If $$\overrightarrow a $$ and $$\overrightarrow b $$ are vectors in space given by $$\overrightarrow a = {{\widehat i - 2\widehat j} \over {\sqrt 5 }}$$ and $$\overrightarrow b = {{2\widehat i + \widehat j + 3\widehat k} \over {\sqrt {14} }},$$ then find the value of $$\,\left( {2\overrightarrow a + \overrightarrow b } \right).\left[ {\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow a - 2\overrightarrow b } \right)} \right].$$

If the distance between the plane $$Ax-2y+z=d$$ and the plane containing the lines $${{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over 4}$$ and $${{x - 2} \over 3} = {{y - 3} \over 4} = {{z - 4} \over 5}\,$$ is $$\sqrt 6 \,\,,$$ then find $$\left| d \right|.$$

Let $$ABC$$ be a triangle such that $$\angle ACB = {\pi \over 6}$$ and let $$a, b$$ and $$c$$ denote the lengths of the sides opposite to $$A$$, $$B$$ and $$C$$ respectively. The value(s) of $$x$$ for which $$a = {x^2} + x + 1,\,\,\,b = {x^2} - 1\,\,\,$$ and $$c = 2x + 1$$ is (are)

A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1 m and its crosssectional area is 4.9 $$ \times $$ 10^-7 m². If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency 140 rad s⁻¹. If the Young’s modulus of the material of the wire is n $$ \times $$ 10⁹ Nm^-2, the value of n is