If the unit cell of a mineral has cubic close packed (ccp) array of oxygen atoms with m fraction of octahedral holes occupied by aluminium ions and n fraction of tetrahedral holes occupied by magnesium ions, m and n, respectively, are

In the scheme given below, \(\text X\) and \(\text Y\), respectively, are
Metal halide \(\quad \xrightarrow{\text { aq. NaOH} } \quad\) White precipitate \((\text P )+\) Filtrate \((\text Q )\)
\(P \xrightarrow[\text { heat }]{\substack{\text { aq. } H _2 SO _4 \\ PbO _2 \text { (excess) }}}\text X\) (a coloured species in solution)
\(Q \xrightarrow[\text { warm }]{ MnO ( OH )_2, \text { Conc. } H _2 SO _4}\text Y\) (gives blue-coloration with KI-starch paper)

Among the electrolytes \(\mathrm{Na}_2 \mathrm{SO}_4, \mathrm{CaCl}_2 \quad \mathrm{Al}_2\left(\mathrm{SO}_4\right)_3\) and \(\mathrm{NH}_4 \mathrm{Cl}\), the most effective coagulating agent for \(\mathrm{Sb}_2 \mathrm{~S}_3\) sol is

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Redox reaction play a pivotal role in chemistry and biology. The values of standard redox potential \(\left(E^{\circ}\right)\) of two half-cell reactions decide which way the reaction is expected to proceed. A simple example is a Daniel cell in which zinc goes into solution and copper gets deposited. Given below are a set of half-cell reactions (acidic medium) along with their \(E^{\circ}(V\) with respect to normal hydrogen electrode) values. Using this data obtain the correct explanations to Questions 14-19.
\(\mathrm{I}_2+2 e^{-} \longrightarrow 2 \mathrm{I}^{-} E^{\circ}=0.54 \)
\( \mathrm{Cl}_2+2 e^{-} \longrightarrow 2 \mathrm{Cl}^{-} E^{\circ}=1.36 \)
\( \mathrm{Mn}^{3+}+e^{-} \longrightarrow \mathrm{Mn}^{2+} E^{\circ}=1.50 \)
\( \mathrm{Fe}^{3+}+e^{-} \longrightarrow \mathrm{Fe}^{2+} E^{\circ}=0.77 \)
\( \mathrm{O}_2+4 \mathrm{H}^{+}+4 e^{-} \longrightarrow 2 \mathrm{H}_2 \mathrm{O} E^{\circ}=1.23\)
Question:
While \(\mathrm{Fe}^{3+}\) is stable, \(\mathrm{Mn}^{3+}\) is not stable in acid solution because

For the following reaction $2\mathrm{X}+\mathrm{Y}\stackrel{\mathrm{k}}{\to }\mathrm{P}$ the rate of reaction is $\frac{\mathrm{d}\left[\mathrm{P}\right]}{\mathrm{dt}}=\mathrm{k}\left[\mathrm{X}\right]$. Two moles of $\mathrm{X}$ are mixed with one mole of $\mathrm{Y}$ to make $1.0 \mathrm{L}$ of solution. At $50 \mathrm{s}, 0.5$ mole of $\mathrm{Y}$ is left in the reaction mixture. The correct statement(s) about the reaction is(are) $($Use: $\ln 2=0.693)$

Match the compounds in Column I with their characteristic test(s)/reactions(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Consider a reaction \(a G+b H \rightarrow\) products. When concentration of both the reactants \(G\) and \(H\) is doubled, the rate increases by eight times. However, when concentration of \(G\) is doubled keeping the concentration of \(H\) fixed, the rate is doubled. The overall order of the reaction is

A point source \(\mathrm{S}\) emits unpolarized light uniformly in all directions. At two points \(\mathrm{A}\) and \(\mathrm{B}\), the ratio \(r=I_A / I_B\) of the intensities of light is 2. If a set of two polaroids having \(45^{\circ}\) angle between their pass-axes is placed just before point \(\mathrm{B}\), then the new value of \(r\) will be ______

The Fischer projection of $\mathrm{D}$-erythrose is shown below.
$\mathrm{D}$-Erythrose and its isomers are listed as $\mathrm{P},\mathrm{Q},\mathrm{R},$ and $\mathrm{S}$ in Column I. Choose the correct relationship of $\mathrm{P},\mathrm{Q},\mathrm{R},$ and $\mathrm{S}$ with $\mathrm{D}$-erythrose from Column II.

	Column - I		Column -II
P.		1.	Diastereomer
Q.		2.	Identical
R.		3.	Enantiomer
S.

Consider the strong electrolytes ${\mathrm{Z}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}, {\mathrm{U}}_{\mathrm{m}}{\mathrm{Y}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$. Limiting molar conductivity of ${\mathrm{U}}_{\mathrm{m}}{\mathrm{Y}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$ are $250$ and $440 \mathrm{S} {\mathrm{cm}}^2 {\mathrm{mol}}^{-1}$, respectively. The value of $\left(\mathrm{m}+\mathrm{n}+\mathrm{p}\right)$ is Given:

Ion	\(Z ^{ n +}\)	\(U ^{ p +}\)	\(V ^{ n +}\)	\(X ^{ m }\)	\(Y ^{ m -}\)
\(\lambda^0(\text{S cm} ^2\) \(\text{mol}^{-1})\)	50.0	25.0	100.0	80.0	100.0

${\lambda }^0$ is the limiting molar conductivity of ions The plot of molar conductivity $\left(\Lambda \right)$ of ${\mathrm{Z}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$ vs ${\mathrm{c}}^{\frac{1}{2}}$ is given below. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

The treatment of galena with ${\mathrm{HNO}}_3$ produces a gas that is

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that $\overrightarrow{a}.\overrightarrow{b}=0.$ For some $x,y\in \mathrm{R,}$ let $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+\left(\overrightarrow{a}\times \overrightarrow{b}\right).$ If $\left|\overrightarrow{c}\right|=2$ and the vector $\overrightarrow{c}$ is inclined at the same angle $\alpha$ to both $\overrightarrow{a}$ and $\overrightarrow{b},$ then the value of $8{\cos }^2\alpha$ is

For any integer $k,$ let ${\alpha }_k=\cos \left(\frac{k\pi }{7}\right)+i\sin \left(\frac{k\pi }{7}\right), where i= \sqrt{-1}.$ The value of the expression $\frac{\sum _{k=1}^{12}\left|{\alpha }_{k+1}-{\alpha }_k\right|}{\sum _{k=1}^3\left|{\alpha }_{4k-1}-{\alpha }_{4k-2}\right|}$ is