1998 College Scholastic Ability Test

Mathematics·Studies (I)

Hum. & Arts
What is the value of \(\bigg\{\!\left(\dfrac{4}{9}\right)^{\!-\,\begin{array}{c}2 \\\hline 3\end{array}}\bigg\}^{\begin{array}{c}9 \\\hline 4\end{array}}\:\)? [2 points]
  1. \(\dfrac{8}{27}\)
  2. \(\dfrac{16}{81}\)
  3. \(\dfrac{81}{16}\)
  4. \(\dfrac{27}{8}\)
  5. \(\dfrac{64}{81}\)
What is the value of \(\sqrt{4+2\sqrt{3}}-\sqrt{4-2\sqrt{3}}\)? [2 points]
  1. \(-2\)
  2. \(-\sqrt{3}\)
  3. \(1\)
  4. \(\sqrt{3}\)
  5. \(2\)
What is the remainder of the division of the polynomial \(2x^3+x^2+3x\;\) by \(\;x^2+1\)? [3 points]
  1. \(x-1\)
  2. \(x\)
  3. \(1\)
  4. \(x+3\)
  5. \(3x-1\)
What is the value of \(\displaystyle\int_{0}^{1}x(1-x)dx\)? [2 points]
  1. \(0\)
  2. \(\dfrac{1}{2}\)
  3. \(\dfrac{1}{3}\)
  4. \(\dfrac{1}{4}\)
  5. \(\dfrac{1}{6}\)
Which set represents the dark region in the following Venn diagram? (※ \(\mathrm{U}\) is the universe, and \(\mathrm{X}^C\) is the complement of \(\mathrm{X}\).) [2 points]
  1. \(\mathrm{A}\cap(\mathrm{B}\cap \mathrm{C})^C\)
  2. \(\mathrm{A}\cap(\mathrm{B}\cup \mathrm{C})^C\)
  3. \(\mathrm{A}\cap(\mathrm{B}^C \cap \mathrm{C})^C\)
  4. \(\mathrm{A}\cap(\mathrm{B}^C \cap \mathrm{C}^C)^C\)
  5. \(\mathrm{A}\cap(\mathrm{B}^C \cup \mathrm{C}^C)^C\)

Mathematics·Studies (I)

Hum. & Arts
On the \(xy\)-plane, which of the following functions has a graph that always meets with any given line? [3 points]
  1. \(y=|x|\)
  2. \(y=x^2\)
  3. \(y=\sqrt{x}\)
  4. \(y=x^3\)
  5. \(y=\dfrac{1}{x}\)
If \(1\) is a solution to the quadratic equation \(x^2-mx+2m+1=0\), what is the other solution?
(※ \(m\) is a constant) [2 points]
  1. \(3\)
  2. \(2\)
  3. \(0\)
  4. \(-1\)
  5. \(-3\)
Figure below is the net of a cube.
What is the magnitude of \(\angle \mathrm{ABC}\) in the original cube? [3 points]
  1. \(30°\)
  2. \(45°\)
  3. \(60°\)
  4. \(90°\)
  5. \(120°\)
What is the area of the triangle enclosed by the
\(x\)-axis, the \(y\)-axis, and the tangent line to the parabola \(y=- \dfrac{1}{4}x^2\) at point \((2, -1)\)? [3 points]
  1. \(\dfrac{1}{2}\)
  2. \(\dfrac{1}{4}\)
  3. \(\dfrac{3}{4}\)
  4. \(\dfrac{3}{2}\)
  5. \(\dfrac{5}{4}\)
A function \(y=f(x)\) has a derivative \(y=f'(x)\) whose graph is shown below. Which option below is correct? [3 points]
  1. \(f(x)\) increases on the interval \((-2, 1)\).
  2. \(f(x)\) decreases on the interval \((1, 3)\).
  3. \(f(x)\) increases on the interval \((4, 5)\).
  4. \(f(x)\) has a local minimum at \(x=2\).
  5. \(f(x)\) has a local minimum at \(x=3\).

Mathematics·Studies (I)

Hum. & Arts
Consider a target marked with numbers from \(1\) to \(9\) as shown to the right. \(5\) shooters \(\mathrm{A, B, C, D}\) and \(\mathrm{E}\) each shot \(10\) times, and each person hit \(10\) numbers with a mean of \(5\). The results of the \(5\) people are as follows.
Among the \(5\) people, which person hit \(10\) numbers with the smallest standard deviation? [2 points]
  1. \(\mathrm{A}\)
  2. \(\mathrm{B}\)
  3. \(\mathrm{C}\)
  4. \(\mathrm{D}\)
  5. \(\mathrm{E}\)
What is the value of \(\left(\dfrac{1+i}{1-i}\right)^{\!1998}\)? (※ \(i=\sqrt{-1}\).) [2 points]
  1. \(-1\)
  2. \(1\)
  3. \(-i\)
  4. \(i\)
  5. \(1998\)
If the sum of all solutions to the equation \(\big|x^2+(a-2)x-2\big|=1\) is \(0\), what is the value of the constant \(a\)? [2 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
What is the list of correct statements in the <List>? (※ A coin lands on heads or tails with equal probability.) [2 points]
  1. If a coin is tossed \(10\) times, the probability that it lands on heads \(4\) times is equal to the probability that it lands on heads \(6\) times.
  2. The probability that a coin tossed \(10\) times lands on heads \(5\) times, is equal to the probability that a coin tossed \(20\) times lands on heads \(10\) times.
  3. The probability that a coin tossed \(10\) times lands on heads \(5\) times or less, is greater than \(0.5\).
  1. a
  2. c
  3. a, b
  4. a, c
  5. a, b, c

Mathematics·Studies (I)

Hum. & Arts
Consider a sequence \(\{a_n\}\) whose first \(6\) terms \(a_1, a_2, a_3, a_4, a_5\) and \(a_6\) are all distinct, and \(a_{n+6}=a_n\) holds for \(n=1,2,3,\cdots\).
In which of the sequences \(\{b_n\}\) defined below do the values \(a_1, a_2, a_3, a_4, a_5\) and \(a_6\) all appear? [3 points]
  1. \(b_n=a_{2n+1}\)
  2. \(b_n=a_{3n+1}\)
  3. \(b_n=a_{4n+1}\)
  4. \(b_n=a_{5n+1}\)
  5. \(b_n=a_{6n+1}\)
For all positive integers \(n\), let \(f(n)\) be the sum of all positive divisors of \(n\). For example, \(f(3)=4\) and \(f(4)=7\). What is the list of correct statements in the <List>? [2 points]
  1. \(f(10)=18\)
  2. If \(f(n)=n+1\), then \(n\) is prime.
  3. For all positive integers \(m\) and \(n\), \(f(mn)=f(m)f(n)\).
  1. a
  2. b
  3. a, b
  4. b, c
  5. a, b, c
The following is a part of a proof of the statement ‘If \(\,3m^2-n^2=1\), then \(\fbox{\(\quad(\alpha)\quad\)}\).’
<Proof>
\(\cdots\) (omitted) \(\cdots\)
Since \(m\) and \(n\) are integers and \(3m^2=n^2+1\),
\(n^2+1\) has to be a multiple of \(3\).
One of the following holds for some integer \(k\).
If \(n=3k\), then
\(n^2=(3k)^2=9k^2=3(3k^2).\)
If \(n=3k+1\), then
\(\begin{align} n^2 &=(3k+1)^2=9k^2+6k+1\\ &=3(3k^2+2k)+1. \end{align}\)
If \(n=3k+2\), then
\(\begin{align} n^2 &=(3k+2)^2=9k^2+12k+4\\ &=3(3k^2+4k+1)+1. \end{align}\)
Therefore, in the division of \(n^2\) by \(3\), the remainder is either \(0\) or \(1\).
It follows that in the division of \(n^2+1\) by \(3\), the remainder is either \(1\) or \(2\).
\(\cdots\) (omitted) \(\cdots\)
What is appropriate for \(\fbox{\(\;(\alpha)\;\)}\) above? [2 points]
  1. at least one of \(m\) or \(n\) is an integer
  2. neither \(m\) nor \(n\) is an integer
  3. there is at least one solution where \(m\) and \(n\) are both integers
  4. there is only one solution where \(m\) and \(n\) are both integers
  5. there are no solutions where \(m\) and \(n\) are both integers

Mathematics·Studies (I)

Hum. & Arts
On the \(xy\)-plane, let \(l\) be a line that is not parallel to any axis. Let \(\mathrm{P}(x_1, y_1)\) be a point not on line \(l\), and let \(\mathrm{H}(x_2, y_2)\) be the perpendicular foot from point \(\mathrm{P}\) to line \(l\). The following is a process computing the length of the line segment \(\mathrm{PH}\).
Suppose the equation of line \(l\) is
\(ax+by+c=0\).
\((1)\)
By the given condition, \(a \ne 0\) and \(b \ne 0\).
Since line \(l\) has a slope of \(- \dfrac{a}{b}\),
the equation of line \(\mathrm{PH}\) is
\(y-y_1=\fbox{\(\;(\alpha)\;\)}\)
\((2)\)
Using \((1)\) and \((2)\), we have the following.
\(x_2-x_1=\dfrac{-a(ax_1+by_1+c)}{a^2+b^2}\)
\(y_2-y_1=\dfrac{-b(ax_1+by_1+c)}{a^2+b^2}\)
Therefore the length of the line segment \(\mathrm{PH}\) is
\(\begin{align} \overline{\mathrm{PH}} &= \fbox{\(\;(\beta)\;\)}\\ &= \dfrac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}. \end{align}\)
In the process above, what are appropriate for \((\alpha)\) and \((\beta)\) in this order? [3 points]
  1. \(\dfrac{a}{b}(x-x_1), \: |x_2-x_1| + |y_2-y_1|\)
  2. \(\dfrac{b}{a}(x-x_1), \: (x_2-x_1)^2 + (y_2-y_1)^2\)
  3. \(-\dfrac{b}{a}(x-x_1), \: \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
  4. \(\dfrac{b}{a}(x-x_1), \: \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
  5. \(-\dfrac{a}{b}(x-x_1), \: |x_2-x_1| + |y_2-y_1|\)
Let us list all positive integers in the following pattern.
Row \(1\) \(1\)
Row \(2\) \(2\) \(3\)
Row \(3\) \(4\) \(5\) \(6\)
Row \(4\) \(7\) \(8\) \(9\) \(10\)
Row \(5\) \(11\) \(12\) \(13\) \(14\) \(15\)
\(\;\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\)
Row \(10\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\cdot\) \(\large\square\)
What would be the last number in row \(10\)? [3 points]
  1. \(45\)
  2. \(50\)
  3. \(55\)
  4. \(60\)
  5. \(65\)
Let \(\{a_n\}\) be a sequence such that
\(a_1=1, a_2=2\) and
\(\,a_{n+2}=a_{n+1}+a_n \,(n=1,2,3,\cdots)\).
What is the value of \(\displaystyle\sum_{n\;\!=\;\!1}^{\infty}\dfrac{a_n}{a_{n+1}a_{n+2}}\)? [3 points]
  1. \(\dfrac{1}{2}\)
  2. \(1\)
  3. \(\dfrac{3}{2}\)
  4. \(2\)
  5. \(3\)

Mathematics·Studies (I)

Hum. & Arts
The following is a hypothetical model of artificial nuclear fission.
Each unstable nucleus splits into two nuclei, which can either be stable or unstable.
Afterwards, each unstable nucleus splits into two nuclei again, which continues until all remaining nuclei are stable.
Whenever an unstable nucleus splits, an energy of \(100\text{MeV}\) is produced.
Suppose an unstable nucleus has repeatedly split until all of the \(8\) remaining nuclei are stable. What is the total energy produced in this process, in \(\text{MeV}\)? [3 points]
  1. \(800\)
  2. \(700\)
  3. \(600\)
  4. \(500\)
  5. \(400\)
The biological index is an index used to scale the degree of water pollution. Given that \(\mathrm{X}\) is the number of colored organisms and \(\mathrm{Y}\) is the number of colorless organisms, the biological index is defined as
\(\dfrac{\mathrm{X}}{\mathrm{X+Y}} \times 100(\%).\)
Suppose the biological index of some lake was \(10(\%)\) in last month. When the water of this lake was examined in this month, the number of colored organisms was \(2\) times that of last month, and the number of colorless organisms was \(3\) times that of last month. What is the biological index of this lake in this month? [3 points]
  1. About \(14.3\%\)
  2. About \(15.2\%\)
  3. About \(16.4\%\)
  4. About \(17.1\%\)
  5. About \(18.5\%\)
Suppose the Korean government saves a part of its budget in every January \(1\)st starting from the year \(2001\), to prepare for unification costs after the reunification of Korea. To account for the economic growth rate, the amount to be saved every year will be increased by \(6\%\) compared to the previous year. Suppose the amount saved in January \(1\)st, \(2001\), is \(10\) trillion won. What is the total amount of the saved money and interest accumulated until December
\(31\)st, \(2010\), in trillion won?
(※ Assume an yearly compound interest of \(6\%\), and approximate \((1.06)^{10}\) as \(1.8\).) [4 points]
  1. \(160\)
  2. \(162\)
  3. \(180\)
  4. \(198\)
  5. \(220\)

Mathematics·Studies (I)

Hum. & Arts
Suppose a baseball player hits the ball with a probability of \(0.2\) while facing against pitcher \(\mathrm{A}\), and hits the ball with a probability of \(0.25\) while facing against pitcher \(\mathrm{B}\). Suppose this player faced against pitcher \(\mathrm{A}\) twice, and then faced against pitcher \(\mathrm{B}\) once in some game. What is the probability that the player hit the ball twice or more out of the \(3\) times? [3 points]
  1. \(0.10\)
  2. \(0.12\)
  3. \(0.14\)
  4. \(0.15\)
  5. \(0.16\)
Short Answers (25~30)
Let \(A= \begin{pmatrix} 0&1 \\ 2&3 \end{pmatrix}\). Compute the sum of all components of \(A^2\). [2 points]
Given \(\triangle \mathrm{ABC}\) with \(b=8, c=7\) and \(\angle \mathrm{A}=120°\), compute \(a\). [3 points]
Figure shows a region in the \(1\)st quadrant of the \(xy\)-plane enclosed by the \(x\)-axis and two curves \(y= 3 - \dfrac{1}{2}x^2\) and \(x^2+y^2=9\). Let \(V\) be the volume of the solid formed by rotating this region about the \(y\)-axis. Compute \(\dfrac{1}{\pi}V\). (※ \(\pi\) is the ratio of a circle's circumference to it's diameter.) [3 points]

Mathematics·Studies (I)

Hum. & Arts
There are \(4\) islands as shown to the right. Compute the number of ways to build \(3\) bridges to make all \(4\) islands connected. [3 points]
Figure below shows a portion of the graphs of functions \(y=1\) and \(y=0\). Suppose we connect the points \(\mathrm{A}(0,1)\) and \(\mathrm{B}(1,0)\) in this figure with a graph of the function \(y=ax^3+bx^2+cx+1\) defined on \(0\leq x\leq 1\). Let us choose the value of constants \(a, b\) and \(c\) such that the function representing the resulting graph is differentiable on \((-\infty, \infty)\). Compute \(a^2+b^2+c^2\). [4 points]
Let \(\mathrm{P_1}(0)\) and \(\mathrm{P_2}(80)\) be points on the number line. Let \(\mathrm{P_3}(x_3)\) be the midpoint of the line segment \(\mathrm{P_{1}P_{2}}\), let \(\mathrm{P_4}(x_4)\) be the midpoint of the line segment \(\mathrm{P_{2}P_{3}}\), \(\cdots\), let \(\mathrm{P}_{n+2}(x_{n+2})\) be the midpoint of the line segment \(\mathrm{P}_{n} \mathrm{P}_{n+1}\).
Compute \(\displaystyle\lim_{n\;\!\to\;\!\infty}x_n\) and round the result to the nearest hundredth. [3 points]