Molten sodium chloride conducts electricity due to the presence of

If $\alpha$ is a non-real root of the equation $x^6-1=0$,then $\frac{\alpha^2+\alpha^3+\alpha^4+\alpha^5}{\alpha+1}$ is equal to

The $pH$ of $0.05 \ M$ acetic acid is $(K_a = 2 \times 10^{-5})$

Two identical piano wires have a fundamental frequency of $600 \text{ Hz}$ when kept under the same tension. What fractional increase in the tension of one wire will lead to the occurrence of $6 \text{ beats per second}$ when both wires vibrate simultaneously?

Let $C$ be the centre of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. The tangent at any point $P$ on this hyperbola meets the straight lines $bx - ay = 0$ and $bx + ay = 0$ at points $Q$ and $R$ respectively. Then $CQ \cdot CR = $