If $\frac{1}{2} \leq x \leq 1$,then $\cos ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{1}{2} \sqrt{3-3 x^2}\right)$ is equal to

The half-life of a radioactive substance against $\alpha$-decay is $1.2 \times 10^7 \, s$. What is the decay rate for $4.0 \times 10^{15}$ atoms of the substance?

If $\int(1-\cos x) \operatorname{cosec}^2 x \, dx = f(x) + c$, then $f(x)$ is equal to

In which of the following production processes is limestone $(CaCO_3)$ $NOT$ used?

If all interior angles of a quadrilateral are in $AP$ and the common difference is $10^{\circ}$,then find the smallest angle in degrees.

Two identical condensers $M$ and $N$ are connected in series with a battery. The space between the plates of $M$ is completely filled with a dielectric medium of dielectric constant $8$,and a copper plate of thickness $d/2$ is introduced between the plates of $N$ ($d$ is the distance between the plates). Then the potential differences across $M$ and $N$ are,respectively,in the ratio: