Which of the following is an emulsifier?

The vectors $b$ and $c$ are in the direction of north-east and north-west respectively and $|b|=|c|= 4$. The magnitude and direction of the vector $d = c - b$ are:

In a meter bridge,a $30 \Omega$ resistance is connected in the left gap and a pair of resistances $P$ and $Q$ in the right gap. Measured from the left,the balance point is $37.5 \text{ cm}$ when $P$ and $Q$ are in series,and $71.4 \text{ cm}$ when they are in parallel. The values of $P$ and $Q$ (in $\Omega$) are:

$4 \ g$ of a hydrocarbon on complete combustion gave $12.571 \ g$ of $CO_2$ and $5.143 \ g$ of water. What is the empirical formula of the hydrocarbon?

If the chord of contact of tangents from a point on the circle $x^2 + y^2 = r_1^2$ to the circle $x^2 + y^2 = r_2^2$ touches the circle $x^2 + y^2 = r_3^2$,then $r_1, r_2, r_3$ are in:

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