Find the length of the intercept cut by the pair of lines $2x^2+4xy-4y^2-6x-8y+7=0$ on the $Y$-axis.

$A$ conducting rod with resistance $r$ per unit length is moving inside a vertical magnetic field $\vec B$ at constant speed $v$ on two horizontal parallel ideal conductor rails. The ends of the rails are connected to a resistor $R$. The separation between the rails is $d$. The rod maintains a tilted angle $\theta$ to the rails. Find the external force required to keep the rod moving.

$A$ light wave has a frequency of $4 \times 10^{14} \text{ Hz}$ and a wavelength of $5 \times 10^{-7} \text{ m}$ in a medium. The refractive index of the medium is

If $\log (1+x) - \frac{2x}{2+x}$ is increasing, then

$A$ body of mass $M$ at rest explodes into three pieces,two of which of mass $M/4$ each are thrown off in perpendicular directions with velocities of $3 \ m/s$ and $4 \ m/s$ respectively. The third piece will be thrown off with a velocity of.......... $m/s$.

$Br_2 + 2X^{-} \to 2Br^{-} + X_2$
$Cl_2 + 2Y^{-} \to 2Cl^{-} + Y_2$
The incorrect match regarding the above reactions is: