Select the correct statement:

The complete set of values of $x$ for which the function $f(x) = 2 \log_e(x-2) - x^2 + 4x + 1$ is increasing is:

Aqueous solutions of two compounds $M_1-OH$ and $M_2-OH$ are prepared in two different beakers. If the electronegativity of $M_1 = 3.4, M_2 = 1.2, O = 3.5$ and $H = 2.1$,then the nature of the two solutions will be respectively:

The force $\vec{F} = 7\widehat{i} + 3\widehat{j} - 5\widehat{k}$ acts on a particle whose position vector is $\vec{r} = \widehat{i} - \widehat{j} + \widehat{k}$. What is the torque of the given force about the origin?

The coefficient of $t^{50}$ in $(1 + t^2)^{25} (1 + t^{25}) (1 + t^{40}) (1 + t^{45}) (1 + t^{47})$ is

For $100\%$ modulation $(AM)$,the useful part of the total power radiated is