The value of $a = - λ - \frac{π}{μ}$ , such that $x^{2} + a x + s i n^{- 1} (x^{2} - 4 x + 5) + c o s^{- 1} (x^{2} - 4 x + 5) = 0$ has atleast one solution. Then $λ + μ$ is equal to

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Solution

The correct option is C
6

Since $s i n^{- 1} y$ is defined for $- 1 \leq y \leq 1$
$- 1 \leq x^{2} - 4 x + 5 \leq 1$
$\Rightarrow x^{2} - 4 x + 4 \leq 0$ and $x^{2} - 4 x + 6 \geq 0$
$\Rightarrow (x - 2)^{2} \leq 0$ and $(x - 2)^{2} + 2 \geq 0$
$\Rightarrow x = 2$ is the only solution
On putting x = 2 in the original equation we get
$4 + 2 a + \frac{π}{2} = 0 \Rightarrow 2 a = - 4 - \frac{π}{2}$
$\Rightarrow a = - 2 - \frac{π}{4} \Rightarrow λ = 2$ and $μ = 4$
Hence $λ + μ = 2 + 4 = 6$

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