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QJEEM21Mac14c: JEE Main 2021 Mathematics - Class 11 JEEM21Mac14c Solution: The number of integral values of 'k' for which the equation has …

The number of integral values of 'k' for which the equation 3sin⁡x+4cos⁡x=k+13\sin x + 4\cos x = k + 1 has a solution, k ∈\in R is ___________.

Step-by-Step Verified Solution

We know,

−a2+b2≤acos⁡x+bsin⁡x≤a2+b2- \sqrt {{a^2} + {b^2}} \le a\cos x + b\sin x \le \sqrt {{a^2} + {b^2}}

∴\therefore −32+42≤3cos⁡x+4sin⁡x≤32+42- \sqrt {{3^2} + {4^2}} \le 3\cos x + 4\sin x \le \sqrt {{3^2} + {4^2}}

−5≤k+1≤5- 5 \le k + 1 \le 5

−6≤k≤4- 6 \le k \le 4

∴\therefore Set of integers = −6,−5,−4,−3,−2,−1,0,1,2,3,4- 6, - 5, - 4, - 3, - 2, - 1,0,1,2,3,4 = Total 11 intergers.

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