Consider a complex reaction taking place in three steps with rate constants \(k_1, k_2\) and \(k_3\) respectively. The overall rate constant \(k\) is given by the expression \(k=\sqrt{\frac{k_1 k_3}{k_2}}\). If the activation energies of the three steps are 60,30 and \(10 \mathrm{~kJ} \mathrm{~mol}{ }^{-1}\) respectively, then the overall energy of activation in \(\mathrm{kJ} \mathrm{mol}^{-1}\) is _______. (Nearest integer)

Consider the following reactions. From these reactions which reaction will give carboxylic acid
as a major product ?
(A) \(\mathrm{R}-\mathrm{C} \equiv \mathrm{N} \xrightarrow[\text { mild condition }]{\text { (i) } \mathrm{H}^{+} / \mathrm{H}_2 \mathrm{O}}\)
(B) \(\mathrm{R}-\mathrm{MgX} \xrightarrow[\text { (ii) } \mathrm{H}_3 \mathrm{O}^{+}]{\text {(i) } \mathrm{CO}_2}\)
(C) \(\mathrm{R}-\mathrm{C} \equiv \mathrm{N} \xrightarrow[\text { (ii) } \mathrm{H}_3 \mathrm{O}^{+}]{\text {(i) } \mathrm{SnCl}_2 / \mathrm{HCl}}\)
(D) \(\mathrm{R} \cdot \mathrm{CH}_2 \cdot \mathrm{OH} \xrightarrow{\mathrm{PCC}}\)
(E)

Choose the correct answer from the options given below:

Match the Compounds (List - I) with the appropriate Catalyst/Reagents (List - II) for their reduction into corresponding amines.

Choose the correct answer from the options given below :

\(NO_2\) required for a reaction is produced by the decomposition of \(N_2O_5\) in \(CCl_4\) as per the equation  \(2N_2O_5 (g) \to 4NO_2 (g) + O_2 (g)\) The initial concentration of \(N_2O_5\) is \(3.00\, mol\, L^{-1}\) and it is \(2.75\, mol\, L^{-1}\) after \(30\, minutes\). The rate of formation of \(NO_2\) is ............... \(mol\, L^{-1}\, min^{-1}\)

The reaction of propene with \(HOCl (Cl_2 + H_2O)\) proceeds through the intermediate :

Number of stereo centers present in linear and cyclic structures of glucose are respectively

The oxygen dissolved in water exerts a partial pressure of \(20\, kPa\) in the vapour above water. The molar solubility of oxygen in water is ............ \(-\times 10^{-5}\, mol\, dm ^{-3}\) (Round off to the Nearest Integer). [Given : Henry's law constant \(= K _{ H }=8.0 \times 10^{4} kPa\) for \(O _{2}\) Density of water with dissolved oxygen \(=1.0\, kg\, dm ^{-3}\) ]

Number of lone pair (s) of electrons on central atom and the shape of \(BrF _{3}\) molecule respectively, are ..... .

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \quad \vec{b}=-\hat{i}-8 \hat{j}+2 \hat{k} \quad\) and \(\overrightarrow{\mathrm{c}}=4 \hat{\mathrm{i}}+\mathrm{c}_2 \hat{\mathrm{j}}+\mathrm{c}_3 \hat{\mathrm{k}}\) be three vectors such that \(\vec{b} \times \vec{a}=\vec{c} \times \vec{a}\). If the angle between the vector \(\vec{c}\) and the vector \(3 \hat{i}+4 \hat{j}+\hat{k}\) is \(\theta\), then the greatest integer less than or equal to \(\tan ^2 \theta\) is :

A perfectly dimagnetic sphere has a small spherical cavity at its centre, which is filled with a paramagnetic substance. The whole system is placed in a uniform magnetic field \(\overrightarrow{ B }\). Then the field inside the paramagnetic substance is:

A capacitor of capacitance \(100\  \mu \mathrm{F}\) is charged to a potential of \(12  \mathrm{~V}\) and connected to a \(6.4 \ \mathrm{mH}\) inductor to produce oscillations. The maximum current in the circuit would be :

A satellite of \(10^3 \mathrm{~kg}\) mass is revolving in circular orbit of radius \(2 \mathrm{R}\). If \(\frac{10^4 \mathrm{R}}{6} \mathrm{~J}\) energy is supplied to the satellite, it would revolve in a new circular orbit of radius _______.
(use \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2, \mathrm{R}=\) radius of earth)

let \(y = y\left( x \right)\) be the solution of the differential equation \(\sin x\frac{{dy}}{{dx}} + ycos\;x = 4x\;\), \(x \in \left( {0,\pi } \right)\) . If  \(y\left( {\frac{\pi }{2}} \right) = 0\) then \(y\left( {\frac{\pi }{6}} \right) = .\;.\;..\;\) .