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

Assertion (A) : At 10$^\circ$C, the density of a 5 M solution of KCl [atomic masses of K & Cl are 39 & 35.5 g mol^$-$1 respectively], is 'x' g ml^$-$1. The solution is cooled to $-$21$^\circ$C. The molality of the solution will remain unchanged.

Reason (R) : The molality of a solution does not change with temperature as mass remains unaffected with temperature.

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

Based upon VSEPR theory, match the shape (geometry) of the molecules in List-I with the molecules in List-II and select the most appropriate option.

Which of the following reactions will yield benzaldehyde as a product?

L-isomer of a compound 'A' (C₄H₈O₄) gives a positive test with [Ag(NH₃)₂]⁺. Treatment of 'A' with acetic anhydride yields triacetate derivative. Compound 'A' produces an optically active compound (B) and an optically inactive compound (C) on treatment with bromine water and HNO₃ respectively. Compound (A) is :

2 g of a non-volatile non-electrolyte solute is dissolved in 200 g of two different solvents A and B whose ebullioscopic constants are in the ratio of 1 : 8. The elevation in boiling points of A and B are in the ratio ${x \over y}$ (x : y). The value of y is ______________. (Nearest integer)

Acidified potassium permanganate solution oxidises oxalic acid. The spin-only magnetic moment of the manganese product formed from the above reaction is ____________ B.M. (Nearest integer)

Total number of possible stereoisomers of dimethyl cyclopentane is ____________.

The area of the polygon, whose vertices are the non-real roots of the equation $\overline z = i{z^2}$ is :

The number of distinct real roots of x⁴ $-$ 4x + 1 = 0 is :

If two straight lines whose direction cosines are given by the relations $l + m - n = 0$, $3{l^2} + {m^2} + cnl = 0$ are parallel, then the positive value of c is :

The major product of the following reaction is :

Given below are two statements :

Statement I : In Hofmann degradation reaction, the migration of only an alkyl group takes place from carbonyl carbon of the amide to the nitrogen atom.

Statement II : The group is migrated in Hofmann degradation reaction to electron deficient atom.

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

If the uncertainty in velocity and position of a minute particle in space are, 2.4 $\times$ 10^$-$26 (m s^$-$1) and 10^$-$7 (m) respectively. The mass of the particle in g is ____________. (Nearest integer)

(Given : h = 6.626 $\times$ 10^$-$34 Js)

2NOCl(g) $\rightleftharpoons$ 2NO(g) + Cl₂(g)

In an experiment, 2.0 moles of NOCl was placed in a one-litre flask and the concentration of NO after equilibrium established, was found to be 0.4 mol/L. The equilibrium constant at 30$^\circ$C is ______________ $\times$ 10^$-$4.

Two elements A and B which form 0.15 moles of A₂B and AB₃ type compounds. If both A₂B and AB₃ weigh equally, then the atomic weight of A is _____________ times of atomic weight of B.

If ${{dy} \over {dx}} + {{{2^{x - y}}({2^y} - 1)} \over {{2^x} - 1}} = 0$, x, y > 0, y(1) = 1, then y(2) is equal to :

In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie on the line x + 3y = 7. If ($\alpha$, $\beta$) is the centroid of $\Delta$ABC, then 15($\alpha$ + $\beta$) is equal to :

Let the eccentricity of an ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, $a > b$, be ${1 \over 4}$. If this ellipse passes through the point $\left( { - 4\sqrt {{2 \over 5}} ,3} \right)$, then ${a^2} + {b^2}$ is equal to :

${\sin ^1}\left( {\sin {{2\pi } \over 3}} \right) + {\cos ^{ - 1}}\left( {\cos {{7\pi } \over 6}} \right) + {\tan ^{ - 1}}\left( {\tan {{3\pi } \over 4}} \right)$ is equal to :

Let f : R $\to$ R be a function defined by $f(x) = {{2{e^{2x}}} \over {{e^{2x}} + e}}$. Then $f\left( {{1 \over {100}}} \right) + f\left( {{2 \over {100}}} \right) + f\left( {{3 \over {100}}} \right) + \,\,\,.....\,\,\, + \,\,\,f\left( {{{99} \over {100}}} \right)$ is equal to ______________.

Which of the following will have maximum stabilization due to crystal field?

The rate constant for a first order reaction is given by the following equation:

$\ln k = 33.24 - {{2.0 \times {{10}^4}\,K} \over T}$

The activation energy for the reaction is given by ____________ kJ mol^$-$1. (In nearest integer) (Given : R = 8.3 J K^$-$1 mol^$-$1)

If ${\cos ^{ - 1}}\left( {{y \over 2}} \right) = {\log _e}{\left( {{x \over 5}} \right)^5},\,|y| < 2$, then :

Five numbers ${x_1},{x_2},{x_3},{x_4},{x_5}$ are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order $({x_1} < {x_2} < {x_3} < {x_4} < {x_5})$. The probability that ${x_2} = 7$ and ${x_4} = 11$ is :

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is _____________.

If the coefficient of x¹⁰ in the binomial expansion of ${\left( {{{\sqrt x } \over {{5^{{1 \over 4}}}}} + {{\sqrt 5 } \over {{x^{{1 \over 3}}}}}} \right)^{60}}$ is ${5^k}\,.\,l$, where l, k $\in$ N and l is co-prime to 5, then k is equal to _____________.

${A_1} = \left\{ {(x,y):|x| \le {y^2},|x| + 2y \le 8} \right\}$ and

${A_2} = \left\{ {(x,y):|x| + |y| \le k} \right\}$. If 27 (Area A₁) = 5 (Area A₂), then k is equal to :

Match List-I with List-II.

Choose the correct answer from the options given below :

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

Assertion (A) : The ionic radii of O^2$-$ and Mg²⁺ are same.

Reason (R) : Both O^2$-$ and Mg²⁺ are isoelectronic species.

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

'A' and 'B' respectively are:

The limiting molar conductivities of NaI, NaNO₃ and AgNO₃ are 12.7, 12.0 and 13.3 mS m² mol^$-$1, respectively (all at 25$^\circ$C). The limiting molar conductivity of AgI at this temperature is ____________ mS m² mol^$-$1.

The number of statements correct from the following for Copper (at. no. 29) is/are ____________.

(A) Cu(II) complexes are always paramagnetic.

(B) Cu(I) complexes are generally colourless

(C) Cu(I) is easily oxidized

(D) In Fehling solution, the active reagent has Cu(I)

Let the system of linear equations
$x + 2y + z = 2$,
$\alpha x + 3y - z = \alpha$,
$- \alpha x + y + 2z = - \alpha$
be inconsistent. Then $\alpha$ is equal to :

$x = \sum\limits_{n = 0}^\infty {{a^n},y = \sum\limits_{n = 0}^\infty {{b^n},z = \sum\limits_{n = 0}^\infty {{c^n}} } }$, where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc $\ne$ 0, then :

Let a be an integer such that $\mathop {\lim }\limits_{x \to 7} {{18 - [1 - x]} \over {[x - 3a]}}$ exists, where [t] is greatest integer $\le$ t. Then a is equal to :

The value of the integral

$\int\limits_{ - 2}^2 {{{|{x^3} + x|} \over {({e^{x|x|}} + 1)}}dx}$ is equal to :

Let $\overrightarrow a = \widehat i + \widehat j - \widehat k$ and $\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$. Then the number of vectors $\overrightarrow b$ such that $\overrightarrow b \times \overrightarrow c = \overrightarrow a$ and $|\overrightarrow b | \in$ {1, 2, ........, 10} is :

The value of $\cos \left( {{{2\pi } \over 7}} \right) + \cos \left( {{{4\pi } \over 7}} \right) + \cos \left( {{{6\pi } \over 7}} \right)$ is equal to :

If the sum of all the roots of the equation

${e^{2x}} - 11{e^x} - 45{e^{ - x}} + {{81} \over 2} = 0$ is ${\log _e}p$, then p is equal to ____________.

A rectangle R with end points of one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x $-$ y + 4 = 0, then the area of R is ____________.

A system of two blocks of masses m = 2 kg and M = 8 kg is placed on a smooth table as shown in figure. The coefficient of static friction between two blocks is 0.5. The maximum horizontal force F that can be applied to the block of mass M so that the blocks move together will be :

Two blocks of masses 10 kg and 30 kg are placed on the same straight line with coordinates (0, 0) cm and (x, 0) cm respectively. The block of 10 kg is moved on the same line through a distance of 6 cm towards the other block. The distance through which the block of 30 kg must be moved to keep the position of centre of mass of the system unchanged is :

The susceptibility of a paramagnetic material is 99. The permeability of the material in Wb/A-m, is :

[Permeability of free space $\mu$₀ = 4$\pi$ $\times$ 10^$-$7 Wb/A-m]

Identify the correct Logic Gate for the following output (Y) of two inputs A and B.

A capacitor of capacitance 50 pF is charged by 100 V source. It is then connected to another uncharged identical capacitor. Electrostatic energy loss in the process is ___________ nJ.

A silver wire has a mass (0.6 $\pm$ 0.006) g, radius (0.5 $\pm$ 0.005) mm and length (4 $\pm$ 0.04) cm. The maximum percentage error in the measurement of its density will be :

What percentage of kinetic energy of a moving particle is transferred to a stationary particle when it strikes the stationary particle of 5 times its mass?

(Assume the collision to be head-on elastic collision)

A force of 10 N acts on a charged particle placed between two plates of a charged capacitor. If one plate of capacitor is removed, then the force acting on that particle will be.

A hydrogen atom in its ground state absorbs 10.2 eV of energy. The angular momentum of electron of the hydrogen atom will increase by the value of :

(Given, Planck's constant = 6.6 $\times$ 10^$-$34 Js).

A beam of monochromatic light is used to excite the electron in Li^{+ +} from the first orbit to the third orbit. The wavelength of monochromatic light is found to be x $\times$ 10^$-$10 m. The value of x is ___________.

[Given hc = 1242 eV nm]

A pendulum of length 2 m consists of a wooden bob of mass 50 g. A bullet of mass 75 g is fired towards the stationary bob with a speed v. The bullet emerges out of the bob with a speed ${v \over 3}$ and the bob just completes the vertical circle. The value of v is ___________ ms^$-$1. (if g = 10 m/s²).

A projectile is launched at an angle '$\alpha$' with the horizontal with a velocity 20 ms^$-$1. After 10 s, its inclination with horizontal is '$\beta$'. The value of tan$\beta$ will be : (g = 10 ms^$-$2).

A girl standing on road holds her umbrella at 45$^\circ$ with the vertical to keep the rain away. If she starts running without umbrella with a speed of 15$\sqrt2$ kmh^$-$1, the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is :

A 72 $\Omega$ galvanometer is shunted by a resistance of 8 $\Omega$. The percentage of the total current which passes through the galvanometer is :

Given below are two statements :

Statement I : The law of gravitation holds good for any pair of bodies in the universe.

Statement II : The weight of any person becomes zero when the person is at the centre of the earth.

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

The velocity of a small ball of mass 'm' and density d₁, when dropped in a container filled with glycerin, becomes constant after some time. If the density of glycerin is d₂, then the viscous force acting on the ball, will be :

An $\alpha$ particle and a carbon 12 atom has same kinetic energy K. The ratio of their de-Broglie wavelengths $({\lambda _\alpha }:{\lambda _{C12}})$ is :

The displacement of simple harmonic oscillator after 3 seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is :

A mixture of hydrogen and oxygen has volume 2000 cm³, temperature 300 K, pressure 100 kPa and mass 0.76 g. The ratio of number of moles of hydrogen to number of moles of oxygen in the mixture will be:

[Take gas constant R = 8.3 JK^$-$1mol^$-$1]

A 220 V, 50 Hz AC source is connected to a 25 V, 5 W lamp and an additional resistance R in series (as shown in figure) to run the lamp at its peak brightness, then the value of R (in ohm) will be _____________.

The current density in a cylindrical wire of radius 4 mm is 4 $\times$ 10⁶ Am^$-$2. The current through the outer portion of the wire between radial distances ${R \over 2}$ and R is ____________ $\pi$ A.

The current flowing through an ac circuit is given by

I = 5 sin(120$\pi$t)A

How long will the current take to reach the peak value starting from zero?

Match List-I with List-II :

Choose the correct answer from the options given below :

Consider a light ray travelling in air is incident into a medium of refractive index $\sqrt{2n}$. The incident angle is twice that of refracting angle. Then, the angle of incidence will be :

In Young's double slit experiment the two slits are 0.6 mm distance apart. Interference pattern is observed on a screen at a distance 80 cm from the slits. The first dark fringe is observed on the screen directly opposite to one of the slits. The wavelength of light will be ____________ nm.

The area of cross-section of a large tank is 0.5 m². It has a narrow opening near the bottom having area of cross-section 1 cm². A load of 25 kg is applied on the water at the top in the tank. Neglecting the speed of water in the tank, the velocity of the water, coming out of the opening at the time when the height of water level in the tank is 40 cm above the bottom, will be ___________ cms^$-$1. [Take g = 10 ms^$-$2]