What is the maximum molarity of \(MgCl_2\) in aqueous solution,
if its solubility product is \(4 \times 10^{-28}\)? 

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The degeneracy of the \(d\) orbitals has been removed due to ligand electron-metal electron repulsions in the octahedral complex to yield three orbitals of lower energy, \(t_{2 g}\) set and two orbitals of higher energy, \(e_g\) set. This splitting of the degenerate levels due to the presence of ligands in a definite geometry is termed as crystal field splitting and the energy separation is denoted by \(\Delta_0\). Thus energy of the two \(e_g\) orbitals will increase by \((3 / 5) \Delta_0\) and that of three \(t_{2 g}\) will decrease by \((2 / 5) \Delta_0\). The crystal field splitting \(\Delta_0\) depends upon the field produced by the ligand and charge on the metal ion. Some ligands are able to produce strong fields, in which case the splitting will be large, whereas others produce weak fields and consequently result in small splitting of \(d\) orbitals. Relative magnitude of crystal field splitting energy \(\Delta_{\circ}\) and pairing energy, \(P\) (energy required for electron pairing in a single orbital) determine the formation of low spin ( \(\Delta_{\circ}>P\) ) or high spin ( \(\Delta_{\circ}<P\) ) complex. In tetrahedral coordination entity formation, the \(d\)-orbital splitting is inverted and is smaller as compared to the octahedral field splitting. The crystal field theory attributes the colour of the complex to \(d-d\) transition of the electron. The colour of the coordination compounds depends on the crystal field splitting. In the absence of ligand, crystal field splitting does not occur and hence the substance is colourless
What will be the crystal field splitting energy for tetrahedral \(\left[ CoCl _4\right]^{2-}\) if its value for octahedral \(\left[ CoCl _6\right]^{4-}\) is \(18,000 cm^{-1}\) ?

A galvanic cell behaves as electrolytic cell when?

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There are mainly four series of the transition metals, \(3 d\) series ( \(S c\) to Zn), \(4 d\) series (Y to Cd), 5d series (La and Hf to Hg) and 6d series which has \(A c\) and elements from Rf to Cn. The two series of the inner transition metals; \(4 f\) (Ce to Lu) and \(5 f\) (Th to Lr) are known as lanthanoids and actinoids respectively. Their chemical properties are transitional between those of \(s\) and p-block elements. The transition metals (with the exception of Zn, Cd and Hg) are very hard and have low volatility. Their melting and boiling points are high. Transition metals have incomplete d subshell either in neutral atom or in their ions. With partly filled d orbitals these elements exhibit certain characteristic properties such as display of a variety of oxidation states, formation of coloured ions and entering into complex formation with a variety of ligands. The transition metals and their compounds are known for their catalytic activity.
In series Sc to Zn, the lowest enthalpy of atomization is shown by

The increasing order of magnetic behaviour for the following complexes is
A. \([Ti(H_2O)_6]^{3+}\)
B. \([V(H_2O)_6]^{3+}\)
C. \([MnCl_4]^{2-}\)
D. \([MnF_6]^{3-}\)
E. \([Cr(H_2O)_6]^{3+}\)
Choose the correct answer from the options given below :

Read the passage carefully and answer the Questions:
According to Valence Bond theory, the metal atom or ion under the influence of ligands can use its ( \(n-1\) )d, \(n s, n p\) or \(n s, n p, n d\) orbitals for hybridisation to yield a set of equivalent orbitals of definite geometry such as octahedral, tetrahedral, square planar and so on. These hybridised orbitals are allowed to overlap with ligand orbitals that can donate electron pairs for bonding. While the VB theory, to a larger extent, explains the formation, structures and magnetic behaviour of coordination compounds, it suffers from the many shortcomings. The Crystal field theory (CFT) is an electrostatic model which considers the metal-ligand bond to be ionic arising purely from electrostatic interactions between the metal ion and the ligand. The five \(d\) orbitals in an isolated gaseous metal atom/ion have same energy, i.e., they are degenerate. This degeneracy is maintained if a spherically symmetrical field of negative charges surrounds the metal atom/ion. However, when this negative field is due to ligands in a complex, it becomes asymmetrical and the degeneracy of the \(d\) orbitals is lifted. It results in splitting of the \(d\) orbitals. The pattern of splitting depends upon the nature of the crystal field.
Select the incorrect statement about the Crystal field splitting in octahedral complexes.

The correct statement (s) for cubic close packed (ccp) three dimensional structure are :
(A) The number of the neighbours of an atom present in the top most layer is 12.
(B) The efficiency of atom packing is \(74\%\).
(C) The number of octahedral and tetrahedral voids per atom are 1 & 2, respectively.
(D) The unit cell edge length is \(2\sqrt{2}\) times the radius of the atom.
(E) The metals like Be, Cd, Mg, Zn have cubic close packed (ccp) structure.
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In the 3d series of transition elements, the element more likely to exhibit +1 oxidation state is :

The function \(f(x)=x^3, x \in R\), has:

A company has been producing steel tubes of mean inner diameter 2 cm.
A sample of 10 tubes gives a mean inner diameter of 2.01 cm and a variance of \(0.0004 cm^2\).
The value of the test statistic is :

Match List-I (Name of the Base) with List-II (Structure)

List - I (Name of the Base)	List - II (Structure)
(A). Adenine (A)	(I)
(B). Cytosine (C)	(II)
(C). Thymine (T)	(III)
(D). Guanine (G)	(IV)

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In the Cell reaction :
\(Mg ( s )+2 Ag ^{+}( aq ) \rightarrow Mg ^{2+}( aq )+2 Ag ( s )\)
\(E _{\text {cell }}^{\circ}=3.17 V\). By doubling the concentration of \(Mg ^{2+}, E _{\text {cell }}\) is :

Solution of \(\frac{d y}{d x}=\left(1+x^2\right)\left(1+y^2\right)\) is :