Let k be a positive real number. If the roots of the equation
$$2 x ^ { 2 } + k x - 1 = 0$$
have a difference of 2, what is k?
A) 1
B) 2
C) $\sqrt { 2 }$
D) $2 \sqrt { 2 }$
E) $\sqrt { 3 }$

Let b and c be non-zero real numbers such that the roots of the equation
$$x ^ { 2 } + b x + c = 0$$
are b and c. Accordingly, what is the product $b \cdot c$?
A) $- 6$
B) $- 5$
C) $- 4$
D) $- 3$
E) $- 2$

The sum of the roots of the equation $x ^ { 2 } - a x + 1 = 0$, which has two real roots, is a root of the equation $$x ^ { 2 } + 6 x + a = 0$$ Accordingly, what is a?\ A) - 3\ B) - 4\ C) - 5\ D) - 6\ E) - 7

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

For the equation $x^2 - 2x + c = 0$, the discriminant is also a root of this equation. What is the product of the possible values of the real number $c$?
A) 1
B) 2
C) 4
D) $\frac{1}{2}$
E) $\frac{1}{4}$

Let $a$ and $b$ be positive real numbers. The equations
$$\begin{aligned} & x^{2} + ax + b = 0 \\ & ax^{2} + (b + 3)x + a = 0 \end{aligned}$$
are given. Given that the solution set of each of these equations has exactly 1 element, what is the product of the different values that the sum $a + b$ can take?
A) 24 B) 32 C) 45 D) 72 E) 120