If A,B,C are real and A+B+C=π and for k>12 the maximum value of cosA+cosB+k.cosC is 12αk+βk where α,β∈N,G.C.Dα,β=1 then the median of α,β,α+2,β+2,α+β equals

y=cosA+cosB+kcosC⇒ g=2sinC2cosA−B2−k1−2sin2C2⇒ 2kx2−2cosA−B2x+y−k=0 sinC2=k⇒ Δ≥0⇒ 8ky−k≤4cos2A−B2 ymin=121k+2k ⇒ A=1,B=2 ⇒A+B=3

If A,B,C are real and A+B+C=π and for k>12 the maximum value of cosA+cosB+k.cosC is 12αk+βk where α,β∈N,G.C.Dα,β=1 then the median of α,β,α+2,β+2,α+β equals

y=cosA+cosB+kcosC⇒ g=2sinC2cosA−B2−k1−2sin2C2⇒ 2kx2−2cosA−B2x+y−k=0 sinC2=k⇒ Δ≥0⇒ 8ky−k≤4cos2A−B2 ymin=121k+2k ⇒ A=1,B=2 ⇒A+B=3

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If $A, B, C$ are real and $A + B + C = π$ and for $k > \frac{1}{2}$ the maximum value of $\cos A + \cos B + k . \cos C$ is $\frac{1}{2} (\frac{α}{k} + β k)$ where $α, β \in N, G . C . D (α, β) = 1$ then the median of $α, β, α + 2, β + 2, α + β$ equals

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$y = \cos A + \cos B + k \cos C$

$\Rightarrow g = 2 \sin \frac{C}{2} \cos (\frac{A - B}{2}) - k (1 - 2 \sin^{2} \frac{C}{2})$

$\Rightarrow 2 k x^{2} - 2 \cos (\frac{A - B}{2}) x + (y - k) = 0 (\sin \frac{C}{2} = k)$ $\Rightarrow Δ \geq 0$

$\Rightarrow 8 k (y - k) \leq 4 \cos^{2} (\frac{A - B}{2})$ $y_{\min} = \frac{1}{2} (\frac{1}{k} + 2 k)$ $\Rightarrow$ $A = 1, B = 2$ $\Rightarrow$ $A + B = 3$

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If A,B,C are real and A+B+C=π and for k>12 the maximum value of cosA+cosB+k.cosC is 12αk+βk where α,β∈N,G.C.Dα,β=1 then the median of α,β,α+2,β+2,α+β equals

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If $A, B, C$ are real and $A + B + C = π$ and for $k > \frac{1}{2}$ the maximum value of $\cos A + \cos B + k . \cos C$ is $\frac{1}{2} (\frac{α}{k} + β k)$ where $α, β \in N, G . C . D (α, β) = 1$ then the median of $α, β, α + 2, β + 2, α + β$ equals