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Find the no. of integral values of k for...

Find the no. of integral values of k for which the equation `7cosx+5sinx=2k+1` has atleast one solution.

Text Solution

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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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Knowledge Check

  • The number of integer values of k, for which the equation 2 sin x =2k+1 has a solution, is

    A
    zero
    B
    one
    C
    two
    D
    four

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