The value of `alpha` in `[0,2pi]` so that `x^(2)+y^(2)+2sqrt(sin alpha )x+(cos alpha-1)=0` having intercept on x-axis always greater than 2, is/are

The value of theta in [0, 2pi] so that circle x^(2)+y^(2)+2 (sin alpha)x+2(cos alpha)y+sin^(2)theta=0 always lies inside the square of unit side length, is/are

The value of alpha in [0,2pi] which does not satify the equation int_(pi//2)^(alpha) sin x dx = sin 2 alpha, is

The number of values of alpha in [-10pi, 10pi] for which the equations (sin alpha)x-(cos alpha)y+3z=0, (cos alpha)x+(sin alpha)y-2z=0 and 2x+3y+(cos alpha)z=0 have nontrivial solution is

The value of alpha in (-pi, 0) satisfying sin alpha+int_(alpha)^(2alpha) cos2 x dx=0 , is

The sum of value of alpha, 0 le alpha le 2pi , such that x^(2) -2x cos alpha +1=0 has real roots is ……….. Times pi .

If x^(2)-2cos alpha x+1=0 and y^(2)-2cos alpha y+1=0 then 2cos(alpha+beta)=

If (sin alpha)^(x)+(cos alpha)^(x) ge 1,0 lt a lt (pi)/(2) then

The number of values of the parameter alpha in [0, 2pi] for which the quadratic function (sin alpha)x^(2)+(2cosalpha)x+(1)/(2)(cos alpha+sinalpha) is the square of a linear funcion is

If (cos3 alpha)/(cos alpha)=(1)/(3),0

If (cos3 alpha)/(cos alpha)=(1)/(3),0