The number of $5$-digit numbers that are not divisible by $5$ and consist of different odd digits is:

If $x=a\left\{\cos \theta+\log \tan \left(\frac{\theta}{2}\right)\right\}$ and $y=a \sin \theta$,then $\frac{d y}{d x}$ is equal to

The distance of the point $(1, 1, 9)$ from the point of intersection of the line $\frac{x-3}{1} = \frac{y-4}{2} = \frac{z-5}{2}$ and the plane $x+y+z=17$ is

$Ge$ and $Si$ diodes conduct at $0.3\,V$ and $0.7\,V$ respectively. In the following figure,if the $Ge$ diode connection is reversed,the value of $V_0$ changes by......$V$.

Which of the following reactions is correct?

If the ratio of diameters,lengths and Young's moduli of steel and brass wires shown in the figure are $p, q$ and $r$ respectively,then the corresponding ratio of increase in their lengths would be