Q. The equation$\ \sin^{4}{x + \cos^{4}{x = \ \alpha\ }}$has a real solution, if
A) $0 < a \leq 1$
B) $\frac{1}{2} \leq a \leq 1$
C) $\frac{1}{4} \leq a \leq \frac{1}{2}$
D) $- 1 \leq a \leq 1$
Explanation
(b)
Given, $\sin^{4}{x + \cos^{4}{x = a}}$
$\Rightarrow \sin^{4}{x + (1 - {\sin^{2}{x)}}^{4} = a}$
$2\sin^{4}{x - 2\sin^{2}{x + (1 - a) = 0}}$
For real solution, $D \geq 0$
$\Rightarrow ( - 2)^{2} - 4 \times 2(1 - a) \geq 0$
$\Rightarrow 1 - 2 + 2a\ \geq 0$
$\Rightarrow a \geq \frac{1}{2}$
Hence, option (b) is true
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.