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Reasoning Skills

Checking Cases

         

Which of the following could be an internal angle in a regular polygon?

\(\begin{array}{r r l} &\text{I.}\ & 108 ^ \circ\\
& \text{II.}\ & 120 ^ \circ\\
&\text{III.} &167 ^ \circ\\
\end{array}\)

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

If \(n\) is a perfect square, which of the following could be true?

\(\begin{array}{r r l} &\text{I.}\ & n\ \text{is a perfect cube.}\\
& \text{II.}\ & n^2 + 6\ \text{is a perfect square.}\\
& \text{III.}\ & n^2 + 1\ \text{is a multiple of 3.}\\
\end{array}\)

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

If \( x^3 < 8 x + 9 \), which of the following could be a value of \(x\)?

\(\begin{array}{r r l} & \text{I.} & 8\\
& \text{II.} & 0\\ & \text{III.} & -8\\
\end{array}\)

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

For all numbers \(a\) and \(b\), let \( a \heartsuit b \) be defined by \( a \heartsuit b = ab + a -b \). For all numbers \(x\) and \(y\), which of the following must be true?

\(\begin{array}{r r l} &\text{I.}\ & x \heartsuit y = y \heartsuit x\\
&\text{II.}\ & x \heartsuit y = (-y) \heartsuit (-x)\\ &\text{III.}\ & x \heartsuit y = (y+3) \heartsuit (x-3)\\
\end{array}\)

(A)\(\ \ \) II only
(B)\(\ \ \) III only
(C)\(\ \ \) II and III only
(D)\(\ \ \) I, II, and III
(E)\(\ \ \) None of the statements

If \(P\) is a prime number greater than 10, which of the following could represent another prime number for some value of \(P\)?

\(\begin{array}{r r l} &\text{I.}\ &P+26\\
&\text{II.}\ & 5P\\
&\text{III.}\ & P^2 + 1\\ \end{array}\)

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

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