2} = {{y + 1} \over { - 1}} = {{z - 2} \over 0}$$ is $$\alpha$$, then 28$$\alpha^2$$ is equal to ____________.

Let $$\alpha \in\mathbb{R}$$ and let $$\alpha,\beta$$ be the roots of the equation $${x^2} + {60^{{1 \over 4}}}x + a = 0$$. If $${\alpha ^4} + {\beta ^4} = - 30$$, then the product of all possible values of $$a$$ is ____________.

The light rays from an object have been reflected towards an observer from a standard flat mirror, the image observed by the observer are :-

A. Real

B. Erect

C. Smaller in size then object

D. Laterally inverted

Choose the most appropriate answer from the options given below :

A particle executes simple harmonic motion between $$x=-A$$ and $$x=+A$$. If time taken by particle to go from $$x=0$$ to $$\frac{A}{2}$$ is 2 s; then time taken by particle in going from $$x=\frac{A}{2}$$ to A is

The distance travelled by a particle is related to time t as $$x=4\mathrm{t}^2$$. The velocity of the particle at t=5s is :-

Match List I with List II

Choose the correct answer from the options given below :

A wire of length 1m moving with velocity 8 m/s at right angles to a magnetic field of 2T. The magnitude of induced emf, between the ends of wire will be __________.

Consider a block kept on an inclined plane (incline at 45$$^\circ$$) as shown in the figure. If the force required to just push it up the incline is 2 times the force required to just prevent it from sliding down, the coefficient of friction between the block and inclined plane($$\mu$$) is equal to :

Match List I with List II

Choose the correct answer from the options given below :

Every planet revolves around the sun in an elliptical orbit :-

A. The force acting on a planet is inversely proportional to square of distance from sun.

B. Force acting on planet is inversely proportional to product of the masses of the planet and the sun.

C. The Centripetal force acting on the planet is directed away from the sun.

D. The square of time period of revolution of planet around sun is directly proportional to cube of semi-major axis of elliptical orbit.

Choose the correct answer from the options given below :

Two objects are projected with same velocity 'u' however at different angles $$\alpha$$ and $$\beta$$ with the horizontal. If $$\alpha+\beta=90^\circ$$, the ratio of horizontal range of the first object to the 2nd object will be :

The graph between two temperature scales P and Q is shown in the figure. Between upper fixed point and lower fixed point there are 150 equal divisions of scale P and 100 divisions on scale Q. The relationship for conversion between the two scales is given by :-

A point charge of 10 $$\mu$$C is placed at the origin. At what location on the X-axis should a point charge of 40 $$\mu$$C be placed so that the net electric field is zero at $$x=2$$cm on the X-axis?

The resistance of a wire is 5 $$\Omega$$. It's new resistance in ohm if stretched to 5 times of it's original length will be :

Match List I with List II

Choose the correct answer from the options given below :

Statement I : When a Si sample is doped with Boron, it becomes P type and when doped by Arsenic it becomes N-type semi conductor such that P-type has excess holes and N-type has excess electrons.

Statement II : When such P-type and N-type semi-conductors, are fused to make a junction, a current will automatically flow which can be detected with an externally connected ameter.

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

According to law of equipartition of energy the molar specific heat of a diatomic gas at constant volume where the molecule has one additional vibrational mode is :-

A body of mass is taken from earth surface to the height h equal to twice the radius of earth (R$$_e$$), the increase in potential energy will be :

(g = acceleration due to gravity on the surface of Earth)

For a moving coil galvanometer, the deflection in the coil is 0.05 rad when a current of 10 mA is passes through it. If the torsional constant of suspension wire is $$4.0\times10^{-5}\mathrm{N~m~rad^{-1}}$$, the magnetic field is 0.01T and the number of turns in the coil is 200, the area of each turn (in cm$$^2$$) is :

Given below are two statements :

Statement I : Stopping potential in photoelectric effect does not depend on the power of the light source.

Statement II : For a given metal, the maximum kinetic energy of the photoelectron depends on the wavelength of the incident light.

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

The energy levels of an atom is shown in figure.

Which one of these transitions will result in the emission of a photon of wavelength 124.1 nm?

Given (h = 6.62 $$\times$$ 10$$^{-34}$$ Js)

Two long parallel wires carrying currents 8A and 15A in opposite directions are placed at a distance of 7 cm from each other. A point P is at equidistant from both the wires such that the lines joining the point P to the wires are perpendicular to each other. The magnitude of magnetic field at P is _____________ $$\times~10^{-6}$$ T.

(Given : $$\sqrt2=1.4$$)

A capacitor has capacitance 5$$\mu$$F when it's parallel plates are separated by air medium of thickness d. A slab of material of dielectric constant 1.5 having area equal to that of plates but thickness $$\frac{d}{2}$$ is inserted between the plates. Capacitance of the capacitor in the presence of slab will be __________ $$\mu$$F.

An object is placed on the principal axis of convex lens of focal length 10cm as shown. A plane mirror is placed on the other side of lens at a distance of 20 cm. The image produced by the plane mirror is 5cm inside the mirror. The distance of the object from the lens is ___________ cm.

Two cells are connected between points A and B as shown. Cell 1 has emf of 12 V and internal resistance of 3$$\Omega$$. Cell 2 has emf of 6V and internal resistance of 6$$\Omega$$. An external resistor R of 4$$\Omega$$ is connected across A and B. The current flowing through R will be ____________ A.

A series LCR circuit is connected to an AC source of 220 V, 50 Hz. The circuit contains a resistance R = 80$$\Omega$$, an inductor of inductive reactance $$\mathrm{X_L=70\Omega}$$, and a capacitor of capacitive reactance $$\mathrm{X_C=130\Omega}$$. The power factor of circuit is $$\frac{x}{10}$$. The value of $$x$$ is :

If a solid sphere of mass 5 kg and a disc of mass 4 kg have the same radius. Then the ratio of moment of inertia of the disc about a tangent in its plane to the moment of inertia of the sphere about its tangent will be $$\frac{x}{7}$$. The value of $$x$$ is ___________.

A body of mass 1 kg collides head on elastically with a stationary body of mass 3 kg. After collision, the smaller body reverses its direction of motion and moves with a speed of 2 m/s. The initial speed of the smaller body before collision is ___________ ms$$^{-1}$$.

A spherical drop of liquid splits into 1000 identical spherical drops. If u$$_\mathrm{i}$$ is the surface energy of the original drop and u$$_\mathrm{f}$$ is the total surface energy of the resulting drops, the (ignoring evaporation), $${{{u_f}} \over {{u_i}}} = \left( {{{10} \over x}} \right)$$. Then value of x is ____________ :