The number of integral values of $k$,for which the equation $7\cos x + 5\sin x = 2k + 1$ has a solution,is

The height of the cone of maximum volume inscribed in a sphere of radius $R$ is

The vector sum of two forces is perpendicular to their vector difference. In that case,the forces

The first minima due to a single slit diffraction is at $\theta = 30^o$ for a light of wavelength $5000\,\mathring{A}$. The width of the slit is:

The narrower distal end of the stomach is called:

When metal $X$ is treated with sodium hydroxide,a white precipitate $(A)$ is obtained,which is soluble in excess of $NaOH$ to give soluble complex $(B)$. Compound $(A)$ is soluble in dilute $HCl$ to form compound $(C)$. The compound $(A)$ when heated strongly gives $(D)$,which is used to extract metal. Identify $(X), (A), (B), (C)$ and $(D)$. Write suitable equations to support their identities.