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Find the number of integral values of k ...

Find the number of integral values of k for which the equation `7 cos x+5 sin x=2k+1` has at least one solution.

Text Solution

Verified by Experts

The correct Answer is:
Eight values
We know that `a cos theta+b sin theta=c` has solution only when `|c| le sqrt(a^(2)+b^(2))`.
Then for the given equation, we must have
`|2k+1| le sqrt(74)`
`rArr -sqrt(74) lt 2k +1 lt sqrt(74)`
`rArr -8 lt 2k +1 lt 8` (for integral solutions)
`rArr k=-4, -3, -2, -1, 0, 1, 2, 3`
Thus, eight values of k will satisfy the given inequality.
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