Extra pure \(\mathrm{N}_2\) can be obtained by heating

\(75.2 \mathrm{~g}\) of \(\mathrm{C}_6 \mathrm{H}_5 \mathrm{OH}\) (phenol) is dissolved in a solvent of \(K_f=14\). If the depression in freezing point is \(7 \mathrm{~K}\), then find the \(\%\) of phenol that dimerises.

Paragraph :
Treatment of compound \(\mathbf{O}\) with \(\mathrm{KMnO}_{4} / \mathrm{H}^{+}\)gave \(\mathbf{P}\), which on heating with ammonia gave \(\mathbf{Q}\). The compound \(\mathbf{Q}\) on treatment with \(\mathrm{Br}_{2} / \mathrm{NaOH}\) produced \(\mathbf{R}\). On strong heating, \(\mathbf{Q}\) gave \(\mathbf{S}\), which on further treatment with ethyl \(2-\)bromopropanoate in the presence of \(\mathrm{KOH}\) followed by acidification, gave a compound \(\mathbf{T}\).


Question :
The compound \(\mathbf{T}\) is

Total number of isomers, considering both structural and stereoisomers, of cyclic ethers with the molecular formula ${\mathrm{C}}_4{\mathrm{H}}_8\mathrm{O}$ is ________

Amongst the following, the total number of compounds soluble in aqueous \(\mathrm{NaOH}\) is

Paragraph I: An organic compound \(\mathbf{P}\) with molecular formula \(\mathrm{C}_9 \mathrm{H}_{18} \mathrm{O}_2\) decolorizes bromine water and also shows positive iodoform test. \(\mathbf{P}\) on ozonolysis followed by treatment with \(\mathrm{H}_2 \mathrm{O}_2\) gives \(\mathbf{Q}\) and \(\mathbf{R}\). While compound \(\mathbf{Q}\) shows positive iodoform test, compound \(\mathbf{R}\) does not give positive iodoform test. \(\mathbf{Q}\) and \(\mathbf{R}\) on oxidation with pyridinium chlorochromate (PCC) followed by heating give \(\mathbf{S}\) and \(\mathbf{T}\), respectively. Both \(\mathbf{S}\) and \(\mathbf{T}\) show positive iodoform test.
Complete copolymerization of 500 moles of \(\mathbf{Q}\) and 500 moles of \(\mathbf{R}\) gives one mole of a single acyclic copolymer \(\mathbf{U}\).
[Given, atomic mass: \(\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16\)]
Question: Sum of number of oxygen atoms in \(\mathbf{S}\) and \(\mathbf{T}\) is

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

For the reaction:
${\mathrm{I}}^{-}+\mathrm{C}\mathrm{l}{\mathrm{O}}_3^{-}+{\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4\to \mathrm{C}{\mathrm{l}}^{-}+\mathrm{H}\mathrm{S}{\mathrm{O}}_4^{-}+{\mathrm{I}}_2\mathrm{ }$
The correct statement(s) in the balanced equation is/are:

The number of water molecule(s) directly bonded to the metal centre in \(\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}\) is

Match the following for the given process

A block of mass \(5 \mathrm{~kg}\) moves along the \(x\)-direction subject to the force \(F=(-20 x+10) \mathrm{N}\), with the value of \(x\) in metre. At time \(t=0 \mathrm{~s}\), it is at rest at position \(x=1 \mathrm{~m}\). The position and momentum of the block at \(t=(\pi / 4) \mathrm{s}\) are

Most materials have the refractive index, \(n>1\). So, when a light ray from air enters a naturally occurring material, then by Snell's

law, \(\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{n_{2}}{n_{1}}\), it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation, \(\mathrm{n}=c / v=\pm \sqrt{\varepsilon_{r} \mu_{r}}\), where \(c\) is the speed of electromagnetic waves in vacuum, \(v\) its speed in the medium, \(\varepsilon_{r}\) and \(\mu_{r}\) are the relative permittivity and permeability of the medium respectively. In normal materials, both \(\varepsilon_{r}\) and \(\mu_{r}\), are positive, implying positive \(n\) for the medium. When both \(\varepsilon_{r}\) and \(\mu_{r}\) are negative, one must choose the negative root of \(n\). Such negative refractive index materials can now be artificially prepared and are called metamaterials. They exhibit significantly different optical behavior, without violating any physical laws. Since \(n\) is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.
Question : Choose the correct statement.

Paragraph:
Read the following passage and answer the questions.
For every function \(f(x)\) which is twice differentiable, these will be good approximation of \(\int_a^b f(x) d x \cong\left(\frac{b-a}{2}\right)\{f(a)+f(b)\}\). Now, if we take \(c=\frac{a+b}{2}\), then using above again, we get \(\int_a^b f(x) d x=\int_a^c f(x) d x+\int_c^b f(x) d x \cong \frac{b-a}{4}\{f(a)+f(b)+2 f(c)\}\) and so on.
We get approximation for value of \(\int_a^b f(x) d x\).Question:
Good approximation of \(\int_0^{\pi / 2} \sin x d x\), is