Number of values of p for which equation `sin ^(3) x + 1 + p^(3) - 3 p sin x = 0 (p gt 0) ` has a root is

The smallest positive real value of p for which the equation cos (p sin x) = sin (p cos x) has a solution where x in [0, 2pi] is

The set of values of p for which roots of the equation 3x ^(2) + 2x +p (p-1) =0 are of opposite sign is

Solve the equation 8x ^(3) - 36 x ^(2) - 18 x + 81 = 0 the roots being in A.P.

if the equation (3x ) ^(2) +(27 xx 3 ^(1//p) - 15 ) x+4=0 has equal roots then p=

If 2, 3 are the roots of the equation 2x^(3) + px^(2) - 13x + q = 0, then (p , q) =

The roots of a quadratic equation x/p = p/x are

If alpha,beta are the roots of the equation x^(2) + Px + P^(3) = 0, P ne 0 such that alpha =beta^(2) then the roots of the given equation are

Show that the roots of the equation x^3 +px^2 +qx +r=0 are in A.P 2p^3 - 9 pq + 27 r=0

Find the value of p for which the fucntion f(x) = ((4^(x) -1)^(3))/(sin ((x)/(p)) log_(e)(1 + (x^(2))/(3))), x ne 0 " and " f(0) = 12 (log4)^(3) , is continuous at x = 0 .