Let A = cos ⁡ α sin ⁡ α - sin ⁡ α cos ⁡ α and matrix B is defined such that B = A + 3 A 2 + 3 A 3 + A 4 . If B = 8 , then the number of values of α in 0 , 10 π is

Step-by-step solution

B = A I + 3 A + 3 A 2 + A 3 = A I + A 3 B = A    I + A 3 = 8 … i A = cos ⁡ α sin ⁡ α - sin ⁡ α cos ⁡ α = c o s 2 α + s i n 2 α = 1 I + A = 1 + cos ⁡ α sin ⁡ α - sin ⁡ α 1 + cos ⁡ α = 1 + cos ⁡ α 2 + s i n 2 α   = 2 + 2 cos ⁡ α From i 1 2 + 2 cos ⁡ α 3 = 8 ⇒ 2 + 2 cos ⁡ α = 2 ⇒ cos ⁡ α = 0 α = π 2 ,   3 π