Calamine, malachite, magnetite and cryolite, respectively are:

The thermal dissociation equilibrium of ${\mathrm{CaCO}}_3\left(\mathrm{s}\right)$ is studied under different conditions.
${\text{CaCO}}_3\left(s\right)\rightleftharpoons \text{CaO}\left(s\right)+{\text{CO}}_2\left(\text{g}\right)$
For this equilibrium, the correct statement(s) is/are

Consider an ionic solid $\mathrm{MX}$, with $\mathrm{NaCl}$ structure. Construct a new structure $\left(\mathrm{Z}\right)$, whose unit cell is constructed from the unit cell of $\mathrm{MX}$, following the sequential instructions given below. Neglect the charge balance.
$\left(\mathrm{i}\right)$ Remove all the anions $\left(\mathrm{X}\right)$, except the central one.
$\left(\mathrm{ii}\right)$ Replace all the face centered cations $\left(\mathrm{M}\right)$, by anions $\left(\mathrm{X}\right)$.
$\left(\mathrm{iii}\right)$ Remove all the corner cations $\left(\mathrm{M}\right)$.
$\left(\mathrm{iv}\right)$ Replace the central anion $\left(\mathrm{X}\right)$, with cation $\left(\mathrm{M}\right)$.
The value of $\left(\frac{\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}}{\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}}\right)$ in $\mathrm{Z}$ is _____.

For the reaction, \(2 \mathrm{CO}+\mathrm{O}_2 \rightarrow 2 \mathrm{CO}_2 ; \Delta H=-560 \mathrm{~kJ}\). Two moles of \(\mathrm{CO}\) and one mole of \(\mathrm{O}_2\) are taken in a container of volume \(1 \mathrm{~L}\). They completely form two moles of \(\mathrm{CO}_2\), the gases deviate appreciably from ideal behaviour. If the pressure in the vessel changes from 70 to \(40 \mathrm{~atm}\), find the magnitude (absolute value) of \(\Delta U\) at \(500 \mathrm{~K} .(1 \mathrm{~L}-\mathrm{atm}=0.1 \mathrm{~kJ})\)

An organic compound $\left({\mathrm{C}}_8{\mathrm{H}}_{10}{\mathrm{O}}_2\right)$ rotates plane-polarized light. It produces pink color with neutral ${\mathrm{FeCl}}_3$ solution. What is the total number of all the possible isomers for this compound?

The solubility of a salt of weak acid $\left(\mathrm{AB}\right)\mathrm{ }\mathrm{at}\mathrm{ }\mathrm{pH}3\mathrm{ }\mathrm{is}\mathrm{ }\mathrm{Y}\times {10}^{-3}\mathrm{ }\mathrm{mol}\mathrm{ }{\mathrm{L}}^{-1}$ . The value of $\mathrm{Y}$ is ____. (Given that the value of solubility product of $\mathrm{AB}\left({\mathrm{K}}_{\mathrm{sp}}\right)=2\times {10}^{-10}\mathrm{ }$, and the value of ionization constant of $\mathrm{HB}\left(\mathrm{Ka}\right)=1\times {10}^{-8}$)

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There are some deposits of nitrates and phosphates in earth's crust. Nitrates are more soluble in water. Nitrates are difficult to reduce under the laboratory conditions but microbes do it easily. Ammonia forms large number of complexes with transition metal ions. Hybridisation easily explains the ease of sigma donation capability of \(\mathrm{NH}_3\) and \(\mathrm{PH}_3\). Phosphine is a flammable gas and is prepared from white phosphorus.
Question:
Among the following, the correct statement is

The largest value of the non - negative integer a for which $\underset{x\to 1}{\lim }{\left\{\frac{-ax+\sin \left(x-1\right)+a}{x+\sin \left(x-1\right)-1}\right\}}^{\frac{1-x}{1- \sqrt{x}}}=\frac{1}{4}$ is _________

If the line $x=\alpha$ divides the area of the region $R=\{\left(x,y\right)\in R^2:x^3\le y\le x, 0\le x\le 1\}$ into two equal parts, then

Some standard electrode potentials at $298 \mathrm{K}$ are given below: $\begin{array}{ll}{\mathrm{Pb}}^{2+}/\mathrm{Pb}& -0.13 \mathrm{V}\end{array}$ $\begin{array}{ll}{\mathrm{Ni}}^{2+}/\mathrm{Ni}& -0.24 \mathrm{V}\end{array}$ $\begin{array}{ll}{\mathrm{Cd}}^{2+}/\mathrm{Cd}& -0.40 \mathrm{V}\end{array}$ $\begin{array}{ll}{\mathrm{Fe}}^{2+}/\mathrm{Fe}& -0.44 \mathrm{V}\end{array}$ To a solution containing $0.001 \mathrm{M}$ of ${\mathbf{X}}^{2+}$ and $0.1 \mathrm{M}$ of ${\mathbf{Y}}^{2+}$, the metal rods $\mathbf{X}$ and $\mathbf{Y}$ are inserted $($at $298 \mathrm{K})$ and connected by a conducting wire. This resulted in dissolution of $\mathbf{X}$. The correct combination(s) of $\mathbf{X}$ and $\mathbf{Y}$, respectively, is(are) (Given: Gas constant, $\mathrm{R}=8.314 \mathrm{J}{ \mathrm{K}}^{-1}{ \mathrm{mol}}^{-1}$, Faraday constant, $\mathrm{F}=96500 \mathrm{C} {\mathrm{mol}}^{-1}$)

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

Consider the following reaction,
\(2 \mathrm{H}_2(\mathrm{~g})+2 \mathrm{NO}(\mathrm{g}) \rightarrow \mathrm{N}_2(\mathrm{~g})+2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})\)
which follows the mechanism given below:
\(2 \mathrm{NO}(\mathrm{g}) \stackrel{k_1}{\underset{k_{-1}}{\rightleftharpoons}} \mathrm{N}_2 \mathrm{O}_2(\mathrm{~g})\) \(\quad\) (fast equlibrium)
\(\mathrm{N}_2 \mathrm{O}_2(\mathrm{~g})+\mathrm{H}_2(\mathrm{~g}) \xrightarrow{k_2} \mathrm{~N}_2 \mathrm{O}(\mathrm{g})+\mathrm{H}_2 \mathrm{O}(\mathrm{g})\) \(\quad\)(slow reaction)
\(\mathrm{N}_2 \mathrm{O}(\mathrm{g})+\mathrm{H}_2(\mathrm{~g}) \xrightarrow{k_3} \mathrm{~N}_2(\mathrm{~g})+\mathrm{H}_2 \mathrm{O}(\mathrm{g})\) \(\quad\) (fast reaction)
The order of the reaction is_____

Consider the hyperbola $H :x^2-y^2=1$ and a circle S with center $N\left(x_2, 0\right).$ Suppose that H and S touch each other at point $P\left(x_1, y_1\right)$ with $x_1>1 and y_1>0.$ The common tangent to H and S at P intersects the x - axis at point M. If $(l, m)$ is the centroid of the triangle $\mathrm{\Delta }PMN,$ then the correct expression(s) is(are)