The absolute value of the tangent of the difference of the angles made by the lines $4x^2 - 24xy + 11y^2 = 0$ with the $X$-axis is

The structure of a mixed oxide is cubic close packed $(ccp)$. The cubic unit cell of the mixed oxide is composed of oxide ions. One-fourth of the tetrahedral voids are occupied by divalent metal $A$ and the octahedral voids are occupied by a monovalent metal $B$. The formula of the oxide is

In the visible region, the dispersive powers and the mean angular deviations for crown and flint glass prisms are $\omega, \omega^{\prime}$ and $d, d^{\prime}$ respectively. The condition for obtaining dispersion without deviation, when the two prisms are combined, is:

The function $f(x) = |x| + \frac{|x|}{x}$ is

For a suitably chosen real constant $a$,let a function $f: R-\{-a\} \rightarrow R$ be defined by $f(x) = \frac{a-x}{a+x}$. Further,suppose that for any real number $x \neq -a$ and $f(x) \neq -a$,$(f \circ f)(x) = x$. Then $f(-\frac{1}{2})$ is equal to

The natural frequency of an $L-C$ circuit is $1,25,000 \text{ cycles/s}$. Then the capacitor $C$ is replaced by another capacitor with a dielectric medium of dielectric constant $K$. In this case,the frequency decreases by $25 \text{ kHz}$. The value of $K$ is