Let M = sin 4 ⁡ θ - 1 - sin 2 ⁡ θ 1 + cos 2 ⁡ θ cos 4 ⁡ θ = α I + β M - 1 , where α = α θ and β = β θ are real number, and I is the 2 × 2 identity matrix. If α * is the minimum of the set α θ : θ ∈ 0 ,   2 π and β * is the minimum of the set β θ : θ ∈ 0 ,   2 π , then the value of α * + β * is
C. - 29 16
Method I As given, M = s i n 4 θ - 1 - s i n 2 θ 1 + c o s 2 θ c o s 4 θ = α I + β M - 1 a d j   M = c o s 4 θ 1 + s i n 2 θ - 1 - c o s 2 θ s i n 4 θ M - 1 = 1 | M | c o s 4 θ 1 + s i n 2 θ - 1 - c o s 2 θ s i n 4 θ s i n 4 θ - 1 - s i n 2 θ 1 + c o s 2 θ c o s 4 θ = α 1 0 0 1 + β M c o s 4 θ 1 + s i n 2 θ - 1 - c o s 4 θ s i n 4 θ Compare element a 12 in L.H.S. and R.H.S. - 1 - s i n 2 θ = 0 + β M 1 + s i n 2 θ β =   | M | ...(iii) Compare element a 11 in LHS. and R.H.S. s i n 4 θ = α + β M c o s 4 θ From equation (iii) s i n 4 θ = α - c o s 4 θ α = s i n 4 θ + c o s 4 θ ...(iv) Now from equation (iv) α = s i n 4 θ + c o s 4 θ α = s i n 2 θ + c o s 2 θ 2 - 2 s i n 2 θ c o s 2 θ α = 1 - s i n 2 2 θ 2 0 ≤ s i n 2 2 θ ≤ 1 - 1 2 ≤ - s i n 2 2 θ 2 ≤ 0 1 2 ≤ 1 - s i n 2 2 θ 2 ≤ 1 α ∈ 1 2 , 1 , α m i n = 1 2 = α * For β , consider M = s i n 4 θ c o s 4 θ + s i n 2 θ c o s 2 θ + 2 M s i n 2 θ c o s 2 θ + 1 2 2 + 7 4 M s i n 2 2 θ 4 + 1 2 2 + 7 4 0 ≤ s i n 2 2 θ ≤ 1 Hence, M   ∈ 2 , 37 16 β = - M β ∈ - 37 16 , - 2 β m i n = - 37 16 = β * α * + β * = 1 2 - 37 16 = - 29 16 Method II M = α I + β M - 1 Multiply with M both side M 2 = α M + β I M 2 - α M - B I = 0 ...(i) Equation (i) represents characteristic equation of matrix ‘ M ’ Hence, Trace ( M ) = α M   = - β Sum of roots of characteristic equation = Trace of matrix Product of roots of characteristic equation = determined of matrix ∴   α = s i n 4 θ + c o s 4 θ β = - s i n 4 θ c o s 4 θ + s i n 2 θ c o s 2 θ + 2 Now proceed as Method 1
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