Let $S$ be the set of all $\alpha  \in  R$ such that the equation, $cos\,2 x + \alpha  \,sin\, x = 2\alpha  -7$ has a solution. Then $S$ is equal to

For what value of $\lambda$ the sum of the squares of the roots of ${x^2} + (2 + \lambda )\,x - \frac{1}{2}(1 + \lambda ) = 0$ is minimum

If $x$ is real , the maximum value of $\frac{{3{x^2} + 9x + 17}}{{3{x^2} + 9x + 7}}$ is 

The number of real solutions of the equation $|x{|^2}$-$3|x| + 2 = 0$ are

The sum of all real values of $x$ satisfying the equation ${\left( {{x^2} - 5x + 5} \right)^{{x^2} + 4x - 60}} = 1$ is ;

The number of solutions of the equation $\log _{(x+1)}\left(2 x^{2}+7 x+5\right)+\log _{(2 x+5)}(x+1)^{2}-4=0, x\,>\,0$, is $....$