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

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:

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 option.
$I.$ $36 x^{2}+47 \sqrt{7} x+105=0$
$II.$ $35 y^{2}+20 \sqrt{3} y+63 \sqrt{2} y+36 \sqrt{6}=0$

If $a < b < c < d$,then the roots of the equation $(x - a)(x - c) + 2(x - b)(x - d) = 0$ are

The set of values of $a$ such that $x^2 - 2ax + a^2 - 6a \leqslant 0$ for all $x \in [1, 2]$ is: