$$\mathrm{SO}_{2} \mathrm{Cl}_{2}$$ on reaction with excess of water results into acidic mixture

$$\mathrm{SO}_{2} \mathrm{Cl}_{2}+2 \mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{H}_{2} \mathrm{SO}_{4}+2 \mathrm{HCl}$$

16 moles of $$\mathrm{NaOH}$$ is required for the complete neutralisation of the resultant acidic mixture. The number of moles of $$\mathrm{SO}_{2} \mathrm{Cl}_{2}$$ used is :

Which of the following sets of quantum numbers is not allowed?

The depression in freezing point observed for a formic acid solution of concentration $$0.5 \mathrm{~mL} \mathrm{~L}^{-1}$$ is $$0.0405^{\circ} \mathrm{C}$$. Density of formic acid is $$1.05 \mathrm{~g} \mathrm{~mL}^{-1}$$. The Van't Hoff factor of the formic acid solution is nearly : (Given for water $$\mathrm{k}_{\mathrm{f}}=1.86\, \mathrm{k} \,\mathrm{kg}\,\mathrm{mol}^{-1}$$ )

$$20 \mathrm{~mL}$$ of $$0.1\, \mathrm{M} \,\mathrm{NH}_{4} \mathrm{OH}$$ is mixed with $$40 \mathrm{~mL}$$ of $$0.05 \mathrm{M} \mathrm{HCl}$$. The $$\mathrm{pH}$$ of the mixture is nearest to :

(Given : $$\mathrm{K}_{\mathrm{b}}\left(\mathrm{NH}_{4} \mathrm{OH}\right)=1 \times 10^{-5}, \log 2=0.30, \log 3=0.48, \log 5=0.69, \log 7=0.84, \log 11= 1.04)$$

The IUPAC nomenclature of an element with electronic configuration [Rn] $$5 \mathrm{f}^{14} 6 \mathrm{d}^{1} 7 \mathrm{s}^{2}$$ is :

A compound ‘$$\mathrm{A}$$’ on reaction with ‘$$\mathrm{X}$$’ and ‘$$\mathrm{Y}$$’ produces the same major product but different by product '$$a$$' and '$$b^{\prime}$$. Oxidation of '$$a$$' gives a substance produced by ants.

'X' and 'Y' respectively are

Most stable product of the following reaction is:

Which one of the following reactions does not represent correct combination of substrate and product under the given conditions?

An organic compound 'A' on reaction with NH₃ followed by heating gives compound B. Which on further strong heating gives compound C (C₈H₅NO₂). Compound C on sequential reaction with ethanolic KOH, alkyl chloride and hydrolysis with alkali gives a primary amine. The compound A is :

During the denaturation of proteins, which of these structures will remain intact?

Given below are two statements :

Statement I : On heating with $$\mathrm{KHSO}_{4}$$, glycerol is dehydrated and acrolein is formed.

Statement II : Acrolein has fruity odour and can be used to test glycerol's presence.

Choose the correct option.

Among the following species

$$\mathrm{N}_{2}, \mathrm{~N}_{2}^{+}, \mathrm{N}_{2}^{-}, \mathrm{N}_{2}^{2-}, \mathrm{O}_{2}, \mathrm{O}_{2}^{+}, \mathrm{O}_{2}^{-}, \mathrm{O}_{2}^{2-}$$

the number of species showing diamagnesim is _______________.

The enthalpy of combustion of propane, graphite and dihydrogen at $$298 \mathrm{~K}$$ are $$-2220.0 \mathrm{~kJ} \mathrm{~mol}^{-1},-393.5 \mathrm{~kJ} \mathrm{~mol}^{-1}$$ and $$-285.8 \mathrm{~kJ} \mathrm{~mol}^{-1}$$ respectively. The magnitude of enthalpy of formation of propane $$\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)$$ is _______________ $$\mathrm{kJ} \,\mathrm{mol}^{-1}$$. (Nearest integer)

The cell potential for $$\mathrm{Zn}\left|\mathrm{Zn}^{2+}(\mathrm{aq})\right|\left|\mathrm{Sn}^{x+}\right| \mathrm{Sn}$$ is $$0.801 \mathrm{~V}$$ at $$298 \mathrm{~K}$$. The reaction quotient for the above reaction is $$10^{-2}$$. The number of electrons involved in the given electrochemical cell reaction is ____________.

$$\left(\right.$$ Given $$: \mathrm{E}_{\mathrm{Zn}^{2+} \mid \mathrm{Zn}}^{\mathrm{o}}=-0.763 \mathrm{~V}, \mathrm{E}_{\mathrm{Sn}^{x+} \mid \mathrm{Sn}}^{\mathrm{o}}=+0.008 \mathrm{~V}$$ and $$\left.\frac{2.303 \mathrm{RT}}{\mathrm{F}}=0.06 \mathrm{~V}\right)$$

The half life for the decomposition of gaseous compound $$\mathrm{A}$$ is $$240 \mathrm{~s}$$ when the gaseous pressure was 500 Torr initially. When the pressure was 250 Torr, the half life was found to be $$4.0$$ min. The order of the reaction is ______________. (Nearest integer)

Consider the following metal complexes :

$$\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}$$

$$\left[\mathrm{CoCl}\left(\mathrm{NH}_{3}\right)_{5}\right]^{2+}$$

$$\left[\mathrm{Co}(\mathrm{CN})_{6}\right]^{3-}$$

$$\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5}\left(\mathrm{H}_{2} \mathrm{O}\right)\right]^{3+}$$

The spin-only magnetic moment value of the complex that absorbes light with shortest wavelength is _____________ B. M. (Nearest integer)

Among Co³⁺, Ti²⁺, V²⁺ and Cr²⁺ ions, one if used as a reagent cannot liberate H₂ from dilute mineral acid solution, its spin-only magnetic moment in gaseous state is ___________ B.M. (Nearest integer)

While estimating the nitrogen present in an organic compound by Kjeldahl's method, the ammonia evolved from $$0.25 \mathrm{~g}$$ of the compound neutralized $$2.5 \mathrm{~mL}$$ of $$2 \,\mathrm{M} \,\mathrm{H}_{2} \mathrm{SO}_{4}$$. The percentage of nitrogen present in organic compound is ______________.

The number of sp³ hybridised carbons in an acyclic neutral compound with molecular formula C₄H₅N is ___________.

In the given reaction

The number of chiral carbon/s in product A is ___________.

The total number of functions,

$$ f:\{1,2,3,4\} \rightarrow\{1,2,3,4,5,6\} $$ such that $$f(1)+f(2)=f(3)$$, is equal to :

If $$\alpha, \beta, \gamma, \delta$$ are the roots of the equation $$x^{4}+x^{3}+x^{2}+x+1=0$$, then $$\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$$ is equal to :

For $$\mathrm{n} \in \mathbf{N}$$, let $$\mathrm{S}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-3+2 i|=\frac{\mathrm{n}}{4}\right\}$$ and $$\mathrm{T}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-2+3 i|=\frac{1}{\mathrm{n}}\right\}$$. Then the number of elements in the set $$\left\{n \in \mathbf{N}: S_{n} \cap T_{n}=\phi\right\}$$ is :

The number of $$\theta \in(0,4 \pi)$$ for which the system of linear equations

$$ \begin{aligned} &3(\sin 3 \theta) x-y+z=2 \\\\ &3(\cos 2 \theta) x+4 y+3 z=3 \\\\ &6 x+7 y+7 z=9 \end{aligned} $$

has no solution, is :

If $$\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} - n - 1} + n\alpha + \beta } \right) = 0$$, then $$8(\alpha+\beta)$$ is equal to :

If the absolute maximum value of the function $$f(x)=\left(x^{2}-2 x+7\right) \mathrm{e}^{\left(4 x^{3}-12 x^{2}-180 x+31\right)}$$ in the interval $$[-3,0]$$ is $$f(\alpha)$$, then :

The curve $$y(x)=a x^{3}+b x^{2}+c x+5$$ touches the $$x$$-axis at the point $$\mathrm{P}(-2,0)$$ and cuts the $$y$$-axis at the point $$Q$$, where $$y^{\prime}$$ is equal to 3 . Then the local maximum value of $$y(x)$$ is:

The area of the region given by

$$A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}$$ is :

For any real number $$x$$, let $$[x]$$ denote the largest integer less than equal to $$x$$. Let $$f$$ be a real valued function defined on the interval $$[-10,10]$$ by $$f(x)=\left\{\begin{array}{l}x-[x], \text { if }[x] \text { is odd } \\ 1+[x]-x, \text { if }[x] \text { is even } .\end{array}\right.$$ Then the value of $$\frac{\pi^{2}}{10} \int_{-10}^{10} f(x) \cos \pi x \,d x$$ is :

The slope of the tangent to a curve $$C: y=y(x)$$ at any point $$(x, y)$$ on it is $$\frac{2 \mathrm{e}^{2 x}-6 \mathrm{e}^{-x}+9}{2+9 \mathrm{e}^{-2 x}}$$. If $$C$$ passes through the points $$\left(0, \frac{1}{2}+\frac{\pi}{2 \sqrt{2}}\right)$$ and $$\left(\alpha, \frac{1}{2} \mathrm{e}^{2 \alpha}\right)$$, then $$\mathrm{e}^{\alpha}$$ is equal to :

The general solution of the differential equation $$\left(x-y^{2}\right) \mathrm{d} x+y\left(5 x+y^{2}\right) \mathrm{d} y=0$$ is :

A line, with the slope greater than one, passes through the point $$A(4,3)$$ and intersects the line $$x-y-2=0$$ at the point B. If the length of the line segment $$A B$$ is $$\frac{\sqrt{29}}{3}$$, then $$B$$ also lies on the line :

Let the locus of the centre $$(\alpha, \beta), \beta>0$$, of the circle which touches the circle $$x^{2}+(y-1)^{2}=1$$ externally and also touches the $$x$$-axis be $$\mathrm{L}$$. Then the area bounded by $$\mathrm{L}$$ and the line $$y=4$$ is:

Let $$\mathrm{ABC}$$ be a triangle such that $$\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{CA}}=\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{c}},|\overrightarrow{\mathrm{a}}|=6 \sqrt{2},|\overrightarrow{\mathrm{b}}|=2 \sqrt{3}$$ and $$\vec{b} \cdot \vec{c}=12$$. Consider the statements :

$$(\mathrm{S} 1):|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})+(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}})|-|\vec{c}|=6(2 \sqrt{2}-1)$$

$$(\mathrm{S} 2): \angle \mathrm{ACB}=\cos ^{-1}\left(\sqrt{\frac{2}{3}}\right)$$

Then

If the numbers appeared on the two throws of a fair six faced die are $$\alpha$$ and $$\beta$$, then the probability that $$x^{2}+\alpha x+\beta>0$$, for all $$x \in \mathbf{R}$$, is :

Let $$A=\left(\begin{array}{rrr}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$$ and $$B=A-I$$. If $$\omega=\frac{\sqrt{3} i-1}{2}$$, then the number of elements in the $$\operatorname{set}\left\{n \in\{1,2, \ldots, 100\}: A^{n}+(\omega B)^{n}=A+B\right\}$$ is equal to ____________.

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is _____________.

If the maximum value of the term independent of $$t$$ in the expansion of $$\left(\mathrm{t}^{2} x^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{\mathrm{t}}\right)^{15}, x \geqslant 0$$, is $$\mathrm{K}$$, then $$8 \mathrm{~K}$$ is equal to ____________.

Let $$a, b$$ be two non-zero real numbers. If $$p$$ and $$r$$ are the roots of the equation $$x^{2}-8 \mathrm{a} x+2 \mathrm{a}=0$$ and $$\mathrm{q}$$ and s are the roots of the equation $$x^{2}+12 \mathrm{~b} x+6 \mathrm{~b}=0$$, such that $$\frac{1}{\mathrm{p}}, \frac{1}{\mathrm{q}}, \frac{1}{\mathrm{r}}, \frac{1}{\mathrm{~s}}$$ are in A.P., then $$\mathrm{a}^{-1}-\mathrm{b}^{-1}$$ is equal to _____________.

Let $$f(x)=\left\{\begin{array}{l}\left|4 x^{2}-8 x+5\right|, \text { if } 8 x^{2}-6 x+1 \geqslant 0 \\ {\left[4 x^{2}-8 x+5\right], \text { if } 8 x^{2}-6 x+1<0,}\end{array}\right.$$ where $$[\alpha]$$ denotes the greatest integer less than or equal to $$\alpha$$. Then the number of points in $$\mathbf{R}$$ where $$f$$ is not differentiable is ___________.

If momentum [P], area $$[\mathrm{A}]$$ and time $$[\mathrm{T}]$$ are taken as fundamental quantities, then the dimensional formula for coefficient of viscosity is :

Which of the following physical quantities have the same dimensions?

A person moved from A to B on a circular path as shown in figure. If the distance travelled by him is $$60 \mathrm{~m}$$, then the magnitude of displacement would be :

(Given $$\left.\cos 135^{\circ}=-0.7\right)$$

A body of mass $$0.5 \mathrm{~kg}$$ travels on straight line path with velocity $$v=\left(3 x^{2}+4\right) \mathrm{m} / \mathrm{s}$$. The net workdone by the force during its displacement from $$x=0$$ to $$x=2 \mathrm{~m}$$ is :

A solid cylinder and a solid sphere, having same mass $$M$$ and radius $$R$$, roll down the same inclined plane from top without slipping. They start from rest. The ratio of velocity of the solid cylinder to that of the solid sphere, with which they reach the ground, will be :

Three identical particles $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}$$ of mass $$100 \mathrm{~kg}$$ each are placed in a straight line with $$\mathrm{AB}=\mathrm{BC}=13 \mathrm{~m}$$. The gravitational force on a fourth particle $$\mathrm{P}$$ of the same mass is $$\mathrm{F}$$, when placed at a distance $$13 \mathrm{~m}$$ from the particle $$\mathrm{B}$$ on the perpendicular bisector of the line $$\mathrm{AC}$$. The value of $$\mathrm{F}$$ will be approximately :

A certain amount of gas of volume $$\mathrm{V}$$ at $$27^{\circ} \mathrm{C}$$ temperature and pressure $$2 \times 10^{7} \mathrm{Nm}^{-2}$$ expands isothermally until its volume gets doubled. Later it expands adiabatically until its volume gets redoubled. The final pressure of the gas will be (Use $$\gamma=1.5)$$ :

Following statements are given :

(A) The average kinetic energy of a gas molecule decreases when the temperature is reduced.

(B) The average kinetic energy of a gas molecule increases with increase in pressure at constant temperature.

(C) The average kinetic energy of a gas molecule decreases with increase in volume.

(D) Pressure of a gas increases with increase in temperature at constant pressure.

(E) The volume of gas decreases with increase in temperature.

Choose the correct answer from the options given below :

In figure $$(\mathrm{A})$$, mass '$$2 \mathrm{~m}^{\text {' }}$$ is fixed on mass '$$\mathrm{m}$$ ' which is attached to two springs of spring constant $$\mathrm{k}$$. In figure (B), mass '$$\mathrm{m}$$' is attached to two springs of spring constant '$$\mathrm{k}$$' and '$$2 \mathrm{k}^{\prime}$$. If mass '$$\mathrm{m}$$' in (A) and in (B) are displaced by distance '$$x^{\prime}$$ horizontally and then released, then time period $$\mathrm{T}_{1}$$ and $$\mathrm{T}_{2}$$ corresponding to $$(\mathrm{A})$$ and (B) respectively follow the relation.

A condenser of $$2 \,\mu \mathrm{F}$$ capacitance is charged steadily from 0 to $$5 \,\mathrm{C}$$. Which of the following graph represents correctly the variation of potential difference $$(\mathrm{V})$$ across it's plates with respect to the charge $$(Q)$$ on the condenser?

Two charged particles, having same kinetic energy, are allowed to pass through a uniform magnetic field perpendicular to the direction of motion. If the ratio of radii of their circular paths is $$6: 5$$ and their respective masses ratio is $$9: 4$$. Then, the ratio of their charges will be :

To increase the resonant frequency in series LCR circuit,

A small square loop of wire of side $$l$$ is placed inside a large square loop of wire $$\mathrm{L}(\mathrm{L}>>l)$$. Both loops are coplanar and their centres coincide at point $$\mathrm{O}$$ as shown in figure. The mutual inductance of the system is :

The rms value of conduction current in a parallel plate capacitor is $$6.9 \,\mu \mathrm{A}$$. The capacity of this capacitor, if it is connected to $$230 \mathrm{~V}$$ ac supply with an angular frequency of $$600 \,\mathrm{rad} / \mathrm{s}$$, will be :

Which of the following statement is correct?

Time taken by light to travel in two different materials $$A$$ and $$B$$ of refractive indices $$\mu_{A}$$ and $$\mu_{B}$$ of same thickness is $$t_{1}$$ and $$t_{2}$$ respectively. If $$t_{2}-t_{1}=5 \times 10^{-10}$$ s and the ratio of $$\mu_{A}$$ to $$\mu_{B}$$ is $$1: 2$$. Then, the thickness of material, in meter is: (Given $$v_{\mathrm{A}}$$ and $$v_{\mathrm{B}}$$ are velocities of light in $$A$$ and $$B$$ materials respectively.)

A metal exposed to light of wavelength $$800 \mathrm{~nm}$$ and and emits photoelectrons with a certain kinetic energy. The maximum kinetic energy of photo-electron doubles when light of wavelength $$500 \mathrm{~nm}$$ is used. The workfunction of the metal is : (Take hc $$=1230 \,\mathrm{eV}-\mathrm{nm}$$ ).

The momentum of an electron revolving in $$\mathrm{n}^{\text {th }}$$ orbit is given by :

(Symbols have their usual meanings)

The magnetic moment of an electron (e) revolving in an orbit around nucleus with an orbital angular momentum is given by :

In the circuit, the logical value of $$A=1$$ or $$B=1$$ when potential at $$A$$ or $$B$$ is $$5 \mathrm{~V}$$ and the logical value of $$A=0$$ or $$B=0$$ when potential at $$A$$ or $$B$$ is $$0 \mathrm{~V}$$.

The truth table of the given circuit will be :

A car is moving with speed of $$150 \mathrm{~km} / \mathrm{h}$$ and after applying the break it will move $$27 \mathrm{~m}$$ before it stops. If the same car is moving with a speed of one third the reported speed then it will stop after travelling ___________ m distance.

Four forces are acting at a point $$\mathrm{P}$$ in equilibrium as shown in figure. The ratio of force $$\mathrm{F}_{1}$$ to $$\mathrm{F}_{2}$$ is $$1: x$$ where $$x=$$ _____________.

A wire of length $$\mathrm{L}$$ and radius $$\mathrm{r}$$ is clamped rigidly at one end. When the other end of the wire is pulled by a force $$\mathrm{F}$$, its length increases by $$5 \mathrm{~cm}$$. Another wire of the same material of length $$4 \mathrm{L}$$ and radius $$4 \mathrm{r}$$ is pulled by a force $$4 \mathrm{F}$$ under same conditions. The increase in length of this wire is __________________ $$\mathrm{cm}$$.

A unit scale is to be prepared whose length does not change with temperature and remains $$20 \mathrm{~cm}$$, using a bimetallic strip made of brass and iron each of different length. The length of both components would change in such a way that difference between their lengths remains constant. If length of brass is $$40 \mathrm{~cm}$$ and length of iron will be __________ $$\mathrm{cm}$$. $$\left(\alpha_{\text {iron }}=1.2 \times 10^{-5} \mathrm{~K}^{-1}\right.$$ and $$\left.\alpha_{\text {brass }}=1.8 \times 10^{-5} \mathrm{~K}^{-1}\right)$$.

The volume charge density of a sphere of radius $$6 \mathrm{~m}$$ is $$2 \,\mu \mathrm{C} \,\mathrm{cm}^{-3}$$. The number of lines of force per unit surface area coming out from the surface of the sphere is _______________ $$\times 10^{10} \,\mathrm{NC}^{-1}$$.

[Given : Permittivity of vacuum $$\epsilon_{0}=8.85 \times 10^{-12} \,\mathrm{C}^{2}\, \mathrm{~N}^{-1}-\mathrm{m}^{-2}$$ )

In the given figure, the value of V_o will be _____________ V.

Eight copper wire of length $$l$$ and diameter $$d$$ are joined in parallel to form a single composite conductor of resistance $$R$$. If a single copper wire of length $$2 l$$ have the same resistance $$(R)$$ then its diameter will be ____________ d.

The energy band gap of semiconducting material to produce violet (wavelength = 4000$$\mathop A\limits^o $$ ) LED is ______________ $$\mathrm{eV}$$. (Round off to the nearest integer).