The range of value's of $k$ for which the equation $2\cos^4 x - \sin^4 x + k = 0$ has atleast one solution is $[\lambda, \mu]$. Find the value of $(9\mu + \lambda)$
Step-by-Step Solution
Key Concept: Polynomial equations in trigonometric functions reduce to algebraic problems by substitution with domain restrictions.
Substituting $t = \sin^2 x$ where $t \in [0,1]$, the equation $\sin^4 x - 4\sin^2 x + (2+k) = 0$ becomes $t^2 - 4t + (2+k) = 0$. For real solutions in $[0,1]$, the discriminant and root constraints give $f(0)f(1) \leq 0$ and $(k+2)(k-1) \leq 0$, yielding $-2 \leq k \leq 1$.
Correct Answer: 7