Consider the cubic equation x3−(1+cos⁡θ+sin⁡θ)x2+(cos⁡θsin⁡θ+cos⁡θ+sin⁡θ)x−sin⁡θcos⁡θ=0whose roots are x1,x2, and x3.The value of x12+x22+x32equalsThe number of values of θ in [0,2π]for which at least two roots are equalThe greatest possible difference between two of the roots if θ∈[0,2π]is

x3−(1+cos⁡θ+sin⁡θ)x2+(cos⁡θsin⁡θ+cos⁡θ+sin⁡θ)x−sin⁡θcos⁡θ=0Given cubic function isf(x)=(x−1)(x−cos ⁡θ)(x−sin ⁡θ)Therefore, roots are 1,sin⁡ θ, and cos⁡ θ. Hence, x12+x22+x32=1+sin2⁡ θ+cos2⁡ θ=2. Now if 1=sin⁡θ, we get θ=π/2If 1=cos⁡θ, then θ=0,2π and sin⁡θ=cos⁡θ, we get tan⁡θ=1.Therefore, θ=π4,5π4Therefore, the number of values of θ in [0,2π] is 5.Again the maximum possible difference between the two roots is 2. or  1−sin⁡θ=2, when θ=3π/21−cos⁡θ=2, when θ=π