Equation of the plane containing the straight line $${x \over 2} = {y \over 3} = {z \over 4}$$ and perpendicular to the plane containing the straight lines $${x \over 3} = {y \over 4} = {z \over 2}$$ and $${x \over 4} = {y \over 2} = {z \over 3}$$ is

For any real number $$x,$$ let $$\left[ x \right]$$ denote the largest integer less than or equal to $$x.$$ Let $$f$$ be a real valued function defined on the interval $$\left[ { - 10,10} \right]$$ by $$$f\left( x \right) = \left\{ {\matrix{ {x - \left[ x \right]} & {if\left[ x \right]is\,odd,} \cr {1 + \left[ x \right] - x} & {if\left[ x \right]is\,even} \cr } } \right.$$$

Then the value of $${{{\pi ^2}} \over {10}}\int\limits_{ - 10}^{10} {f\left( x \right)\cos \,\pi x\,dx} $$ is

Let $$\omega $$ be a complex cube root of unity with $$\omega \ne 1.$$ A fair die is thrown three times. If $${r_1},$$ $${r_2}$$ and $${r_3}$$ are the numbers obtained on the die, then the probability that $${\omega ^{{r_1}}} + {\omega ^{{r_2}}} + {\omega ^{{r_3}}} = 0$$ is

The value of $$\int\limits_0^1 {{{{x^4}{{\left( {1 - x} \right)}^4}} \over {1 + {x^2}}}dx} $$ is (are)