If [x] denotes the greatest integer `le x`, then the system of liner equations `["sin" theta]x + [-"cos"theta]y= 0, ["cot" theta]x + y =0`

Options

A. have infinitely many solutions if `theta in ((pi)/(2), (2pi)/(3))` and has a unique solution if `theta in (pi, (7pi)/(6))`
B. has a unique solution if
`theta in ((pi)/(2), (2pi)/(3)) cup (pi, (7pi)/(6))`
C. has a unique solution if `theta ((pi)/(2), (2pi)/(3))` and have infinitely many solutions if `theta in (pi, (7pi)/(6))`
D. have infinitely many solutions if
`theta in ((pi)/(2), (2pi)/(3)) cup (pi, (7pi)/(6))`

Explanation

Given system of linear equations is
`["sin" theta ] x + [-"cos" theta] y = 0 " "… (i)`
`"and "["cot" theta] x +y = 0 " "…(ii)`
where, [x] denotes the greatest integer `le` x.
`"Here, "Delta = |{:(["sin" theta], [-"cos" theta]), (["cot" theta], " "1):}|`
`rArr Delta = ["sin" theta] - [-"cos" theta] ["cot" theta]`
When `theta in ((pi)/(2), (2pi)/(3))`
`"sin theta in ((sqrt(3))/(2), 1)`
`rArr ["sin" theta] = 0 " "...(iii)`
`-"cos" theta in (0, (1)/(2))`
`rArr [-"cos" theta] = 0 " "...(iv)`
`"and cot" theta in (-(1)/(sqrt(3)), 0)`
`rArr ["cot" theta] = -1 " "... (v)`
`"So, " Delta = ["sin" theta] - [-"cos" theta]["cot" theta]`
`-(0 xx (-1)) =0 " " ["from Eqs. (iii), (iv) and (v)"]`
Thus, for `theta in ((pi)/(2), (2pi)/(3))`, the given system have infinitely many solutions.
`"When "theta in (pi, (7pi)/(6)), "sin" theta in (-(1)/(2), 0)`
`rArr ["sin" theta] = -1`
`-"cos" theta in ((sqrt(3))/(2), 1) rArr ["cos" theta] = 0`
`"and cot" theta in (sqrt(3), oo) rArr ["cot" thea] = n, n in N`
`"So, "Delta = -1 - (0 xx n) = -1`
Thus, for `theta in (pi, (7pi)/(6))`, the given system has a unique solution.