Let \(A\) be the intersection point of the curve \(y=x^2\) and the line \(y=x~(x>0),\) and \(B\) the intersection point of \(y=x^2\) and \(y=-2x~(x<0).\) If \(O=(0, 0)\) is the origin, what is the area of \(\triangle OAB?\)

(A)\(\ \ 2\)

(B)\(\ \ 3\)

(C)\(\ \ 4\)

(D)\(\ \ 5\)

(E)\(\ \ 6\)

(A)\(\ \ 5.5\)

(B)\(\ \ 6\)

(C)\(\ \ 6.5\)

(D)\(\ \ 7\)

(E)\(\ \ 8\)

For all real numbers \(m\) and \(n,\) let \(\oplus\) be defined as \[m\oplus n=(1-m)(2+n).\] Then what is the sum of all the values of \(x\) that satisfy \[(x\oplus x)\oplus(x-1)=(x-1)\oplus (x\oplus x)?\]

(A)\(\ \ -2\)

(B)\(\ \ -1\)

(C)\(\ \ 0\)

(D)\(\ \ 1\)

(E)\(\ \ 2\)

\(\begin{array}{r r l}
& \text{I.} & a+b-c<0\\

& \text{II.} & b-2a>0\\

& \text{III.} & 4a+b^2-4ac>0\\

\end{array}\)

(A)\(\ \ \) I only

(B)\(\ \ \) II only

(C)\(\ \ \) I and III only

(D)\(\ \ \) II and III only

(E)\(\ \ \) I, II and III

(A)\(\ \ -\frac{1}{\sqrt{3}}\)

(B)\(\ \ -\frac{\sqrt{3}}{2}\)

(C)\(\ \ \frac{\sqrt{3}-\sqrt{15}}{2}\)

(D)\(\ \ -\frac{2\sqrt{3}}{3}\)

(E)\(\ \ \frac{-\sqrt{3}-\sqrt{15}}{2}\)

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