If ${x^2} + px + q = 0$ is the quadratic equation whose roots are $a - 2$ and $b - 2$,where $a$ and $b$ are the roots of ${x^2} - 3x + 1 = 0$,then

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + 6x + \lambda = 0$ and $3\alpha + 2\beta = -20$,then $\lambda = $

Solve the given two equations and select the correct answer from the given options.
$I.$ $36 x^{4}+369 x^{2}+900=0$
$II.$ $144 y^{4}+337 y^{2}+144=0$

The number of solutions of $\frac{\log 5 + \log (x^2 + 1)}{\log (x - 2)} = 2$ is

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}-7 \sqrt{3} x+35 \sqrt{15}=5 \sqrt{5} x$
$II.$ $y^{2}-5 \sqrt{5} y+30=0$

Let $a, b, c$ be real numbers with $a \ne 0$. If $\alpha$ is a root of $a^2x^2 + bx + c = 0$,$\beta$ is a root of $a^2x^2 - bx - c = 0$,and $0 < \alpha < \beta$,then the equation $a^2x^2 + 2bx + 2c = 0$ has a root $\gamma$ that always satisfies: