If $\sin 2\theta$ and $\cos 2\theta$ are solutions of $x^2 + ax - c = 0$,then

If $2, 3, 6$ are the roots of the polynomial $f(x) = x^3 + ax^2 + bx + c$,where $a, b, c \in \mathbb{C}$. Then,the value of $a - c$ is

The condition that one root of the equation $ax^2 + bx + c = 0$ is three times the other is:

The condition that $x^3 - p x^2 + q x - r = 0$ may have two of its roots equal to each other but of opposite sign is

Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up with roots $(4, 3)$. Rahul made a mistake in writing down the coefficient of $x$ and ended up with roots $(3, 2)$. The correct roots of the equation are:

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 5x^2 - 7x - 1 = 0$,then find the equation whose roots are $\alpha\beta, \beta\gamma, \gamma\alpha$.