Let `2sin^2x+3sinx-2>0a n dx^2-x-2<0(x` is measured in radians). Then `x` lies in the interval `(pi/6,(5pi)/6)` (b) `(-1,(5pi)/6)` `(-1,2)` (d) `(pi/6,2)`

Options

A. `(pi//6,5pi//6)`
B. `(-1,5pi//6)`
C. `(-1,2)`
D. `(pi//6,2)`

Answer

D

Explanation

We have,
`2sin^(2)x+3 sin x-2 gt 0`
`implies(2sinx-1)(sinx+2) gt 0`
`implies 2 sin x-1 gt 0" "[:' sin x+2 gt 0 "for all "x]`
`implies sin x gt (1)/(2) rArr x in (pi//6,5pi//6)" "`....................(i)
Also,
`x^(2)-x-2 lt 0`
`implies (x-2)(x+1) lt 0`
`implies -1 lt x lt 2 implies in (-1,2)" "`..........(ii)
Since` 2 lt (5pi)/(6)`, therefore values of x satisfying (i) and (ii) lie in the interval `(pi//6,2)`.