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

Find the complete set of values of $k$ for which the equation $4^x - (k + 2)2^x + 2k = 0$ has exactly one positive root.

The value of $2 + \frac{1}{2 + \frac{1}{2 + \dots \infty}}$ is

The area of a rectangle is the same as that of a circle of radius $\frac{\sqrt{35}}{\sqrt{11}} \text{ cm}$. If the length of the rectangle exceeds its breadth by $3 \text{ cm}$,find the dimensions of the rectangle.

If $x = 2^{1/3} - 2^{-1/3}$,find the value of $2x^3 + 6x$.

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}-16x+63=0$
$II.$ $y^{2}-2y-35=0$