The Number Of Irrational Solutions Of The Equation                        

The number of integral roots of the equation 

is

The product of roots of 

Let S denote the set of all values of the parameter a for which

has no solution, then S equals

The number of roots of the equation 

If all the roots of x³ + px + q = 0 p, q Ïµ R q ≠ 0 are real, then

Let S denote the set of all real value of q for which the roots of the equation

                          x² – 2ax + a² – 1 = 0          ...(1)

lie between 5 and 10, then S equals

Let S denote the set of all values of a for which the roots of the equation (1 + a)x² – 3ax + 4a = 0 exceed 1, then S equals

Let S denote the set of all values of S for which the equation

         2x² – 2(2a + 1) x + a(a + 1) = 0

has one root less than a and root greater than a, then S equals

Let a, b, p, q Ïµ Q and suppose that f (x) = x² + ax + b = 0 and g (x) = x³+ px + q = 0 have a common irrational root, then