Match List-I with List-II and select the correct option.
Step 1: Analyze the given complexes and their coordination geometry
\([NiCl_4]^{2-}\) is a tetrahedral complex, typically exhibiting \(sp^3\) hybridization.
It also has a magnetic moment of 3.87 BM.
\([Ni(CN_4)]^{2-}\) is a square planar complex, usually exhibiting \(dsp^2\) hybridization.
It shows no unpaired electrons, so the magnetic moment is 0 BM.
\([CoCl_4]^{2-}\) is an octahedral complex, usually with \(sp^3d^2\) hybridization.
Its magnetic moment is 2.82 BM.
\([Ni(H_2O)_6]^{2+}\) is a tetrahedral complex, exhibiting \(sp^3\) hybridization and a magnetic moment of 2.82 BM.
Step 2: Match the complexes with their characteristics
A.
\([NiCl_4]^{2-}\) matches with IV.
\(sp^3\), tetrahedral, 3.87 BM.
B.
\([Ni(CN_4)]^{2-}\) matches with II.
\(dsp^2\), square planar, 0 BM.
C.
\([CoCl_4]^{2-}\) matches with I.
\(sp^3d^2\), octahedral, 2.82 BM.
D.
\([Ni(H_2O)_6]^{2+}\) matches with III.
\(sp^3\), tetrahedral, 2.82 BM.
(b) If \( \vec{L} \) is the angular momentum of the electron, show that:
\[ \vec{\mu} = -\frac{e}{2m} \vec{L} \]
The sum of the spin-only magnetic moment values (in B.M.) of $[\text{Mn}(\text{Br})_6]^{3-}$ and $[\text{Mn}(\text{CN})_6]^{3-}$ is ____.
Let $ f: \mathbb{R} \to \mathbb{R} $ be a twice differentiable function such that $$ f''(x)\sin\left(\frac{x}{2}\right) + f'(2x - 2y) = (\cos x)\sin(y + 2x) + f(2x - 2y) $$ for all $ x, y \in \mathbb{R} $. If $ f(0) = 1 $, then the value of $ 24f^{(4)}\left(\frac{5\pi}{3}\right) $ is: