Two roots of the cubic equation $x^3 + 3x^2 + kx + 12 = 0$ are real and unequal but have the same absolute value. Find the value of $k$.

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5x^3-4x^2+3x-2=0$,then find the value of $\alpha^3+\beta^3+\gamma^3$.

If $\alpha, \beta, \gamma, \delta$ are in geometric progression,where $\alpha, \beta$ are the roots of the equation $ax^2 + 2bx + c = 0$ and $\gamma, \delta$ are the roots of the equation $px^2 + 2qx + r = 0$,then:

If the roots of the quadratic equation $x^2 + px + q = 0$ are $\tan 30^\circ$ and $\tan 15^\circ$ respectively,then the value of $2 + q - p$ is

Let $f(x)$ be a quadratic polynomial such that $f(-2) + f(3) = 0$. If one of the roots of $f(x) = 0$ is $-1$,then the sum of the roots of $f(x) = 0$ is equal to

If one root of the equation $ax^2 + bx + c = 0$ is the square of the other,then $a(c - b)^3 = cX$,where $X$ is