The string of a pendulum of length $l$ is displaced through $90^o$ from the vertical and released. Then,the minimum strength of the string in order to withstand the tension as the pendulum passes through the mean position 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:

An electron in a Hydrogen atom first jumps from the third excited state to the second excited state and then from the second excited state to the first excited state. The ratio of the wavelengths $\lambda_{1} : \lambda_{2}$ emitted in the two cases respectively is:

The length of the $x$-intercept made by the pair of lines $2x^2 + xy - 6y^2 - 2x + 17y - 12 = 0$ is

If $\sin 6 \theta = 32 \cos^5 \theta \sin \theta - 32 \cos^3 \theta \sin \theta + 3x$,then $x$ is equal to:

In the reaction: $P + Q \longrightarrow R + S$,the time taken for $75\%$ reaction of $P$ is twice the time taken for $50\%$ reaction of $P$. The concentration of $Q$ varies with reaction time as shown in the figure. The overall order of the reaction is: