$$x ^ { 2 } - ( \sin a ) x - \frac { 1 } { 4 } \left( \cos ^ { 2 } a \right) = 0$$
One root of the equation is $\frac { 2 } { 3 }$. Accordingly, what is $\sin a$?
A) $\frac { \sqrt { 2 } } { 2 }$
B) $\frac { \sqrt { 2 } } { 3 }$
C) $\frac { \sqrt { 2 } } { 6 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 1 } { 3 }$

Let k be a positive real number. If one root of the equation
$$3 x ^ { 2 } + k x - 2 = 0$$
is k, what is the other root?
A) $\frac { \sqrt { 2 } } { 3 }$
B) $\frac { 2 \sqrt { 3 } } { 3 }$
C) $\frac { - 2 \sqrt { 2 } } { 3 }$
D) $\frac { - \sqrt { 2 } } { 6 }$
E) $\frac { - \sqrt { 3 } } { 6 }$

A second-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with real coefficients whose leading coefficient is 1 has two distinct roots that are $P ( 0 )$ and $P ( - 1 )$. Accordingly, what is the value of $\mathbf { P } ( 2 )$?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 5 } { 2 }$
D) 1
E) 2

Where $a$ and $b$ are positive real numbers,
$$2ax^2 - 5bx + 8b = a$$
the roots of the equation are $a$ and $b$. Accordingly, what is the sum $a + b$?
A) 5
B) 6
C) 10
D) 12
E) 15