If $a$ and $b$ are odd integers,then the roots of the equation $2ax^2 + (2a + b)x + b = 0, a \ne 0,$ will be

$A$ cylindrical rod with one end in a steam chamber and the other end in ice results in the melting of $0.1 \, g$ of ice per second. If the rod is replaced by another with half the length and double the radius of the first,and if the thermal conductivity of the material of the second rod is $1/4$ that of the first,the rate at which ice melts in $g/s$ will be:

The network shown in the figure is part of a complete circuit. If at a certain instant,the current $I$ is $5 \ A$ and it is decreasing at a rate of $10^3 \ A \ s^{-1}$,and if $I$ is reversed in direction,then $V_B - V_A$ equals .......... $V$.

The condition for the lines $lx + my + n = 0$ and $l_1x + m_1y + n_1 = 0$ to be conjugate with respect to the circle $x^2 + y^2 = r^2$ is

$A$ molecule that can act as a genetic material must fulfill the traits given below,except.

Consider a block of conducting material of resistivity ' $\rho$ ' as described. Current ' $I$ ' enters at ' $A$ ' and leaves from ' $D$ '. We apply the superposition principle to find the voltage ' $\Delta V$ ' developed between ' $B$ ' and ' $C$ '. The calculation is done in the following steps:
$(i)$ Take current ' $I$ ' entering from ' $A$ ' and assume it to spread over a hemispherical surface in the block.
(ii) Calculate the field $E(r)$ at distance ' $r$ ' from $A$ by using Ohm's law $E = \rho j$,where $j$ is the current per unit area at ' $r$ '.
(iii) From the ' $r$ ' dependence of $E(r)$,obtain the potential $V(r)$ at $r$.
(iv) Repeat $(i)$,(ii),and (iii) for current ' $I$ ' leaving ' $D$ ' and superpose results for ' $A$ ' and ' $D$ '.
For current entering at $A$,the electric field at a distance ' $r$ ' from $A$ is: