In the reaction,
\(
2 \mathrm{X}+\mathrm{B}_2 \mathrm{H}_6 \longrightarrow\left[\mathrm{BH}_2(\mathrm{X})_2\right]^{+}\left[\mathrm{BH}_4\right]^{-}
\)
The amine (s) \(X\), is (are)

If the value of Avogadro number is $6.023\times {10}^{23}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{–1}$ and the value of Boltzmann constant is $1.380\times {10}^{-23}\mathrm{ }\mathrm{J}{\mathrm{K}}^{-1}$ , then the number of significant digits in the calculated value of the universal gas constant is

Match the compounds in List-I with the appropriate observations in List-II and choose the correct option.

List - I	List - II
(P)	(1) Reaction with phenyl diazonium salt gives #ff0 dye.
(Q)	(2) Reaction with ninhydrin gives purple color and it also reacts with \(FeCl _3\) to give violet color.
(R)	(3) Reaction with glucose will give corresponding hydrazone.
(S)	(4) Lassiagne extract of the compound treated with dilute HCl followed by addition of aqueous \(FeCl _3\) gives blood red color.
	(5) After complete hydrolysis,it will give ninhydrin test and it DOES NOT give positive phthalein dye test.

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In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' \(r\) '.
Question:
The volume of this \(\mathrm{HCP}\) unit cell is

One mole of an ideal monoatomic gas undergoes two reversible processes $(\mathrm{A}\to \mathrm{B}$ and $\mathrm{B}\to \mathrm{C})$ as shown in the given figure:

$\mathrm{A}\to \mathrm{B}$ is an adiabatic process. If the total heat absorbed in the entire process $(\mathrm{A}\to \mathrm{B}$ and $\mathrm{B}\to \mathrm{C})$ is ${\mathrm{RT}}_2\ln 10$, the value of $2{\mathrm{logV}}_3$ is
[Use molar heat capacity of the gas at constant pressure, ${\mathrm{C}}_{\mathrm{p},\mathrm{m}}=\frac{5}{2}\mathrm{R}$]

The reaction of $4$-methyloct-$1$-ene $(\mathrm{P},2.52 \mathrm{g})$ with $\mathrm{HBr}$ in the presence of ${\left({\mathrm{C}}_6{\mathrm{H}}_5\mathrm{CO}\right)}_2{\mathrm{O}}_2$ gives two isomeric bromides in a $9: 1$ ratio, with a combined yield of $50\%$. Of these, the entire amount of the primary alkyl bromide was reacted with an appropriate amount of diethylamine followed by treatment with aq. ${\mathrm{K}}_2{\mathrm{CO}}_3$ to give a non-ionic product $\mathrm{S}$ in $100\%$ Yield. The mass (in $\mathrm{mg}$ ) of $\mathrm{S}$ obtained is
[Use molar mass (in ${\mathrm{gmol}}^{-1}$ ): $\mathrm{H}=1,\mathrm{C}=12, \mathrm{N}=14,\mathrm{Br}=80$ ]

With reference to the scheme given below, which of the given statement(s) about \(\mathrm{T}, \mathrm{U}, \mathrm{V}\) and \(\mathrm{W}\) is (are) correct?

A block \((B)\) is attached to two unstretched \(S_1\) and \(S_2\) with spring constants \(k\) and \(4 k\), respectively (see figure I). The other ends are attached to identical supports \(M_1\) and \(M_2\) not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block \(B\) is displaced towards wall 1 by a small distance \(x\) (figure II) and released. The block returns and moves a maximum distance \(y\) towards wall 2 . Displacements \(x\) and \(y\) are measured with respect to the equilibrium position of the block \(B\). The ratio \(\frac{y}{-1}\) is

Paragraph:

Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index \(n_{1}\) surrounded by a medium of lower refractive index \(n_{2}\). The light guidance in the structure takes place due to successive total internal reflections at the interface of the media \(n_{1}\) and \(n_{2}\) as shown in the figure. All rays with the angle of incidence \(i\) less than a particular value \(i_{m}\) are confined in the medium of refractive index \(n_{1}\). The numerical aperture (NA) of the structure is defined as \(\sin i_{m}\).



Question:

If two structures of same cross-sectional area, but different numerical apertures \(N A_{1}\) and \(N A_{2}\left(N A_{2} < N A_{1}\right)\) are joined longitudinally, the numerical aperture of the combined structure is

Let $f:R\to R$ be given by $f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(x-5\right).$ Define $F\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt, x>0.$ Then which of the following options is/are correct?

List-I includes starting materials and reagents of selected chemical reactions. List-II gives structures of compounds that may be formed as intermediate products and/or final products from the reactions of List-I.

List-I	List-II
(I)	$\left(\mathrm{P}\right)$
(II)	$\left(\mathrm{Q}\right)$
(III)	$\left(\mathrm{R}\right)$
(IV)	$\left(\mathrm{S}\right)$
	$\left(\mathrm{T}\right)$
	$\left(\mathrm{U}\right)$

Which of the following options has correct combination considering List-I and List-II?

For the reduction of \(\mathrm{NO}_3^{-}\)ion in an aqueous solution \(E^{\circ}\) is \(+0.96 \mathrm{~V}\). Values of \(E^{\circ}\) for some metal ions are given below.
\(
\begin{aligned}
& \mathrm{V}^{2+}(a q)+2 e^{-} \longrightarrow \mathrm{V} ; E^{\circ}=-1.19 \mathrm{~V} \\
& \mathrm{Fe}^{3+}(a q)+3 e^{-} \longrightarrow \mathrm{Fe} ; \mathrm{E}^{\circ}=-0.04 \mathrm{~V} \\
& \mathrm{Au}^{3+}(a q)+3 e^{-} \longrightarrow \mathrm{Au} ; E^{\circ}=+1.40 \mathrm{~V}
\end{aligned}
\)
\(
\mathrm{Hg}^{2+}(a q)+2 e^{-} \longrightarrow \mathrm{Hg} ; E^{\circ}=+0.86 \mathrm{~V}
\)
The pair(s) of metals that is (are) oxidised by \(\mathrm{NO}_3^{-}\)in aqueous solution is (are)

Paragraph :
\(P_{14-16}\) : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius \(R\) and height \(L_0\). The cylinder is open at its bottom and has a small hole at its top. A piston of mass \(M\) is held at a distance \(L\) from the top surface as shown in the figure. The atmospheric pressure is \(p_0\).

Question :
While the piston is at a distance \(2 L\) from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is