If \(x+\frac{1}{x}=2\), the principal value of \(\sin ^{-1} x\) is :

The minimum value of \(a x+b y\), where \(x y=c^2\) and \(a, b, c\) are positive, is:

If \(e^x+e^y=e^{x+y}\), then \(\frac{d y}{d x}=\)

The corner points of the feasible region determined by the set of constraints (linear inequations) are P(0,5), Q(3, 5), R(5, 0) and S(4, 1) and the objective function z = ax + 2by where a, b > 0 The condition on a and b such that maximum z occurs at Q and S is :

Consider the following data

Year (X)	2014	2016	2018	2020	2022	2024
Profit(in Rs. Thousand) (Y)	7	9	10	12	14	14

Then for the above data the equation of straight line trend by method of least square is given by:

The integral \(I=\int e^x\left(\frac{x-1}{3 x^2}\right) d x\) is equal to:

Let a relation \(R=\{(a, b): a\) is a factor of \(b, a, b \in N\}\). Then, \(R\) is __________.

The reverse osmosis occurs when

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In the formation of coordination complexes, if the inner \(d\) orbital \((n-1) d\) is used in hybridisation, the complex, is called an inner orbital or low spin or spin paired complex. And if it uses outer \(d\) orbital ( \(n d\) ) in hybridisation (like \(s p^3 d^2\) ), it is called outer orbital or high spin or spin free complex. The degeneracy of the \(d\) orbitals has been removed due to ligand electron-metal electron repulsions in the complex. This splitting of the degenerate levels due to the presence of ligands in a definite geometry is termed as crystal field splitting. For coordination complexes, the magnetic moment is determined by the number of unpaired electrons and is calculated by using the 'spin-only' formula, \(\mu=\sqrt{n(n+2)}\) where \(n\) is the number of unpaired electrons and \(\mu\) is the magnetic moment in units of Bohr magneton (BM). The coordination compounds are of great importance. These compounds are widely present in the mineral, plant and animal worlds and are known to play many important functions in the area of analytical chemistry, metallurgy, biological systems, industry and medicine.
Fe(II) octahedral complexes can show different 'spin only' magnetic moments for arrangements in the presence of weak and strong ligands. The 'spin only' magnetic moments of these complexes with weak and strong ligands, respectively, are

The molecular formula of dodecacarbonyltrimanganese \((0)\) is:

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Amines are classified as primary \(\left(1^{\circ}\right)\), secondary \(\left(2^{\circ}\right)\) and tertiary \(\left(3^{\circ}\right)\) depending upon the number of hydrogen atoms replaced by alkyl or aryl groups in ammonia molecules. Aliphatic amines are a stronger bases due to \(+I\) effect of an alkyl groups. Intermolecular hydrogen bonding in primary and secondary amines affects the boiling points of amines. The intermolecular association is more in primary amines than in secondary amines as there are more number of hydrogen atoms available for hydrogen bond formation. Nitro compounds are reduced to amines by passing hydrogen gas in the presence of finely divided nickel, palladium or platinum and also by reduction with metals in an acidic medium. Gabriel synthesis is used for the preparation of primary amines. Primary amines react with nitrous acid to give unstable diazonium-salts which can be used to synthesize a variety of compounds.
The compound having the highest boiling point is:

The value of \(\int \tan ^2 x d x\) is :
where \(C\) is the constant of integration.

Value of \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \log (\tan x) d x\) is