Find the equation of the plane containing the line $$2x-y+z-3=0,3x+y+z=5$$ and at a distance of $${1 \over {\sqrt 6 }}$$ from the point $$(2, 1, -1).$$

A person goes to office either by car, scooter, bus or train, the probability of which being $${1 \over 7},{3 \over 7},{2 \over 7}$$ and $${1 \over 7}$$ respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is $${2 \over 9},{1 \over 9},{4 \over 9}$$ and $${1 \over 9}$$ respectively. Given that he reached office in time, then what is the probability that he travelled by a car.

If length of tangent at any point on the curve $$y=f(x)$$ intecepted between the point and the $$x$$-axis is length $$1.$$ Find the equation of the curve.

$$f(x)$$ is a differentiable function and $$g(x)$$ is a double differentiable
function such that $$\left| {f\left( x \right)} \right| \le 1$$ and $$f'(x)=g(x).$$
If $${f^2}\left( 0 \right) + {g^2}\left( 0 \right) = 9.$$ Prove that there exists some $$c \in \left( { - 3,3} \right)$$
such that $$g(c).g''(c)<0.$$