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The solution of the equation kcosx3sinx=k+1 is possible if

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a
k∈(−∞,4]
b
k∈(−∞,∞)
c
k∈[4,∞)
d
None of the above

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detailed solution

Correct option is A

Here, k cos x - 3 sin x = k +1 could be rewritten askk2+9cos⁡x−3k2+9sin⁡x=k+1k2+9⇒ cos⁡(x+ϕ)=k+1k2+9where cos⁡ϕ=kk2+9and sin⁡ϕ=3k2+9which posses solution only, if−1≤k+1k2+9≤1⇒k+1k2+9≤1⇒(k+1)2≤k2+9⇒k2+2k+1≤k2+9⇒k≤4Hence, the interval in which kcos x - 3sin x = k +1 admits solution for k is (−∞,4]

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