If \(x\) and \(y\) are positive numbers and \(x\) is directly proportional to \(\frac{1}{y}\), which of the following is directly proportional to \(\frac{1}{x^{7}}\)?

(A) \(\ \ \frac{1}{y^{7}}\)

(B) \(\ \ \frac{1}{y}\)

(C) \(\ \ y\)

(D) \(\ \ y^{7}\)

(E) \(\ \ y^{8}\)

\(B\) varies directly with the square of \(n\) and the cube of \(m\) and inversely with \(p\). What is the value of \(B\) if \(m=7, n=12\), \(p=84\), and the constant of proportionality, \(k,\) is \(k=\frac{1}{21}\)?

(A) \(\ \ \frac{1}{12348}\)

(B) \(\ \ \frac{1}{21}\)

(C) \(\ \ \frac{12}{21}\)

(D) \(\ \ 28\)

(E) \(\ \ 588\)

In which of the following equations is \(y\) directly proportional to \(x\)?

(A) \(\ \ y=\frac{1}{5}\cdot x^{-2}\)

(B) \(\ \ y=\frac{5}{x}\)

(C) \(\ \ y=\frac{1}{5}\cdot x\)

(D) \(\ \ y=\frac{1}{5}\cdot x^{2}\)

(E) \(\ \ y=5+2x\)

The kinetic energy of an object, \(KE\), is directly proportional to its mass, \(m\), and to the square of its velocity, \(v\). If the mass is divided by \(2\) and the velocity is multiplied by \(11\), and if \(k\) is a proportionality constant, which of the following does NOT represent the object's new kinetic energy, \(KE_{new}\)?

(A) \(\ \ KE_{new}=\frac{121k}{2}\cdot KE_{old}\)

(B) \(\ \ KE_{new}=\frac{121 \cdot k}{2} \cdot m \cdot v^{2}\)

(C) \(\ \ KE_{new}=k\cdot \frac{m}{2} \cdot (v \cdot 11)^{2}\)

(D) \(\ \ KE_{new}=\frac{k\cdot 11}{2} \cdot m \cdot v^{2}\)

(E) \(\ \ KE_{new}= k\cdot 11^{2} \cdot \frac{1}{2}\cdot m \cdot v^{2}\)

\(y\) varies directly with the square of \(x,\) and \(y=196\) when \(x=7\). Given \(x<0\), what is \(x\) when \(y=1764\)?

(A) \(\ \ -63\)

(B) \(\ \ -21\)

(C) \(\ \ 0\)

(D) \(\ \ 21\)

(E) \(\ \ 63\)

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