Solve the given two equations and choose the correct option.
$I.$ $2x^{2} - 13x + 21 = 0$
$II.$ $5y^{2} - 22y + 21 = 0$

If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-3 \lambda x+\lambda^{2}=0,$ find $\lambda$ if $\alpha^{2}+\beta^{2}=\frac{7}{4}$

If the graph of $y = ax^2 - bx + c$ is as shown below,then the signs of $a$,$b$,and $c$ are:

If ${b_1}{b_2} = 2({c_1} + {c_2})$,then at least one of the equations ${x^2} + {b_1}x + {c_1} = 0$ and ${x^2} + {b_2}x + {c_2} = 0$ has

The real number $k$ for which the equation $2x^2 + 3x + k = 0$ has two distinct real roots in $[0, 1]$:

Let $\alpha, \beta$ be the roots of the equation $ax^2 + 2bx + c = 0$ and $\gamma, \delta$ be the roots of the equation $px^2 + 2qx + r = 0$. If $\alpha, \beta, \gamma, \delta$ are in $G.P.$,then