LIST-I | LIST-II | ||
|---|---|---|---|
| I | \(\text{[Cr(CN)}_6\text{]}^{4-}\) | P | t2g orbitals contain 4 electrons |
| II | \(\text{[RuCl}_6\text{]}^{2-}\) | Q | \(\mu\)(spin-only) = 4.9 BM |
| III | \(\text{[Cr(H}_2\text{O)}_6\text{]}^{2+}\) | R | low spin complex ion |
| IV | \(\text{[Fe(H}_2\text{O)}_6\text{]}^{2+}\) | S | metal ion in 4+ oxidation state |
| T | d4 species | ||
I → R, T; II → P, S; III → Q, T; IV → P, Q
I → R, S; II → P, T; III → P, Q; IV → Q, T
I → P, R; II → R, S; III → R, T; IV → P, T
I → Q, T; II → S, T; III → P, T; IV → Q, R
(I) \(\text{[Cr(CN)}_6\text{]}^{4-}\)
The Cr2+ ion, with the electron configuration [Ar] 3d4 4s0, undergoes d2sp3 hybridization due to the strong field ligand CN–.
(II) \(\text{[RuCl}_6\text{]}^{2-}\):
The Ru4+ ion, with the electron configuration [Kr] 4d4 5s0, has a t2g set that contains four electrons.
(III) \(\text{[Cr(H}_2\text{O)}_6\text{]}^{2+}\):
The Cr2+ ion, with the electron configuration [Ar] 3d4 4s0, exhibits four unpaired electrons due to the weak field ligand H2O, resulting in a magnetic moment of 4.9 B.M.
(IV) \(\text{[Fe(H}_2\text{O)}_6\text{]}^{2+}\)
The Fe2+ ion, with the electron configuration [Ar] 3d6 4s0, also possesses four unpaired electrons, resulting in a magnetic moment of 4.9 B.M.
Hence option A is Correct
In the following species, how many species have the same magnetic moment?
(i) Cr\(^{2+}\)
(ii) Mn\(^{3+}\)
(iii) Ni\(^{2+}\)
(iv) Sc\(^{2+}\)
(v) Zn\(^{2+}\)
(vi) V\(^{3+}\)
(vii) Ti\(^{4+}\)
The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity): 
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.