If $a, b$ and $c$ are the roots of $x^3+4x+1=0$,then $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=$

$A$ long straight rod of diameter $4 \ mm$ carries a steady current '$i$'. The current is uniformly distributed across its cross-section. The ratio of the magnetic fields at distances $1 \ mm$ and $4 \ mm$ from the axis of the rod is

The correct order of increasing $s-$ character (in percentage) in the hybrid orbitals of the following molecules/ions is:
$I. CO_3^{2-}$
$II. XeF_4$
$III. I_3^{-}$
$IV. NCl_3$
$V. BeCl_2$

The pulley and spring are massless,and friction is absent everywhere. The $5\,kg$ block is released from rest. The speed of the $5\,kg$ block when the $2\,kg$ block leaves contact with the ground is (take the force constant of the spring $K = 40\,N/m$ and $g = 10\,m/s^2$):

Which of the following is incorrect regarding $H_2O_2$?

Assertion : In the purely resistive element of a series $LCR$ $AC$ circuit,the maximum value of $rms$ current increases with an increase in the angular frequency of the applied $emf$.
Reason : $I_{\max} = \frac{\varepsilon_{\max}}{Z}$,where $Z = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2}$ and $I_{\max}$ is the peak current in a cycle.