1995 College Scholastic Ability Test

Mathematics·Studies (I)

Nat. Sciences
Let \(\alpha\) and \(\beta\) be the two solutions to the quadratic equation \(2x^2-4x-1=0\). What is the value of \(\alpha^3 + \beta^3\)? [1 point]
  1. \(1\)
  2. \(3\)
  3. \(4\)
  4. \(8\)
  5. \(11\)
Let \(\alpha\) be the solution to the equation \(3^{x+2}=96\). Which option below is correct? [1 point]
  1. \(0 < \alpha < 1\)
  2. \(1 < \alpha < 2\)
  3. \(2 < \alpha < 3\)
  4. \(3 < \alpha < 4\)
  5. \(4 < \alpha < 5\)
Consider the following \(2 \times 2\) matrices \(A\) and \(B\).
\(A=\begin{pmatrix*}[r] 2&-4 \\ -1&2 \end{pmatrix*},\; B=\begin{pmatrix} 1&2 \\ 2&4 \end{pmatrix}\)
Which option is equal to the matrix \(\dfrac{1}{3}AB-BA\)? [1 point]
  1. \(\begin{pmatrix*}[r] -2&-4 \\ 1&2 \end{pmatrix*}\)
  2. \(\begin{pmatrix*}[r] -2&8 \\ 2&-4 \end{pmatrix*}\)
  3. \(\begin{pmatrix*}[r] -4&-8 \\ 2&4 \end{pmatrix*}\)
  4. \(\begin{pmatrix*}[r] -6&-12 \\ 3&6 \end{pmatrix*}\)
  5. \(\begin{pmatrix*}[r] 0&0 \\ 0&0 \end{pmatrix*}\)
What is the value of \(\displaystyle\int_{0}^{\pi}(1-\cos^3 x)\cos x\sin x\,dx\)? [1 point]
  1. \(0\)
  2. \(-\dfrac{1}{5}\)
  3. \(-\dfrac{2}{5}\)
  4. \(-\dfrac{3}{5}\)
  5. \(-\dfrac{4}{5}\)

Mathematics·Studies (I)

Nat. Sciences
Consider the universe \(U\) and its subsets \(A\) and \(B\). Given that \(A \subseteq B\), which option below is not always true? (※ \(U\ne\varnothing\)) [1 point]
  1. \(A\cup B=B\)
  2. \(A\cap B=A\)
  3. \((A\cap B)^c=B^c\)
  4. \(B^c \subseteq A^c\)
  5. \(A-B=\varnothing\)
Let \(f(x)=2x-1\). A function \(g(x)\) satisfies
\((h \circ g \circ f)(x)=h(x)\)
for all functions \(h(x)\). What is the value of \(g(3)\)?
(※ \(f(x), g(x)\) and \(h(x)\) are functions from \(\mathbb{R}\) to \(\mathbb{R}\), where \(\mathbb{R}\) is the set of all real numbers.) [1 point]
  1. \(-2\)
  2. \(-1\)
  3. \(0\)
  4. \(1\)
  5. \(2\)
Figure below shows \(7\) points on a semicircle. What is the number of triangles that have three of these points as its vertices? [1 point]
  1. \(34\)
  2. \(33\)
  3. \(32\)
  4. \(31\)
  5. \(30\)
Suppose
\(\dfrac{1}{(x-1)(x-2)\cdots(x-10)}\)
\(=\dfrac{a_1}{x-1} + \dfrac{a_2}{x-2} + \cdots + \dfrac{a_{10}}{x-10}\)
holds for all real numbers \(x\) which does not make the denominator \(0\). What is the value of \(a_1+a_2+\cdots+a_{10}\)? [1.5 points]
  1. \(0\)
  2. \(-1\)
  3. \(1\)
  4. \(-10\)
  5. \(10\)

Mathematics·Studies (I)

Nat. Sciences
Let \(x\) and \(y\) be real numbers such that \((x+y)(x-y)\ne 0\) and
\(\sqrt{\dfrac{x+y}{x-y}}=-\dfrac{\sqrt{x+y}}{\sqrt{x-y}}\).
Which option below depicts the region on the \(xy\)-plane where the point \((x, y)\) can exist?
(※ The dotted lines are not included.) [1 point]
Let \(f(x)\) be a function which is continuous, but not differentiable, at \(x=0\). Which option contains every function in the <List> that is differentiable at \(x=0\)? [1.5 points]
  1. \(y=xf(x)\)
  2. \(y=x^2f(x)\)
  3. \(y=\dfrac{1}{1+xf(x)}\)
  1. a
  2. b
  3. c
  4. a, b
  5. a, b, c
A point \(\mathrm{P}\) starts from the origin and moves on the number line for \(7\) seconds. Its velocity \(v(t)\) at time \(t\) is shown below. What is the list of correct statements in the <List>? [1.5 points]
  1. After point \(\mathrm{P}\) started moving, there was a time where it stopped for \(1\) second.
  2. While point \(\mathrm{P}\) was moving, it changed its direction \(4\) times.
  3. Point \(\mathrm{P}\) was on the starting point
    \(4\) seconds after it started moving.
  1. a
  2. c
  3. a, b
  4. a, c
  5. b, c
† Rephrasing statement a. “Between \(\boldsymbol{0\leq t \leq 7}\), there was a time where point \(\mathrm{P}\) stopped for \(1\) second.”
Let \(f(x)\) and \(g(x)\) be probability density functions defined on the closed interval \([0,1]\). Which option below can always be a probability density function? [1 point]
  1. \(f(x)-g(x)\)
  2. \(f(x)+g(x)\)
  3. \(\dfrac{1}{2}\{f(x)-g(x)\}\)
  4. \(\dfrac{1}{3}\{2f(x)+g(x)\}\)
  5. \(2f(x)-g(x)\)

Mathematics·Studies (I)

Nat. Sciences
Consider a complex number \(\alpha\) such that the value of \(|z-\alpha|\) is constant for all complex numbers \(z\) that satisfy \(|z|=1\). What is the number of all such values of \(\alpha\)? [1.5 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. Infinitely many.
What is the value of \(m, n\) and \(k\) printed by the following flowchart, respectively? [1.5 points]
  1. \(0,2\) and \(5\)
  2. \(0,2\) and \(6\)
  3. \(0,5\) and \(3\)
  4. \(2,3\) and \(6\)
  5. \(2,3\) and \(5\)
Consider two points \(\mathrm{O}(0,0,0)\) and \(\mathrm{A}(1,0,0)\). Let \(\mathrm{P}(x,y,z)\) be a point moving in \(3\)-dimensional space such that the area of \(\triangle \mathrm{OAP}\) is \(2\). Given that \(0\leq x\leq 1\), consider the shape created by the locus of point \(\mathrm{P}\). What is the area of this shape when spread on a plane? [1.5 points]
  1. \(16\pi\)
  2. \(8\pi\)
  3. \(5\pi\)
  4. \(2\pi\)
  5. \(\pi\)
Consider a right triangle \(\mathrm{ABC}\) where \(\angle \mathrm{C}\) is a right angle and the magnitude of \(\angle \mathrm{B}\) is \(\dfrac{\pi}{3}\). Let \(\mathrm{D}\) be a point on the edge \(\mathrm{BC}\), and let \(\theta\) be the magnitude of \(\angle \mathrm{BAD}\). What is \(\dfrac{\:\overline{\mathrm{BD}}\:}{\overline{\mathrm{AB}}}\) expressed in terms of \(\theta\)? [1.5 points]
  1. \(\sin\theta\)
  2. \(\dfrac{\sin\theta}{1+\cos\theta}\)
  3. \(\dfrac{2\sin\theta}{1+2\cos\theta}\)
  4. \(\dfrac{2\sin\theta}{\sin\theta+\sqrt{3}\cos\theta}\)
  5. \(\dfrac{1-\cos\theta}{2}\)

Mathematics·Studies (I)

Nat. Sciences
An object starts from the origin and moves at a constant speed along a spiral shown to the right. Let \(x(t)\) be the \(x\)-coordinate of this object at time \(t\). Which option below is an appropriate graph of the relation between \(t\) and \(x(t)\)? [1 point]
Figure below is the graph of the function \(y=f(x)\). Which option below lists the number of distinct real solutions, and the sum of distinct real solutions, of the equation \(f(f(x+2))=4\) solved for \(x\)?
(※ \(f(x)<0\) when \(x<2\) or \(x>19\).) [1.5 points]
  1. \(2,20\)
  2. \(2,22\)
  3. \(3,30\)
  4. \(4,42\)
  5. \(4,50\)
Let us represent a positive integer \(n\) as \(n=2^p \cdot k\)
(\(p\) is a nonnegative integer and \(k\) is an odd number).
Let \(f(n)=p\). For example, \(f(12)=2\). What is the list of correct statements in the <List>? [1 point]
  1. If \(n\) is an odd number, then \(f(n)=0\).
  2. \(f(8)<f(24)\).
  3. There are infinitely many positive integers \(n\) that satisfy \(f(n)=3\).
  1. a
  2. b
  3. a, b
  4. a, c
  5. b, c
Let the set \(U=\{1,2,3,4,\cdots,100\}\). Consider a set \(A\) which is a subset of \(U\) and satisfies the following conditions (A) and (B). What such set \(A\) has the smallest number of elements? [1 point]
  1. \(3\in A\)
  2. If \(m, n\in A\) and \(m+n \in U\), then \(m+n \in A\).
  1. \(A=\{3,9,15,21,\cdots,99\}\)
  2. \(A=\{3,6,9,12,\cdots,99\}\)
  3. \(A=\{3,4,5,6,\cdots,100\}\)
  4. \(A=\{1,3,5,7,\cdots,99\}\)
  5. \(A=\{1,2,3,4,\cdots,100\}\)

Mathematics·Studies (I)

Nat. Sciences
Figure shows a trapezoid \(\mathrm{ABCD}\) with
\(\overline{\mathrm{AB}}=\overline{\mathrm{AD}}=1\), \(\:\overline{\mathrm{BC}}=2\),
and the magnitude of \(\angle \mathrm{A}\) and \(\angle \mathrm{B}\) are \(\dfrac{\pi}{2}\).
Let \(\mathrm{P}\) be a point on the edge \(\mathrm{AD}\). Let \(\overline{\mathrm{PB}}=x\) and \(\overline{\mathrm{PC}}=y\). What is the list of correct statements in the <List>? [1.5 points]
  1. \(xy\geq 2\).
  2. If \(xy=2\), then \(\triangle \mathrm{BCP}\) is a right triangle.
  3. \(xy\leq \sqrt{5}\).
  1. a
  2. c
  3. a, c
  4. b, c
  5. a, b, c
The following is a proof of the law of cosines relating the side lengths of a triangle to the cosine of an angle, specifically if \(\angle \mathrm{A}\) is obtuse in \(\triangle \mathrm{ABC}\).
(Proof) As the figure shows, let \(\triangle \mathrm{ABC}\) with side lengths \(a, b\) and \(c\) be on the \(xy\)-plane so that point \(\mathrm{A}\) is on the origin. Let \((x,y)\) be the coordinates of point \(\mathrm{C}\). Then
\(x= \fbox{\(\;(\alpha)\;\)}\) and \(y= \fbox{\(\;(\beta)\;\)}\),
so the following holds by the Pythagorean theorem.
\(\begin{align}a^2 &= (\fbox{\(\;(\gamma)\;\)})^2+y^2\\ &= b^2+c^2-2bc \cos A \end{align}\)
In the proof above, what are appropriate for \((\alpha), (\beta)\) and \((\gamma)\) in this order? [1 point]
① \(b\cos A\), \(b\sin A\), \(c+x\)
② \(b\cos A\), \(b\sin A\), \(c-x\)
③ \(b\cos A\), \(-b\sin A\), \(c+x\)
④ \(-b\cos A\), \(-b\sin A\), \(c-x\)
⑤ \(-b\cos A\), \(-b\sin A\), \(c+x\)
Consider a cubic equation \(x^3+ax^2+bx+c=0\) with three real solutions \(\alpha, \beta\) and \(\gamma\). The following is a proof that at least one of the three solutions has an absolute value greater than or equal to \(\dfrac{|a|}{3}\).
(Proof)
Let us assume the conclusion is false and that \(\fbox{ (A) }\). Then
\(|\,\alpha\,|< \dfrac{|a|}{3},\; |\,\beta\,|< \dfrac{|a|}{3}\:\) and \(\:|\,\gamma\,|< \dfrac{|a|}{3}\).
Using the relation between solutions and coefficients,
\(a= \fbox{ (B) }\),
and
\(\begin{align} |a| &\leq |\alpha+\beta| + |\gamma| \\ &\leq \fbox{ (C) }\\ &< \dfrac{|a|}{3}+\dfrac{|a|}{3}+\dfrac{|a|}{3}=|a|. \end{align}\)
This is a contradiction. Therefore at least one of the three solutions has an absolute value greater than or equal to \(\dfrac{|a|}{3}\).
In the proof above, what are appropriate for \(\text{(A)}, \text{(B)}\) and \(\text{(C)}\) in this order? [1 point]
  1. some solutions have an absolute value less than \(\dfrac{|a|}{3}\),
    \(-(\alpha+\beta+\gamma),\quad |\alpha|+|\beta|+|\gamma|\)
  2. some solutions have an absolute value less than or equal to \(\dfrac{|a|}{3}\),
    \(\alpha+\beta+\gamma,\quad |\alpha|+|\beta|+|\gamma|\)
  3. all solutions have an absolute value less than \(\dfrac{|a|}{3}\),
    \(\alpha+\beta+\gamma,\quad |\alpha+\beta+\gamma|\)
  4. some solutions have an absolute value less than \(\dfrac{|a|}{3}\),
    \(-(\alpha+\beta+\gamma),\quad |\alpha|+|\beta|+|\gamma|\)
  5. all solutions have an absolute value less than or equal to \(\dfrac{|a|}{3}\),
    \(\alpha+\beta+\gamma,\quad |\alpha+\beta+\gamma|\)

Mathematics·Studies (I)

Nat. Sciences
The following is a proof of a theorem about the harmonic mean.
(Proof) For positive numbers \(a,b\) and \(H\), suppose a real number \(r\) exists such that
\(a=H+\dfrac{a}{r}\) and \(H=b+\dfrac{b}{r}\).
\((\mathrm{A})\)
Then, \(a\ne b\) and
\(\dfrac{a-H}{a}= \fbox{\(\;(\alpha)\;\)}\)
\((\mathrm{B})\)
therefore \(H=\fbox{\(\;(\beta)\;\)}\).
Conversely, let \(a\) and \(b\) be positive numbers such that \(a\ne b\). Let \(H=\fbox{\(\;(\beta)\;\)}\). Then \((\mathrm{B})\) holds and \(\dfrac{a-H}{a}\ne 0\).
From \((\mathrm{B})\), let \(\dfrac{a-H}{a}=\dfrac{1}{r}\). Then \((\mathrm{A})\) also holds.
Therefore, for positive numbers \(a, b\) and \(H\),
‘\(a\ne b\) and \(H=\fbox{\(\;(\beta)\;\)}\)’ is \(\fbox{\(\;(\gamma)\;\)}\) for the existence of a real number \(r\) such that \((\mathrm{A})\) holds.
In the proof above, what are appropriate for \((\alpha), (\beta)\) and \((\gamma)\) in this order? [1.5 points]
  1. \(\dfrac{H-b}{b}\), \(\dfrac{2ab}{a+b}\), necessary and sufficient
  2. \(\dfrac{H-b}{b}\), \(\dfrac{ab}{a+b}\), necessary and sufficient
  3. \(\dfrac{H-b}{b}\), \(\dfrac{2ab}{a+b}\), sufficient
  4. \(\dfrac{b-H}{b}\), \(\dfrac{2ab}{a+b}\), necessary
  5. \(\dfrac{b-H}{b}\), \(\dfrac{ab}{a+b}\), sufficient
For all positive numbers \(n\), the polynomial function \(f_n(x)\) satisfies the following.
  1. \(f_1(x)=x^2\)
  2. \(f_{n+1}(x)=f_n(x)+f_n'(x)\)
What is the constant term of \(f_{25}(x)\)? [1.5 points]
  1. \(548\)
  2. \(550\)
  3. \(552\)
  4. \(554\)
  5. \(556\)
Consider two points \(\mathrm{O}(0,0)\) and \(\mathrm{A}(2,0)\) on the \(xy\)-plane, and a point \(\mathrm{P}(t,2)\) moving on the line \(y=2\). Let \(\mathrm{Q}\) be the point where the line \(\mathrm{AP}\) meets the line \(y=\dfrac{1}{2}x\). Let \(f(t)\) be the the area of \(\triangle\mathrm{QOA}\) divided by the area of \(\triangle\mathrm{POA}\).
Let \(t_1\) be the value of \(t\) such that \(f(t)= \dfrac{1}{3}\),
let \(t_2\) be the value of \(t\) such that \(f(t)=\dfrac{1}{2}\),
\(\cdots\), let \(t_n\) be the value of \(t\) such that \(f(t)=\dfrac{n}{n+2}\). What is the value of \(\displaystyle\lim_{n\;\!\to\;\!\,\infty}t_n\)? [2 points]
  1. \(0\)
  2. \(1\)
  3. \(2\)
  4. \(3\)
  5. \(4\)
† This question has been rephrased with the inclusion of \(f(t)\), since the original sentence is hard to translate.
What is the maximum value of the function \(f(x)=\log_9 (5-x) + \log_3 (x+4)\)? [1.5 points]
  1. \(\dfrac{7}{2}\)
  2. \(4\)
  3. \(\dfrac{2}{5}+\log_3 4\)
  4. \(\dfrac{3}{2}+\log_3 2\)
  5. \(4+\log_3 6\)

Mathematics·Studies (I)

Nat. Sciences
Points \(\mathrm{P, Q}\) and \(\mathrm{R}\) on the \(xy\)-plane satisfy the following.
  1. Points \(\mathrm{P}\) and \(\mathrm{Q}\) are symmetric about the line \(y=x\).
  2. \(\overrightarrow{\mathrm{OP}}+ \overrightarrow{\mathrm{OQ}} =\overrightarrow{\mathrm{OR}}\) (※ \(\mathrm{O}\) is the origin)
Suppose point \(\mathrm{P}\) moves on a unit circle with the origin as its center. What kind of shape is created by the movement of point \(\mathrm{R}\)? [2 points]
  1. A point
  2. An ellipse
  3. A line segment
  4. A hyperbola
  5. A parallelogram
Let \(x\) be the amount of labor input and \(y\) be the amount of capital input for some industry. Then, the output \(z\) of the industry is known to be as follows.
\(z=2x^\alpha y^{1-\alpha}\) (\(\alpha\) is a constant with \(0<\alpha<1\))
According to some data, the amount of labor input and capital input in the year \(1993\) was \(4\) times, and \(2\) times, that of the year \(1980\), respectively, and the industry output in the year \(1993\) was \(2.5\) times that of the year \(1980\). From this data, what is the value of the constant \(\alpha\) calculated to the hundredth?
(※ \(\log_{10}2=0.30\)) [2 points]
  1. \(0.50\)
  2. \(0.33\)
  3. \(0.25\)
  4. \(0.20\)
  5. \(0.10\)
Suppose we want to buy three rectangular iron plates, and make a gas boiler in the shape of a closed cylinder as shown to the right, using two plates for the top and bottom faces, and the other one for the body. We can buy each iron plate with any horizontal and vertical length as we desire, and the price of an iron plate is \(10,\!000\) won per \(1\text{m}^2\). What is the minimum cost needed to buy the iron plates to make a gas boiler with a volume of \(64\text{m}^3\)? [2 points]
  1. \(1,\!110,\!000\) won
  2. \(1,\!040,\!000\) won
  3. \(1,\!000,\!000\) won
  4. \(960,\!000\) won
  5. \(900,\!000\) won