The equation $$\left( {\cos p - 1} \right){x^2} + \left( {\cos p} \right)x + \sin p = 0\,$$ In the variable x, has real roots. Then p can take any value in the interval

$$ABC$$ is a triangle such that $$$\sin \left( {2A + B} \right) = \sin \left( {C - A} \right) = \, - \sin \left( {B + 2C} \right) = {1 \over 2}.$$$

If $$A,\,B$$ and $$C$$ are in arithmetic progression, determine the values of $$A,\,B$$ and $$C$$.

If $$\,x < 0,\,\,y < 0,\,\,x + y + {x \over y} = {1 \over 2}$$ and $$(x + y)\,{x \over y} = - {1 \over 2}$$, then x =..........and y =.........

The number of solutions of the equation sin$${(e)^x} = {5^x} + {5^{ - x}}$$ is