\(\int_1^2 \tan ^{-1}\left(\frac{x}{x^2+1}\right)+\tan ^{-1}\left(\frac{x^2+1}{x}\right) d x=\)

The coefficient of \(x^3\) in the expansion of \(\frac{x^4+1}{\left(x^2+1\right)(x-1)}\) when it is expressed in terms of positive integral powers of \(x\), is

The sum of integers from 1 to 50 that are divisible by 2 and 3 is

If \(\mathrm{a}\) and \(\mathrm{b}\) are the roots of the equation \(\mathrm{y}^2+\mathrm{y}+1=0\), then the value of \(a^4+b^4+a^{-1} b^{-1}\) is

The solution of \(\frac{d y}{d x}=\frac{y^2}{x y-x^2}\) is

If the chord \(x+y=1\) of the circle \(x^2+y^2=a^2\) subtends a right angle at the origin, then \(a=\)

Identify the ortho and para directing groups towards aromatic electrophilic substitution reactions from the following list

$-\mathrm{OH}$	$-\mathrm{CN}$	$-{\mathrm{CO}}_2\mathrm{H}$	$-{\mathrm{OCH}}_3$	$-{\mathrm{NHCOCH}}_3$	$-\mathrm{CHO}$
I	II	III	IV	V	VI

A bag contains 10 identical pens, of which 4 are red and 6 are blue. 3 pens are taken out at random one after another. Find probability that all 3 are blue

If \(a, b\) and \(c\) are the roots of \(x^3+q x+r=0\), then \((a-b)^2+(b-c)^2+(c-a)^2=\)

When ' \(\mathrm{X}\) ' \(\mathrm{g}\) of graphite is completely burnt in a bomb calorimeter in excess of \(\mathrm{O}_2\) at \(298 \mathrm{~K}\) and \(1 \mathrm{~atm}\) pressure as given in the equation
\(\mathrm{C}\) (graphite) \(+\mathrm{O}_2(\mathrm{~g}) \rightarrow \mathrm{CO}_2(\mathrm{~g})\)
The temperature of calorimeter raised from \(298 \mathrm{~K}\) to \(302 \mathrm{~K}\). If the heat capacity of the calorimeter and molar enthalpy change for the reaction at \(1 \mathrm{~atm}\) and \(298 \mathrm{~K}\) are \(20.7 \mathrm{~kJ} \mathrm{~K}^{-1}\) and \(-248.4 \mathrm{~kJ} \mathrm{~mol}^{-1}\), ' \(\mathrm{X}^{\prime}\) in \(\mathrm{g}\) is

The domain of defined of the function \(f(x)=\sqrt{\frac{1-|x|}{2-|x|}}\) is

Compound \(X\) is anhydride of sulphuric acid. The number of \(\sigma\)-bonds and the number of \(\pi\)-bonds present in \(X\) are, respectively :