Let $r_1, r_2, r_3$ be roots of the equation $x^3 - 2x^2 + 4x + 5074 = 0$. Then the value of $(r_1 + 2)(r_2 + 2)(r_3 + 2)$ is

Consider the two equations $x^2 - 2ax + b^2 = 0$ and $x^2 - 2bx + a^2 = 0$. The arithmetic mean of the roots of the first equation is equal to what?

If the product of the roots of the equation $(a + 1)x^2 + (2a + 3)x + (3a + 4) = 0$ is $2$,then the sum of the roots will be:

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

If $1, 2, 3$ and $4$ are the roots of the equation $x^4+ax^3+bx^2+cx+d=0$,then $a+2b+c$ is equal to

Suppose that the equation $ax^2+bx+c=0$ has roots $\alpha$ and $\beta$,both of which are different from $\frac{1}{3}$. Then,an equation whose roots are $\frac{1}{3\alpha-1}$ and $\frac{1}{3\beta-1}$ is