Let `a ,\ b ,\ c` be three non-zero real numbers such that the equation `sqrt(3)\ acosx+2\ bsinx=c ,\ x in [-pi/2,pi/2]` , has two distinct real roots `alpha` and `beta` with `alpha+beta=pi/3` . Then, the value of `b/a` is _______.

The number of real solutions of the equation sqrt(1+cos2x)=sqrt2cos^(-1)(cosx) in [pi/2,pi] is _____

ax^2 + bx + c = 0(a > 0), has two roots alpha and beta such alpha 2, then

If alpha+beta=pi/2 and beta+ gamma = alpha , then find the value of tan alpha .

If p''(x) has real roots alpha,beta,gamma . Then , [alpha]+[beta]+[gamma] is

Let S be the set of all non-zero numbers alpha such that the quadratic equation alphax^2-x+alpha=0 has two distinct real roots x_1, and x_2 satisfying the inequality |x_1-x_2|lt1 which of the following intervals is(are) a subset of S?

Let alpha and beta be nonzero real numbers such that 2(cosbeta-cosalpha)+cosalphacosbeta=1 . Then which of the following is/are true?

Let a ,b ,c be real. If a x^2+b x+c=0 has two real roots alphaa n dbeta,w h e r ealpha >1 , then show that 1+c/a+|b/a|<0

If the function f(x)=x^(3)-9x^(2)+24x+c has three real and distinct roots alpha, beta and gamma , then the value of [alpha]+[beta]+[gamma] is

Let -pi/6 beta_1 and alpha_2 >beta_2 , then alpha_1 + beta_2 equals

If a, b, c are real and a!=b , then the roots ofthe equation, 2(a-b)x^2-11(a + b + c) x-3(a-b) = 0 are :