taiwan-gsat 2006 Q11

taiwan-gsat · Other · gsat__math 45 marks Not Maths True/False Justification
11. Decomposing the positive integer 18 into a product of two positive integers gives
$$1 \times 18, 2 \times 9, 3 \times 6$$
three ways. Among these three decompositions, $3 \times 6$ has the smallest difference between the two numbers, so we call $3 \times 6$ the optimal decomposition of 18. When $p \times q ( p \leq q )$ is the optimal decomposition of a positive integer $n$, we define the function $F ( n ) = \frac { p } { q }$. For example, $F ( 18 ) = \frac { 3 } { 6 } = \frac { 1 } { 2 }$. Which of the following statements about the function $F ( n )$ are correct?
(1) $F ( 4 ) = 1$.
(2) $F ( 24 ) = \frac { 3 } { 8 }$.
(3) $F ( 27 ) = \frac { 1 } { 3 }$.
(4) If $n$ is a prime number, then $F ( n ) = \frac { 1 } { n }$.
(5) If $n$ is a perfect square, then $F ( n ) = 1$.
Part Two: Fill-in-the-Blank Questions (45 points)
Instructions: 1. For questions A through I, mark your answers on the "Answer Section" of the answer sheet at the indicated row numbers (12–32).
2. Each completely correct answer is worth 5 points. Wrong answers are not penalized. Incomplete answers receive no points.
A. A sample survey of 1000 families with two children in a certain region obtained the following data, where (boy, girl) represents a family where the first child is a boy and the second child is a girl, and so on.
Family TypeNumber of Families
(boy, boy)261
(boy, girl)249
(girl, boy)255
(girl, girl)235

From this data, the estimated ratio of boys to girls in families with two children in this region is approximately (rounded to the nearest integer).
B
& B
11. Decomposing the positive integer 18 into a product of two positive integers gives

$$1 \times 18, 2 \times 9, 3 \times 6$$

three ways. Among these three decompositions, $3 \times 6$ has the smallest difference between the two numbers, so we call $3 \times 6$ the optimal decomposition of 18. When $p \times q ( p \leq q )$ is the optimal decomposition of a positive integer $n$, we define the function $F ( n ) = \frac { p } { q }$. For example, $F ( 18 ) = \frac { 3 } { 6 } = \frac { 1 } { 2 }$. Which of the following statements about the function $F ( n )$ are correct?\\
(1) $F ( 4 ) = 1$.\\
(2) $F ( 24 ) = \frac { 3 } { 8 }$.\\
(3) $F ( 27 ) = \frac { 1 } { 3 }$.\\
(4) If $n$ is a prime number, then $F ( n ) = \frac { 1 } { n }$.\\
(5) If $n$ is a perfect square, then $F ( n ) = 1$.

\section*{Part Two: Fill-in-the-Blank Questions (45 points)}
Instructions: 1. For questions A through I, mark your answers on the "Answer Section" of the answer sheet at the indicated row numbers (12–32).\\
2. Each completely correct answer is worth 5 points. Wrong answers are not penalized. Incomplete answers receive no points.

A. A sample survey of 1000 families with two children in a certain region obtained the following data, where (boy, girl) represents a family where the first child is a boy and the second child is a girl, and so on.

\begin{center}
\begin{tabular}{ | c | c | }
\hline
Family Type & Number of Families \\
\hline
(boy, boy) & 261 \\
\hline
(boy, girl) & 249 \\
\hline
(girl, boy) & 255 \\
\hline
(girl, girl) & 235 \\
\hline
\end{tabular}
\end{center}

From this data, the estimated ratio of boys to girls in families with two children in this region is approximately\\
(rounded to the nearest integer).

B