Let S = { sin 2 2 θ : ( sin 4 θ + cos 4 θ ) x 2 + ( sin 2 θ ) x + ( sin 6 θ + cos 6 θ ) = 0 has real roots } . If α and β be the smallest and largest elements of the set S , respectively, then 3 ( ( α

Question

Let S = { sin 2 2 θ : ( sin 4 θ + cos 4 θ ) x 2 + ( sin 2 θ ) x + ( sin 6 θ + cos 6 θ ) = 0 has real roots } . If α and β be the smallest and largest elements of the set S , respectively, then 3 ( ( α − 2 ) 2 + ( β − 1 ) 2 ) equals _________

TestHub · Accepted Answer

​ D = ( sin 2 θ ) 2 − 4 ( 1 − 2 sin 2 2 θ ​ ) ( 1 − 4 3 ​ sin 2 2 θ ) = ( sin 2 θ ) 2 − 4 ( 1 − 4 5 ​ sin 2 2 θ + 8 3 ​ sin 4 2 θ ) D = − 2 3 ​ sin 4 2 θ + 6 sin 2 2 θ − 4 > 0 3 sin 4 2 θ − 12 sin 2 2 θ + 8 < 0 sin 2 2 θ = 6 12 ± 1 2 2 − 12.8 ​ ​ = 6 12 ± 4 3 ​ ​ = 3 6 ± 2 3 ​ ​ sin 2 2 θ = 2 ± 3 ​ 2 ​ , but sin 2 2 θ ∈ [ 0 , 1 ] ∴ α = 2 − 3 ​ 2 ​ , β = 1 → ( α − 2 ) 2 = 3 4 ​ , ( β − 1 ) 2 = 0 3 ( α − 2 ) 2 + ( β − 1 ) 2 = 4 ​

Mathematics - Trigonometric Equation Question with Solution | TestHub

Doubts & Discussion