Arrange these ions into decreasing order of their ionic radii :
(A) \(\text{Ce}^{3+}\)
(B) \(\text{Pr}^{3+}\)
(C) \(\text{La}^{3+}\)
(D) \(\text{Nd}^{3+}\)
(E) \(\text{Pm}^{3+}\)
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Read the passage carefully and answer the questions:
The degeneracy of the \(d\) orbitals has been removed due to ligand electron-metal electron repulsions in the octahedral complez 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 \(\Delta_{\circ}\) 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 \(\Delta_0\) and pairing energy, \(P\) (energy required for electron pairing in a single orbital) determine the formation of low spin ( \(\Delta_0>P\) ) or high spin ( \(\Delta_0<P\) ) complex. In tetrahedral coordination entity formation, the \(d\)-orbital splitting is inverted and is smaller as compared to the octabedral 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.
Which of the following relations is correct for the splitting of \(d\) orbitals in octahedral and tetrahedral crystal fields, considering same metal, the same ligands and metal-ligand distances?

DNA fingerprinting is used for:
(A) Forensic Laboratories for identification of criminals
(B) To identify paternity of an individual
(C) To identify the dead bodies by comparing the DNA of parents or children
(D) To identify racial groups, to rewrite biological evaluations.
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According to Arrhenius equation \(k=A e^{-\frac{E_a}{R T}}\). If In k vs 1/T graph plotted, what will be the intercept of this plot?

Phenol is brominated in solvent \(\text{CS}_2\) at low temperature. The product formed are :

\(\left[ Ag \left( NH _3\right)_2\right] OH\) forms silver when it reacts with:

Amongst the following, identify the condensation polymer/s:
(A)Bakelite
(B) Teflon
(C) Melamine formaldehyde resin
(D) Nylon-2 nylon-6
(E) Buna-S
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A telescope has an objective of focal length 30 cm and an eyepiece of focal length 3.0 cm. It is focused on a scale distant 2.0 m. For seeing with relaxed eye, the separation between the objective and eyepiece would be:

Match List-I with List-II

List – I (Name of Reaction)	List – II (Reagents)
(A) Stephens reaction	(I) \(CrO _2 Cl _2 / CS _2\)
(B) Rosenmund reduction	(II) Acyl chloride and anhyd. \(AlCl _3\)
(C) Etard reaction	(III) \(SnCl _2, HCl\)
(D) Friedel Craft acylation	(IV) \(H _2 / Pd - BaSO _4\)

Choose the correct answer from the options given below:

The maximum value of the objective function \(Z=8 x+2 y\) of an L.P.P. subject to the constraints: \(2 x+y \leq 3,2 x+3 y \leq 6, x \geq 0, y \geq 0 \) is:

If \(A=\left[\begin{array}{ccc}2 & -2 & 1 \\ 0 & 4 & 2\end{array}\right]\) and \(B=\left[\begin{array}{ccc}1 & -2 & 7 \\ 2 & 0 & 6\end{array}\right]\) are two matrices such that 3A - 2B + 4C = O, then matrix C is equal to:

If \(x=e^{\cos 2 t}, y=e^{\sin 2 t}\), then \(\frac{d y}{d x}\) equals to

Nitration of aniline with nitric acid and sulphuric acid gives major product as