$A$ value of $x$ satisfying $150 x \equiv 35 \pmod{31}$ is

Let $S$ be the set of all possible integral values of $\lambda$ in the interval $(-3, 7)$ for which the roots of the quadratic equation $\lambda x^2 + 13x + 7 = 0$ are all rational numbers. Then the sum of the elements in $S$ is

Which among the following equations has roots that are negatives of the roots of the equation $x^3-x^2+x-4=0$?

If each root of the equation $2x^3 + ax^2 - 8x + b = 0$ is reduced by $1$,then in the transformed equation thus formed,the term containing $x^2$ and the constant term vanish. The roots of the original equation are

The roots of the equation $({a^2} + {b^2}){t^2} - 2(ac + bd)t + ({c^2} + {d^2}) = 0$ are equal. Then:

If the roots of the equation $x^2 + px + q = 0$ are $\alpha$ and $\beta$,and the roots of the equation $x^2 - xr + s = 0$ are $\alpha^4$ and $\beta^4$,then the roots of the equation $x^2 - 4qx + 2q^2 - r = 0$ will be