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SAT Algebra Perfect Score

If the number of integers $$x$$ that satisfy $$\left\lvert 8-\frac{x}{n} \right\rvert<3$$ is $$17,$$ what is the value of the positive integer $$n?$$

(A)$$\ \ 1$$
(B)$$\ \ 2$$
(C)$$\ \ 3$$
(D)$$\ \ 4$$
(E)$$\ \ 5$$

If $$a+b=-2$$ and $$a^2+b^2=5,$$ what is the value of $$a^5+b^5?$$

(A) $$-68$$
(B) $$-55.5$$
(C) $$-54.5$$
(D) $$-24.5$$
(E) $$\ \ 24.5$$

If $$a-b=-4$$ and $$ab=3,$$ what is the value of $\frac{1}{a^3}-\frac{3}{a^2b}+\frac{3}{ab^2}-\frac{1}{b^3}?$

(A) $$\ \ \frac{40}{27}$$
(B) $$\ \ \frac{64}{27}$$
(C) $$\ \ \frac{112}{27}$$
(D) $$\ \ \frac{64}{9}$$
(E) $$\ \ \frac{112}{9}$$

If $$x, y$$ and $$z$$ are positive numbers such that $$x^2=y^5$$ and $$y^{-\frac{1}{3}}=z^{-\frac{1}{5}},$$ what is $$\sqrt{z^3}$$ in terms of $$x?$$

(A) $$\ \ x^{-\frac{1}{2}}$$
(B) $$\ \ x^{\frac{1}{2}}$$
(C) $$\ \ x^{\frac{2}{3}}$$
(D) $$\ \ x$$
(E) $$\ \ x^{\frac{3}{2}}$$

The following two lines in the $$xy$$-plane are perpendicular to each other: $\begin{array} &3^{2a}x+27^{-3a+1}y-7=0, &3^{-5a+1}x-9^{a-3-\frac{1}{2a}}y+4=0.\end{array}$ What is the real number $$a?$$

(A) $$\ \ -1$$
(B) $$\ \ -\frac{1}{2}$$
(C) $$\ \ \frac{1}{2}$$
(D) $$\ \ 1$$
(E) $$\ \ \frac{3}{2}$$

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