Basic Mathematics
# Reasoning Skills

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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