JEE Previous Year Question

Let $S=({\text{sin}}^2\text{⁡}2\theta \text{:}({\text{sin}}^4\text{⁡}\theta +{\text{cos}}^4\text{⁡}\theta )x^2+\text{(}\text{sin}\text{⁡}2\theta \text{)}x+({\text{sin}}^6\text{⁡}\theta +{\text{cos}}^6\text{⁡}\theta )=0)$ has real roots $\text{}}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3(\text{(}\alpha -2{\text{)}}^2+\text{(}\beta -1{\text{)}}^2)$ equals_________

Topic: JEE PYQ

Answer: 4. Full step-by-step solution below.