Number of electrons present in \(4 \mathrm{f}\) orbital of \(\mathrm{Ho}^{3+}\) ion is \(.....\) (Given Atomic No. of \(\mathrm{Ho}=67\) )

Arrange the following in order of magnitude of work done by the system / on the system at constant temperature :
(a) \(\left|\mathrm{w}_{\text {reversible }}\right|\) for expansion in infinite stage.
(b) \(\left|w_{\text {irreversible }}\right|\) for expansion in single stage.
(c) \(\left|w_{\text {reversible }}\right|\) for compression in infinite stage.
(d) \(\left|w_{\text {irreversible }}\right|\) for compression in single stage.
Choose the correct answer from the options given below:

Consider that a \(d ^{6}\) metal ion \(\left( M ^{2+}\right)\) forms a complex with aqua ligands, and the spin only magnetic moment of the complex is \(4.90\) \(BM.\) The geometry and the crystal field stabilization energy of the complex is:

Total number of possible stereoisomers of dimethyl cyclopentane is ....

For the reaction given below : The compound which is not formed as a product in the reaction is a :

The number of non-ionisable hydrogen atoms present in the final product obtained from the hydrolysis of \(\mathrm{PCl}_{5}\) is :

Electricity is passed through an acidic solution of \( Cu^{2+} \) till all the \( Cu^{2+} \) was exhausted, leading to the deposition of 300 mg of Cu metal. However, a current of 600 mA was continued to pass through the same solution for another 28 minutes by keeping the total volume of the solution fixed at 200 mL. The total volume of oxygen evolved at STP during the entire process is ______ mL.
(Nearest integer)
[Given :
\(Cu ^{2+}( aq )+2 e ^{-} \rightarrow Cu ( s ) E _{\text {red }}^0=+0.34 V\)
\(O _2(g)+4 H ^{+}+4 e ^{-} \rightarrow 2 H _2 O E _{ red }^0=+1.23 V\)
Molar mass of \(Cu =63.54~g~mol ^{-1}\)
Molar mass of \(O _2=32~g~mol ^{-1}\)
Faraday Constant \(=96500~C~mol ^{-1}\)
Molar volume at \(S T P=22.4 L\) ]

Let \(S\) be the sum of the first \(9\) terms of the series: \(\{x+k a\}+\left\{x^{2}+(k+2) a\right\}+\left\{x^{3}+(k+4) a\right\}+\) \(\left\{x^{4}+(k+6) a\right\}+\ldots \ldots\) where \(a \neq 0\) and \(x \neq 1 .\) If \(S =\frac{ x ^{10}- x +45 a ( x -1)}{ x -1},\) then \(k\) is equal to

A water tank has the shape of an inverted right circular cone, whose semi vertical angle is \({\tan ^{ - 1}}\,\left( {\frac{1}{2}} \right)\). Water is poured in at a constant rate of \(5\) cubic meter per minute. Then the rate (in \(m/min\)) at which the level of water is rising at the instant when the depth of water in the tank is \(10\, m\) is

Let \(A=\{(a,b,c): a,b,c \text{ are non-negative integers and } a+b+2c=22\}\). Then \(n(A)\) is equal to:

The vernier constant of Vernier callipers is \(0.1 \,mm\) and it has zero error of \((-0.05) \,cm\). While measuring diameter of a sphere, the main scale reading is \(1.7 \,cm\) and coinciding vernier division is \(5\). The corrected diameter will be ........... \(\times 10^{-2} \,cm\)

Given below are two statements: Statement \((I)\) : Fusion of \(\mathrm{MnO}_2\) with \(\mathrm{KOH}\) and an oxidising agent gives dark green \(\mathrm{K}_2 \mathrm{MnO}_4\). Statement \((II)\) : Manganate ion on electrolytic oxidation in alkaline medium gives permanganate ion. In the light of the above statements, choose the correct answer from the options given below.

Given; A circle \(2{x^2} + 2{y^2} = 5\) and parabola \({y^2} = 4\sqrt 5 x\) Statement \(-1\):An equation of a common tangent to these curve is  \(y = x + \sqrt 5 \) Statement \(-2\): If the line, \(y = mx + \frac{{\sqrt 5 }}{m}\left( {m \ne 0} \right)\) is their common tangent , then \(m\) satisfies \({m^4} - 3{m^2} + 2 = 0\).