If one of the roots of the equation $x^2 + ax + 3a = 0$ is $1$, then its other root is

The zeroes of $ax² + bx + c = 0$ are in the ratio $1:2$ (assume the coefficients to be non-zero). Which of these is one of the zeroes?

What must be added to $x^4+2x^3-2x^2+x-1$ so that the result is exactly divisible by $x^2+2x-3$?

If $\sin\theta$ and $\cos\theta$ are the roots of the quadratic equation $px^2+qx+r=0$ $(p \neq 0)$, then which of the following relation holds good?

Say for positive quantities a, b, $a + b$ equals $a^2 + ab + b^2$. Choose the nearest value to b if a is given as 0.5.

What is the condition for $x^2 + (a + b)x + ab$, with a and b non-zero, to have real roots?