In $$\triangle \mathrm{ABC}, \mathrm{m} \angle \mathrm{B}=\frac{\pi}{3}$$ and $$\mathrm{m} \angle \mathrm{C}=\frac{\pi}{4}$$. Let point $$\mathrm{D}$$ divide $$\mathrm{BC}$$ internally in the ratio $$1: 3$$, then $$\frac{\sin (\angle B A D)}{\sin (\angle C A D)}$$ has the value

If $$\tan ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \tan ^{-1} x$$, then $$x$$ is

If

then $$|\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}}| \text { is }$$

Three fair coins with faces numbered 1 and 0 are tossed simultaneously. Then variance (X) of the probability distribution of random variable $$\mathrm{X}$$, where $$\mathrm{X}$$ is the sum of numbers on the upper most faces, is