The real number `x_(1), x_(2), x_(3)` satisfying the equation `x^(3) -x^(2) + betax + gamma = 0` are in AP. Find the intervals in which beta and gamma lie.

The real numbers x_1, x_2, x_3 satisfying the equation x^3-x^2+b x+gamma=0 ar ein A.P. Find the intervals in which betaa n dgamma lie.

The number of value of x in the interval [0,3pi] satisfying the equation 2 sin^2 + 5 sin x -3 =0 is

The number of real solution of the equation x^(2)-3|x|+2=0 is

The number of solutions of the equation 3^(2sec^(2)x)+1=10.3^(tan^(2)x) in the interval [0,2pi] is

If t is a real number satisfying the equation 2t^3-9t^2+30-a=0, then find the values of the parameter a for which the equation x+1/x=t gives six real and distinct values of x .

If alpha, beta and gamma the roots of the equation x^(3) + 3x^(2) - 4x - 2 = 0. then find the values of the following expressions: (i) alpha ^(2) + beta^(2) + gamma^(2) (ii) alpha ^(3) + beta^(3) + gamma^(3) (iii) (1)/(alpha)+(1)/(beta)+(1)/(gamma)

If alpha, beta and gamma are the roots of the equation x^(3) + 3x^(2) - 24x + 1 = 0 thebn find the value of (3sqrt(alpha)+3sqrt(beta)+ 3sqrt(gamma)).