The theory that can completely/properly explain the nature of bonding in \([Ni(CO)_4]\) is

The number of optical isomers possible for \(\left[\mathrm{Cr}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{3}\right]^{3-}\) is ..... .

A solution is prepared by dissolving \(0.6\, g\) of urea (molar mass \(= 60\, g\, mol^{-1}\)) and \(1.8\, g\) of glucose (molar mass \(= 180\, g\, mol^{-1}\)) in \(100\, mL\) of water at \(27\,^oC\). The osmotic pressure of the solution is : ............. \(\mathrm{atm}\) \((R = 0.08206\, L\, atm\, K^{-1}\, mol^{-1})\)

In the estimation of bromine, \(0.5\, g\) of an organic compound gave \(0.40\, g\) of silver bromide. The percentage of bromine in the given compound is \(.....\,\%\) (nearest integer) (Relative atomic masses of \(Ag\) and \(Br\) are \(108\, u\) and \(80\, u\), respectively).

If shortest wavelength of hydrogen atom in Lyman series is \(x\), then longest wavelength in Balmer series of \(\text{He}^+\) is:

40 mL of a mixture of \(\mathrm{CH}_3 \mathrm{COOH}\) and HCl (aqueous solution) is titrated against 0.1 M NaOH solution conductometrically. Which of the following statement is correct?

The gas produced by treating an aqueous solution of ammonium chloride with sodium nitrite is .... .

The bond dissociation enthalpy of \(\mathrm{X}_2 \Delta \mathrm{H}_{\text {bond }}\) calculated from the given data is _______ \(\mathrm{kJ} \mathrm{mol}^{-1}\). (Nearest integer)
\(\begin{aligned}
& \mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s}) \rightarrow \mathrm{M}^{+}(\mathrm{g})+\mathrm{X}^{-}(\mathrm{g}) \Delta \mathrm{H}_{\text {lattice }}^*=800 \mathrm{~kJ} \mathrm{~mol}^{-1} \\
& \mathrm{M}(\mathrm{~s}) \rightarrow \mathrm{M}(\mathrm{~g}) \Delta \mathrm{H}_{\text {sub }}^{\circ}=100 \mathrm{~kJ} \mathrm{~mol}^{-1}
\end{aligned}\)
\(\mathrm{M}(\mathrm{~g}) \rightarrow \mathrm{M}^{+}(\mathrm{g})+\mathrm{e}^{-}(\mathrm{g}) \Delta \mathrm{H}_{\mathrm{i}}=500 \mathrm{~kJ} \mathrm{~mol}^{-1}\)
\(\mathrm{X}(\mathrm{~g})+\mathrm{e}^{-}(\mathrm{g}) \rightarrow \mathrm{X}^{-}(\mathrm{g}) \Delta \mathrm{H}_{\mathrm{eg}}^*=-300 \mathrm{~kJ} \mathrm{~mol}^{-1}\)
\(\mathrm{M}(\mathrm{~s})+\frac{1}{2} \mathrm{X}_2(\mathrm{~g}) \rightarrow \mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s}) \Delta \mathrm{H}_f^{\circ}=-400 \mathrm{~kJ} \mathrm{~mol}^{-1}\)
[Given : \(\mathrm{M}^{+} \mathrm{X}^{-}\)is a pure ionic compound and X forms a diatomic molecule \(\mathrm{X}_2\) in gaseous state]

Let the area of the region \(\{(x, y): x-2 y+4 \geq 0\), \(\left.x+2 y^2 \geq 0, x+4 y^2 \leq 8, y \geq 0\right\}\) be \(\frac{m}{n}\), where \(m\) and \(\mathrm{n}\) are coprime numbers. Then \(\mathrm{m}+\mathrm{n}\) is equal to

Let A and B be two distinct points on the line \(\mathrm{L}: \frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-7}{2}=\frac{\mathrm{z}-7}{-2}\). Both A and B are at a distance \(2 \sqrt{17}\) from the foot of perpendicular drawn from the point \((1,2,3)\) on the line L . If O is the origin, then \(\overrightarrow{O A} \cdot \overrightarrow{O B}\) is equal to:

A particle of mass \(\mathrm{m}\) is fixed to one end of a light spring having force constant \(\mathrm{k}\) and unstretched length \(\ell .\) The other end is fixed. The system is given an angular speed \(\omega\) about the fixed end of the spring such that it rotates in a circle in gravity free space. Then the stretch in the spring is

Let the mean of the data

\(X\)	\(1\)	\(3\)	\(5\)	\(7\)	\(9\)
\((f)\)	\(4\)	\(24\)	\(28\)	\(\alpha\)	\(8\)

be \(5.\) If \(m\) and \(\sigma^2\) are respectively the mean deviation about the mean and the variance of the data, then \(\frac{3 \alpha}{m+\sigma^2}\) is equal to \(..........\).

Let \(S\) be the mirror image of the point \(Q(1,3,4)\) with respect to the plane \(2 x-y+z+3=0\) and let \(\mathrm{R}(3,5, \gamma)\) be a point of this plane. Then the square of the length of the line segment \(SR\) is ..... .