The transformed equation of $3x^2 - 4xy = r^2$ when the coordinate axes are rotated about the origin through an angle of $\operatorname{Tan}^{-1}(2)$ in the positive direction is:

In the given figures,which path is the shortest?

$A$ source of sound of frequency $640 ~Hz$ is moving at a velocity of $\frac{100}{3} ~m/s$ along a road,and is at an instant $30 ~m$ away from a point $A$ on the road (as shown in figure). $A$ person standing at $O$,$40 ~m$ away from the road,hears sound of apparent frequency $v^{\prime}$. The value of $v^{\prime}$ is (velocity of sound $= 340 ~m/s$). (in $~Hz$)

The total number of nitrogenous bases in the human genome is estimated to be about

$\sum_{k=1}^{\infty} \frac{1}{k!}\left(\sum_{n=1}^k 2^{n-1}\right)$ is equal to

If $\alpha+\beta=-2$ and $\alpha^3+\beta^3=-56$,then the quadratic equation whose roots are $\alpha$ and $\beta$ is