Let `theta in [0,4pi]` satisfy the equation `(sintheta+2)(sintheta+3)(sintheta+4)=6.` If the sum of all the values of `theta` is of the form `kpi` , then the value of `k` is 6 (b) 5 (c) 4 (d) 2

Explanation

`(sintheta+2)(sintheta+3)(sintheta+4) = 6`

Only solution, possible for the given equation is `sintheta = -1`, which is minimum value of `sintheta.`

If we even slightly increase the value of `sin theta`, then result becomes greater than `6`.

`:. sin theta = -1`

So, for `theta in [0,4pi]`, two solutions possible for `sin theta = -1`.

Those solutions are , `theta = (3pi)/2 and theta = (7pi)/2.`

`:.` Sum of solutions ` = (3pi)/2+ (7pi)/2 = 5pi`

`:. kpi = 5pi`

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