If L = sin²$$\left( {{\pi \over {16}}} \right)$$ - sin²$$\left( {{\pi \over {8}}} \right)$$ and
M = cos²$$\left( {{\pi \over {16}}} \right)$$ - sin²$$\left( {{\pi \over {8}}} \right)$$, then :

The derivative of
$${\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$$ with
respect to $${\tan ^{ - 1}}\left( {{{2x\sqrt {1 - {x^2}} } \over {1 - 2{x^2}}}} \right)$$ at x = $${1 \over 2}$$ is :

If the system of linear equations
x + y + 3z = 0
x + 3y + k²z = 0
3x + y + 3z = 0
has a non-zero solution (x, y, z) for some k $$ \in $$ R, then x + $$\left( {{y \over z}} \right)$$ is equal to :

If the length of the chord of the circle,
x² + y² = r² (r > 0) along the line, y – 2x = 3 is r,
then r² is equal to :