Let α θ and β θ be the distinct roots of 2 x 2 + ( cos θ ) x − 1 = 0 , θ ∈ ( 0 , 2 π ) . If m and M are the minimum and the maximum values of α θ 4 + β θ 4 , then 16 ( M + m ) equals :

Question

Let α θ ​ and β θ ​ be the distinct roots of 2 x 2 + ( cos θ ) x − 1 = 0 , θ ∈ ( 0 , 2 π ) . If m and M are the minimum and the maximum values of α θ 4 ​ + β θ 4 ​ , then 16 ( M + m ) equals :

TestHub · Accepted Answer

​ 2 x 2 + ( cos θ ) x − 1 = 0 α θ ​ + β θ ​ = 2 − cos θ ​ α θ ​ ⋅ β θ ​ = 2 − 1 ​ α θ 2 ​ + β θ 2 ​ = ( α θ ​ + β θ ​ ) 2 − 2 α θ ​ β θ ​ 4 cos 2 θ ​ + 1 ​ ​ α θ 4 ​ + β θ 4 ​ = ( a θ 2 ​ + β θ 2 ​ ) 2 − 2 α θ 2 ​ β θ 2 ​ = ( 4 cos 2 θ ​ + 1 ) 2 − 4 2 ​ = ( 4 cos 2 θ ​ + 1 ) 2 − 2 1 ​ ​ Maximum when cos θ = 1 ​ M = ( 4 1 ​ + 1 ) 2 − 2 1 ​ M = 16 17 ​ ​ Minimum when cos θ = 0 ​ m = 1 − 2 1 ​ = 2 1 ​ 16 ( M + m ) = 16 ( 16 17 ​ + 2 1 ​ ) = 25 ​

Mathematics - Quadratic Equation Question with Solution | TestHub

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