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# SAT Functions Perfect Score

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)$$\ \ 6$$

In the above diagram, quadrilateral $$ABCD$$ is a square centered at the origin $$O,$$ and quadrilateral $$EFGH$$ is a rectangle. Suppose that $$A=(-1, 1), E=(2, 0), G=(6, -2),$$ and a single line cuts $$\square ABCD$$ into halves. If the same line cuts the area of rectangle $$EFGH$$ at a ratio of $$1:7,$$ the shape with vertex $$F$$ being $$1$$ and the shape with vertex $$H$$ being $$7,$$ what is the sum of the $$x$$-coordinates of the points on rectangle $$EFGH$$ that intersect the line?

(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)$$\ \ 2$$

The above is the graph of the quadratic function $$y=ax^2+bx+c.$$ Which of the following statements is true?

$$\begin{array}{r r l} & \text{I.} & a+b-c<0\\ & \text{II.} & b-2a>0\\ & \text{III.} & 4a+b^2-4ac>0\\ \end{array}$$

(A)$$\ \$$ I only
(B)$$\ \$$ II only
(C)$$\ \$$ I and III only
(D)$$\ \$$ II and III only
(E)$$\ \$$ I, II and III

In the above diagram, quadrilateral $$ABCD$$ is a square with side length $$2$$ and $$A=(-1, 1),$$ and triangle $$EFG$$ is an equilateral triangle with side length $$4$$ and $$E=(2, 0).$$ If a single line cuts into halves both the areas of $$\square ABCD$$ and $$\triangle EFG,$$ what is the slope of the line?

(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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