A crystalline solid of a pure substance has a face-centred cubic structure with a cell edge of $400\mathrm{ }\mathrm{pm}$ . If the density of the substance in the crystal is $8 \mathrm{g}\mathrm{ }{\mathrm{cm}}^{-3},\mathrm{ }$ then the number of atoms present in $256 \mathrm{g}$ of the crystal $\mathrm{is}\mathrm{ }\mathrm{N}\mathrm{ }\times \mathrm{ }{10}^{24}$ . The value of $\mathrm{N}$is

The total number of compounds having at least one bridging oxo group among the molecules given below is
\(\text N _2\text O _5,\)\(\text P _4\text O _6,\)\(\text P _4\text O _8,\)\(\text H _4\text P _2\text O _5,\)\(\text H _5\text P _3\text O _{10},\) \(\text H _2\text S_2\text O _3,\text H _2\text S_2\text O _5\)

The compound(s) with two lone pairs of electrons on the central atom is (are)

Columns 1, 2 and 3 contain starting materials, reaction conditions and type of reactions respectively.

Column 1	Column 2	Column 3
(I) Toulene	(i) \(NaOH\) \( /Br _2\)	(P) Condensation
(II) Acetophenone	(ii) \(B r_2 / h v\)	(Q) Carboxylation
(III) Benzaldehyde	(iii) \(\left( CH _3 CO \right)_2\) \(O / CH _3\) \(COOK\)	(R) Substitution
(IV) Phenol	(iv) \(NaOH\) \(/CO _2\)	(S) Haloform

The only CORRECT combination in which the reaction proceeds through radical mechanism is

Among the following, the surfactant that will form micelles in aqueous solution at the lowest molar concentration at ambient conditions is

The correct statement(s) regarding the binary transition metal carbonyl compounds is (are) (Atomic numbers: $Fe = 26, Ni = 28$ )

Amongst the following, the total number of compounds whose aqueous solution turns red litmus paper blue is
\(\begin{array}{llll}
\mathrm{KCN} & \mathrm{K}_2 \mathrm{SO}_4 & \left(\mathrm{NH}_4\right)_2 \mathrm{C}_2 \mathrm{O}_4 & \mathrm{NaCl} \\
\mathrm{Zn}\left(\mathrm{NO}_3\right)_2 & \mathrm{FeCl}_3 & \mathrm{~K}_2 \mathrm{CO}_3 & \mathrm{NH}_4 \mathrm{NO}_3 \\
\mathrm{LiCN} & & &
\end{array}\)

A student performed the experiment of determination of focal length of a concave mirror by \(u-v\) method using an optical bench of length \(1.5 \mathrm{~m}\). The focal length of the mirror used is 24 \(\mathrm{cm}\). The maximum error in the location of the image can be \(0.2 \mathrm{~cm}\).
The 5 sets of \((u, v)\) values recorded by the student (in \(\mathrm{cm})\) are : \((42,56),(48\), \(48),(60,40),(66,33),(78,39)\). The data set(s) that cannot come from experiment and is (are) incorrectly recorded, is(are)

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A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question :
The work done in bringing the dipole from infinity to a distance$r$from the centre of the loop by the given process is proportional to:

An ideal gas undergoes a four step cycle as shown in the $P-V$ diagram below. During this cycle, heat is absorbed by the gas in:

The number of \(3 \times 3\) matrices \(A\) whose entries are either 0 or 1 and for which the system \(A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) has exactly two distinct solutions, is

A simple telescope used to view distant objects has eyepiece and objective lens of focal lengths \(f_e\) and \(f_0\), respectively. Then

Paragraph:
There are \(n\) urns each containing \((n+1)\) balls such that the ith urn contains \(i\) white balls and \((n+1-i)\) red balls. Let \(u_i\) be the event of selecting ith urn, \(i=1,2,3, \ldots, n\) and \(W\) denotes the event of getting a white balls.Question:
If \(P\left(u_i\right) \propto i\), where \(i=1,2,3, \ldots, n\), then \(\lim _{n \rightarrow \infty} P(W)\) is equal to