If $\alpha, \beta$ are the roots of $x^2 - 3x + a = 0, a \in R$ and $\alpha < 1 < \beta$,then:

Solve the given two equations and select the correct option.
$I.$ $\sqrt{500} x + \sqrt{402} = 0$
$II.$ $\sqrt{360} y + \sqrt{200} = 0$

Solve the given two equations and select the correct answer from the given options.
$I. \quad 3x + 4y = (1681)^{1/2}$
$II. \quad 3x + 2y = (961)^{1/2}$

The equation whose roots are $\frac{1}{3 + \sqrt{2}}$ and $\frac{1}{3 - \sqrt{2}}$ is

If the difference between the roots of the equation $x^2 + ax + b = 0$ is equal to the difference between the roots of the equation $x^2 + bx + a = 0$ $(a \ne b)$,then:

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 got roots $(3, 2).$ The correct roots of the equation are: