`PQ` is a smooth inclined plane whose angle `theta` can be varied in such a way that point `Q` remains fixed and `P` can move on a vertical line `PR`. A particle slides from rest from point `P` at different value of `theta` time for descent from `P` and `Q` is noted. The following statement `(s)` is `//` are correct about the time of descent `:`

Options

A. the minimum time of descent is `2sqrt(l//g)`
B. the time descent is minimum continuously as `theta` is increased
C. the time of descent decreases continuously as `theta` is increased
D. the time of descent first decreases then increases.

Answer

A, D

Explanation

`(l)/(cos theta)=0+(1)/(2)(g sin theta).t^(2)`
`t=sqrt((2l)/(g sin theta. cos theta))=sqrt((4l)/(g sin 2theta))` `t` will be minimum if `sin 2 theta =` maximum
`theta=45^(@)`
`t_(mi n)=2sqrt((l)/(g))`
Since as `theta` increases from `0` to `45^(@)t` decreases and as `theta` increases `45^(@)` to `90^(@)t` increases.