The angle of elevation of the top of a $T.V.$ tower from three points $A, B, C$ in a straight line through the foot of the tower are $\alpha, 2\alpha, 3\alpha$ respectively. If $AB = a$,the height of the tower is

The number of solutions for $z^3+\bar{z}=0$ is

$A$ car $P$ approaching a crossing at a speed of $10\, m/s$ sounds a horn of frequency $700\, Hz$ when $40\, m$ in front of the crossing. The speed of sound in air is $340\, m/s$. Another car $Q$ is at rest on a road which is perpendicular to the road on which car $P$ is reaching the crossing (see figure). The driver of car $Q$ hears the sound of the horn of car $P$ when he is $30\, m$ in front of the crossing. The apparent frequency heard by the driver of car $Q$ is .... $Hz$.

The energy of a photon is equal to the kinetic energy of a proton. If $\lambda_1$ is the de-Broglie wavelength of a proton and $\lambda_2$ is the wavelength associated with the photon,and if the energy of the photon is $E$,then $(\lambda_1 / \lambda_2)$ is proportional to

$DNA$ fingerprinting refers to

Molar volume $( V_m )$ of a van der Waals gas can be calculated by expressing the van der Waals equation as a cubic equation with $V_m$ as the variable. The ratio $($in $mol dm ^{-3} )$ of the coefficient of $V_m^2$ to the coefficient of $V_m$ for a gas having van der Waals constants $a=6.0 dm ^6 atm \ mol ^{-2}$ and $b=0.060 dm ^3 mol^{-1}$ at $300 K$ and $300 atm$ is $....$ .
Use: Universal gas constant $(R)=0.082 dm ^3 atm \ mol { }^{-1} K^{-1}$