For the reaction, \(A(g) + B(g) \to  C(g) + D(g),\)  \(\Delta H^o\) and \(\Delta S^o\) are, respectively, \(-29.8\,\,kJ\, mol^{-1}\)  and \(-0.100\,\,kJ\,K^{-1}\) \(mol^{-1}\) at  \(298\,K.\)  The equilibrium constant for the reaction at  \(298\,K\)  is

In the structure of \(SF _{4}\), the lone pair of electrons on \(S\) is in ....

A molecule undergoes two independent first order reactions whose respective half lives are \(12\,min\) and \(3\,min\). If both the reactions are occurring then the time taken for the \(50 \%\) consumption of the reactant is \(..........\,\min\) (Nearest integer)

Given below are two statements Statement \(I:\) Iron \((III)\) catalyst, acidified \(K _{2} Cr _{2} O _{7}\) and neutral \(KMnO _{4}\) have the ability to oxidise \(I\) to \(I _{2}\) independently. Statement \(II\): Manganate ion is paramagnetic in nature and involves \(p \pi- p \pi\) bonding. In the light of the above statements, choose the correct answer from the options.

Number of molecules having bond order \(2\) from the following molecule is _______.
[\(\mathrm{C}_2, \mathrm{O}_2, \mathrm{Be}_2, \mathrm{Li}_2, \mathrm{Ne}_2, \mathrm{~N}_2, \mathrm{He}_2\)]

\(40\, \mathrm{~g}\) of glucose (Molar mass \(=180\) ) is mixed with \(200\, \mathrm{~mL}\) of water. The freezing point of solution is \(.....\,\mathrm{K}\). (Nearest integer) [Given : \(\mathrm{K}_{\mathrm{f}}=1.86 \,\mathrm{~K} \,\mathrm{~kg} \,\mathrm{~mol}^{-1} ;\) Density of water \(=\) \(1.00 \,\mathrm{~g}\, \mathrm{~cm}^{-3} ;\) Freezing point of water \(\left.=273.15\, \mathrm{~K}\right]\)

Choose the correct answer from the options given below :

The number of 3 -digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is______.

An engine operates by taking \(n\,moles\) of an ideal gas through the cycle \(ABCDA\) shown in figure. The thermal efficiency of the engine is : (Take \(C_v =1 .5\, R\), where \(R\) is gas constant)

Let \(A\) and \(B\) be two events such that the probability that exactly one of them occurs is \(\frac{2}{5}\) and the probability that \(A\) or \(B\) occurs is \(\frac{1}{2}\) then the probability of both of them occur together is

Combustion of \(1\) mole of benzene is expressed at \(\mathrm{C}_6 \mathrm{H}_6(1)+\frac{15}{2} \mathrm{O}_2(\mathrm{~g}) \rightarrow \mathrm{CO}_2(\mathrm{~g})+3 \mathrm{H}_2 \mathrm{O}(1) \text {. }\) The standard enthalpy of combustion of \(2 \mathrm{~mol}\) of benzene is - ' \(x\) ' \(k J\). \(\mathrm{x}=\) _______ \((1)\) standard Enthalpy of formation of \(1 \mathrm{~mol}\) of \(\mathrm{C}_6 \mathrm{H}_6(1)\), for the reaction \(6 \mathrm{C}\) (graphite) \(+3 \mathrm{H}_2(\mathrm{~g}) \rightarrow \mathrm{C}_6 \mathrm{H}_n(1)\) is \(48.5 \mathrm{~kJ} \mathrm{~mol}^{-1}\). \((2)\) Standard Enthalpy of formation of \(1 \mathrm{~mol}\) of \(\mathrm{CO}_2(\mathrm{~g})\), for the reaction \(\mathrm{C}\) (graphite) \(+\mathrm{O}_{2(\mathrm{~g})} \rightarrow \mathrm{CO}_2(\mathrm{~g})\) is \(-393.5 \mathrm{~kJ} \mathrm{~mol}^{-1}\). \((3)\) Standard and Enthalpy of formation of \(1 \mathrm{~mol}\) of \(\mathrm{H}_2 \mathrm{O}(1)\), for the reaction \(\mathrm{H}_2(\mathrm{~g})+\frac{1}{2} \mathrm{O}_2(\mathrm{~g}) \rightarrow \mathrm{H}_2 \mathrm{O}(1)\) is \(-286 \mathrm{~kJ} \mathrm{~mol}^{-1}\).

Consider the system of linear equations in \(x, y, z\):
\(x + 2y + tz = 0\),
\(6x + y + 5tz = 0\),
\(3x + t^2 y + f(t) z = 0\),
where \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function. If this system has infinitely many solutions for all \(t \in \mathbb{R}\), then \(f\)

If the minimum and the maximum values of the function \(f :\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow R ,\) defined by :  \(f (\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta & -1-\sin ^{2} \theta & 1 \\ -\cos ^{2} \theta & -1-\cos ^{2} \theta & 1 \\ 12 & 10 & -2\end{array}\right|\) are \(m\) and \(M\) respectively, then the ordered pair \(( m , M )\) is equal to