Which of the following produces hydrolith with dihydrogen?

If $y=\tan ^{-1}\left(\frac{\sqrt{1+a^2 x^2}-1}{a x}\right)$,then $\left(1+a^2 x^2\right) y^{\prime \prime}+2 a^2 x y^{\prime}$ is equal to

$A$ line segment $AM = a$ moves in the $XOY$ plane such that $AM$ is parallel to the $X$-axis. If $A$ moves along the circle $x^2 + y^2 = a^2$,then the locus of $M$ is

If $f: R \rightarrow C$ is defined by $f(x)=e^{2 i x}$ for $x \in R$, then $f$ is (where $C$ denotes the set of all complex numbers)

Shreya read in her textbook that communicable diseases are caused by pathogens. These pathogens pass from patients to healthy persons causing the disease. She suggested her friends that we can prevent the spread of these diseases and even some of them can be eradicated.

$(a)$ What are the various methods of spread of communicable diseases?

$(b)$ What preventive measures can be taken to prevent the spread of communicable diseases?

$(c)$ Which values are displayed by Shreya in taking initiative ?

$(d)$ Suggest some school activity to promote such values in school students ?

The wavelengths of electron waves in two orbits is $3: 5$. The ratio of kinetic energy of electrons will be