The Number Of Solution Of The Pair Of Equations 2sin2θ – Cos2θ = 0 And 2cos2θ – 3sinθ = 0 In The Interval [0, 2π] is

The arithmetic mean of the roots of the equation 4 cos³ x – 4 cos² x – cos (π + x) – 1 = 0 in the interval (0, 315) is equal to

 then cos 2x is equal to

The value of x between 0 and 2π which satisfy the equation  are in A.P. The common different of the A.P. is

The equation (cos p – 1)x² + (cos p)x + sin p = 0 where x is a variable, has real roots if p lies in the interval

The smallest positive number p for which the equation cos(p sinx) = sin(pcos x) has a solution x ∈ [0, 2π]

Number of solutions of pair x, y of the equation sin x sin y = min{–1, α² – 4α + 5}, α ∈ R (where 0 < x < π, – π < y < 0) is

  then sin x lies in

Number of solution of the equation 

Let α β ∈[– π π] be such that cos (α – β) = 1 and cos(α + β) = 1/e, The number of pairs of α, β satisfying the above system of equation is

If sin 3α = 4 sin α sin (x + α) sin (x – α), then 864 sin² x + 3620 cos² x is equal to.