If the centroid of the triangle formed by the lines $2y^2 + 5xy - 3x^2 = 0$ and $x + y = k$ is $\left(\frac{1}{18}, \frac{11}{18}\right)$,then the value of $k$ equals $............$

If $a, b$ and $c$ form a geometric progression with common ratio $r$,then the sum of the ordinates of the points of intersection of the line $ax + by + c = 0$ and the curve $x + 2y^2 = 0$ is

$A$ $20 \, cm$ length of a certain solution causes right-handed rotation of $38^\circ$. $A$ $30 \, cm$ length of another solution causes left-handed rotation of $24^\circ$. The optical rotation caused by a $30 \, cm$ length of a mixture of the above solutions in the volume ratio $1 : 2$ is:

What effect does branching of an alkane chain have on its boiling point?

During the mean life of a radioactive element,the fraction that disintegrates is

An organ pipe $P_1$,closed at one end and containing a gas of density $\rho_1$ is vibrating in its first harmonic. Another organ pipe $P_2$,open at both ends and containing a gas of density $\rho_2$ is vibrating in its third harmonic. Both the pipes are in resonance with a given tuning fork. If the compressibility of gases is equal in both pipes,the ratio of the lengths of $P_1$ and $P_2$ is (assume the given gases to be monoatomic).