1996 College Scholastic Ability Test

Mathematics·Studies (I)

Hum. & Arts
For \(x=2-\sqrt{3}\) and \(y=2+\sqrt{3}\), what is the value of \(\dfrac{y}{x}+\dfrac{x}{y}\)? [1 point]
  1. \(8\)
  2. \(10\)
  3. \(12\)
  4. \(14\)
  5. \(16\)
The division of the polynomial \(x^4-3x^2+ax+5\) by \(x+2\) has a remainder of \(3\). What is the value of \(a\)? [1 point]
  1. \(0\)
  2. \(2\)
  3. \(3\)
  4. \(-2\)
  5. \(-3\)
For the matrix \(A= \begin{pmatrix*}[r] 1&-1 \\ 0&1 \end{pmatrix*}\), what is the value of \(A^3\)? [1 point]
  1. \(\begin{pmatrix*}[r] 1&-3 \\ 0&1 \end{pmatrix*}\)
  2. \(\begin{pmatrix*}[r] 1&0 \\ -3&1 \end{pmatrix*}\)
  3. \(\begin{pmatrix*}[r] 1&-1 \\ -1&1 \end{pmatrix*}\)
  4. \(\begin{pmatrix*}[r] -1&1 \\ 1&-1 \end{pmatrix*}\)
  5. \(\begin{pmatrix*}[r] 3&-3 \\ 0&3 \end{pmatrix*}\)
What is the value of the integral \(\displaystyle\int_{-1}^{1} x(1-x)^2 dx\)? [1 point]
  1. \(0\)
  2. \(-\dfrac{2}{3}\)
  3. \(\dfrac{2}{3}\)
  4. \(-\dfrac{4}{3}\)
  5. \(\dfrac{4}{3}\)

Mathematics·Studies (I)

Hum. & Arts
What is the number of ways to arrange the alphabets \(P, A, S, S\) in a row? [1 point]
  1. \(6\)
  2. \(8\)
  3. \(12\)
  4. \(18\)
  5. \(24\)
For a triangle \(\mathrm{ABC}\), consider the statement
‘If \(\overline{\mathrm{AB}}=\overline{\mathrm{AC}}\), then \(\angle \mathrm{B} = \angle \mathrm{C}\).’ Which option contains all correct statements among the converse, inverse and contrapositive of this statement? [1 point]
  1. Contrapositive
  2. Converse and inverse
  3. Inverse and contrapositive
  4. Converse and contrapositive
  5. Converse, inverse and contrapositive
Figure to the right is the graph of a differentiable function \(y=f(x)\) and \(y=x\). Which option contains every correct statement in the <List>, given that \(0<a<b\)? [1 point]
  1. \(\dfrac{f(a)}{a}< \dfrac{f(b)}{b}\)
  2. \(f(b)-f(a) > b-a\)
  3. \(f'(a) > f'(b)\)
  1. a
  2. b
  3. c
  4. a, b
  5. b, c
Which option below depicts the region on the
\(xy\)-plane satisfying \((x^2-4y^2)(x^2-6x+y^2+8)\leq 0\)? (※ The boundary of the colored region is included.) [1 point]

Mathematics·Studies (I)

Hum. & Arts
Figure to the right is the graph of a function \(y=f(x)\). Among the functions \(y=g_1(x)\), \(y=g_2(x)\) and \(y=g_3(x)\) shown in the graphs below, which option contains all functions \(g_k(x)\) (\(k=1,2,3\)) for which the function \(y=f(x)g_k(x)\) is continuous on the interval \([-1, 3]\)? [1 point]
  1. \(g_1(x)\)
  2. \(g_2(x)\)
  3. \(g_1(x)\) and \(g_2(x)\)
  4. \(g_1(x)\) and \(g_3(x)\)
  5. \(g_1(x)\), \(g_2(x)\) and \(g_3(x)\)
\(b_k\) equals either \(0\) or \(1\) for \(k=1, 2, 3, 4, \cdots\), and
\(\log_7 2= \dfrac{b_1}{2} + \dfrac{b_2}{2^2} + \dfrac{b_3}{2^3} + \dfrac{b_4}{2^4} + \cdots\).
What are the values of \(b_1, b_2\) and \(b_3\) in this order? [1.5 points]
  1. \(0, 0\) and \(0\)
  2. \(0, 1\) and \(0\)
  3. \(0, 0\) and \(1\)
  4. \(0, 1\) and \(1\)
  5. \(1, 1\) and \(1\)
There is a right triangle \(\mathrm{ABC}\) with \(\overline{\mathrm{AB}}=2\), \(\overline{\mathrm{BC}}=1\) and \(\angle \mathrm{B}= 90°\). As the figure shows, let \(\mathrm{B_1, B_2, B_3,} \cdots, \mathrm{B}_{n-1}\) be the points dividing the edge \(\mathrm{AB}\) into \(n\) equal parts. Let us draw lines at each point parallel to edge \(\mathrm{BC}\), and let their intersections with edge \(\mathrm{AB}\) be points \(\mathrm{C_1, C_2, C_3,} \cdots, \mathrm{C}_{n-1}\), respectively. What is the value of \(\displaystyle\lim_{n\to\infty}\dfrac{2\pi}{n} \displaystyle\sum_{k\,=\,1}^{n\,-\,1} \overline{\mathrm{B}_k \mathrm{C}_k}^2\)? [1 point]
  1. \(\dfrac{\pi}{6}\)
  2. \(\dfrac{\pi}{3}\)
  3. \(\dfrac{\pi}{2}\)
  4. \(\dfrac{2\pi}{3}\)
  5. \(\pi\)
Which of the following data have the greatest standard deviation? [1 point]
  1. \(1,5,1,5,1,5,1,5,1,5\)
  2. \(1,5,1,5,1,5,3,3,3,3\)
  3. \(2,4,2,4,2,4,2,4,2,4\)
  4. \(2,4,2,4,2,4,3,3,3,3\)
  5. \(4,4,4,4,4,4,4,4,4,4\)

Mathematics·Studies (I)

Hum. & Arts
A polynomial function \(g(x)\) satisfies \(g(g(x))=x\) for all real numbers \(x\) and \(g(0)=1\). What is the value of \(g(-1)\)? [1.5 points]
  1. \(-2\)
  2. \(-1\)
  3. \(0\)
  4. \(1\)
  5. \(2\)
Below is the graph of the function \(y=f(x)\) defined on all real numbers.
For \(g(x)=\sin x\), which option below is an appropriate graph of the composite function \(y=(g \circ f)(x)\)? [1 point]
Two cars \(A\) and \(B\) are driving in the same direction on a race course shown in the figure. The speed of cars \(A\) and \(B\) are \(a\,\text{km/minutes}\) and \(b\,\text{km/minutes}\) respectively, and the length of one lap around the race course is \(c\,\text{km}\). Given that
\(3a-3b=2c\),
which option below is correct? [1.5 points]
  1. Every \(3\) minutes, \(A\) does two more laps than \(B\).
  2. Every \(3\) minutes, \(A\) does one more lap than \(B\).
  3. Every \(2\) minutes, \(A\) does three more laps than \(B\).
  4. Every \(2\) minutes, \(B\) does two more laps than \(A\).
  5. Every \(2\) minutes, \(B\) does three more laps than \(A\).
The following is a part of an argument about sets between two students Gap and Eul.
Gap : Suppose \(S\) is a set of <all sets that we can think of>. Then \(S\) itself is an element of \(S\)
\((\alpha)\)
, right?
Eul : That's nonsense. How would such a thing exist?
Gap : Alright. Then, how about a set of <all sets that are not an element of themselves>?
\((\beta)\)
In the argument above, what are the appropriate mathematical expressions for the underlined places \((\alpha)\) and \((\beta)\)? [1.5 points]
\((\alpha)\) \((\beta)\)
① \(S \in S\) \(\{A \,\big|\, A \notin A, \:A\) is a set \(\}\)
② \(S \in S\) \(\{A \,\big|\, A \nsubseteq A, \:A\) is a set \(\}\)
③ \(S \in S\) \(\{A \,\big|\, A \in A, \:A\) is a set \(\}\)
④ \(S \subseteq S\) \(\{A \,\big|\, A \notin A, \:A\) is a set \(\}\)
⑤ \(S \subseteq S\) \(\{A \,\big|\, A \subseteq A, \:A\) is a set \(\}\)

Mathematics·Studies (I)

Hum. & Arts
Let us select three random vertices in the cube to the right and form a triangle. What is the number of ways to form a triangle congruent to the triangle shown in the figure? [1.5 points]
  1. \(4\)
  2. \(6\)
  3. \(8\)
  4. \(12\)
  5. \(24\)
The following is an explanation of the way to produce products \(P_n\) and the required time.
(※ \(n=2^k,\: k=0,1,2,3,\cdots\))
A. The time it takes to produce one product \(P_1\) is \(1\).
B. After producing two products \(P_1\), one at a time, you can attach them to produce the product \(P_2\).
C. After producing two products \(P_n\), one at a time, you can attach them to produce one product \(P_{2n}\). The time it takes to attach two products \(P_n\) is \(2n\).
What is the time it takes to produce one product \(P_{16}\)? [1 point]
  1. \(32\)
  2. \(64\)
  3. \(80\)
  4. \(96\)
  5. \(112\)
Figure below shows some squares attached together. What is the ratio of the edge length of squares \(A\) and \(B\)? [1.5 points]
  1. \(4\,:\,3\)
  2. \(8\,:\,5\)
  3. \(15\,:\,12\)
  4. \(16\,:\,11\)
  5. \(17\,:\,13\)
As the figure shows, let \(\mathrm{C}\) be a point on the line segment \(\mathrm{AB}\), and let us form two equilateral triangles \(\mathrm{ACD}\) and \(\mathrm{BCE}\) above the line segment \(\mathrm{AB}\). The following is a proof that \(\overline{\mathrm{AE}} = \overline{\mathrm{DB}}\).
(Proof)
\(\fbox{\(\;(\alpha)\;\)}\) from triangle \(\mathrm{ACD}\)
\((1)\)
\(\fbox{\(\;(\beta)\;\)}\) from triangle \(\mathrm{BCE}\)
\((2)\)
Since \(\angle \mathrm{ACD} = \angle \mathrm{ECB} = 60°\),
\(\angle \mathrm{ACE} = 60°+ \angle \mathrm{DCE} = \angle \mathrm{DCB}\)
\((3)\)
From \((1), (2)\) and \((3)\),
\(\triangle \mathrm{ACE} \equiv \triangle \mathrm{DCB}\)
since the two sides and the angle between them are equivalent. Therefore \(\overline{\mathrm{AE}} = \overline{\mathrm{DB}}\).
In the proof above, what are appropriate for \((\alpha)\) and \((\beta)\)? [1 point]
\((\alpha)\) \((\beta)\)
① \(\overline{\mathrm{AC}} = \overline{\mathrm{AD}}\) \(\overline{\mathrm{CE}} = \overline{\mathrm{BE}}\)
② \(\overline{\mathrm{AC}} = \overline{\mathrm{DC}}\) \(\overline{\mathrm{CE}} = \overline{\mathrm{BE}}\)
③ \(\overline{\mathrm{AD}} = \overline{\mathrm{CD}}\) \(\overline{\mathrm{CB}} = \overline{\mathrm{BE}}\)
④ \(\overline{\mathrm{AC}} = \overline{\mathrm{AD}}\) \(\overline{\mathrm{CE}} = \overline{\mathrm{CB}}\)
⑤ \(\overline{\mathrm{AC}} = \overline{\mathrm{DC}}\) \(\overline{\mathrm{CE}} = \overline{\mathrm{CB}}\)

Mathematics·Studies (I)

Hum. & Arts
The following is a proof that ‘if \(p\) is even and \(q\) is odd, then the equation \(x^2+px-2q=0\) does not have an integer solution.’
(Proof)
If \(x\) is \(\fbox{\(\;(\alpha)\;\)}\), \(x^2\) is \(\fbox{\(\;(\alpha)\;\)}\) and \(px-2q\) is even.
Therefore the equation \(x^2+px-2q\) is \(\fbox{\(\;(\alpha)\;\)}\), and thus cannot be \(\fbox{\(\;(\beta)\;\)}\).
If \(x\) is \(\fbox{\(\;(\gamma)\;\)}\), then \(x^2+px\) is a multiple of \(4\) and \(2q\) is not a multiple of \(4\).
However, this is a contradiction since \(\fbox{\(\;(\delta)\;\)}\).
Therefore, this equation does not have an integer solution.
In the proof above, what are appropriate for \((\alpha)\) ~ \((\delta)\)? [1.5 points]
\((\alpha)\) \((\beta)\) \((\gamma)\) \((\delta)\)
① even \(0\) odd \(x^2+px=2q\)
② even quadratic odd \(2q\) is even
③ an integer \(0\) even \(x^2+px=2q\)
④ odd quadratic even \(2q\) is even
⑤ odd \(0\) even \(x^2+px=2q\)
There is a box containing ten balls marked with integers from \(1\) to \(10\). Suppose we mix them well and take two balls out, one by one. The probability that the number marked on the second ball is greater than the first, is \(\dfrac{1}{2}\). The following is a proof of this.
(※ The ball taken out is not put in again)
(Proof)
Let \(X_1\) and \(X_2\) be the numbers marked on the first and second ball respecitvely, and \(p\) be the probability to be calculated. For integers \(n\) from \(1\) to \(10\), let \(A_n\) be the event where \(X_1=n\), and \(B_n\) be the event where \(X_2 \geq n+1\). Then
\(\begin{align} p &= \displaystyle\sum_{n\,=\,1}^{10} \fbox{\(\;(\alpha)\;\)} \, \cdot \mathrm{P}(A_n) \\ &= \displaystyle\sum_{n\,=\,1}^{9} \dfrac{10-n}{9} \cdot \fbox{\(\;(\beta)\;\)} \; = \dfrac{1}{2}. \end{align}\)
In the proof above, what are appropriate for \((\alpha)\) and \((\beta)\)? [1.5 points]
\((\alpha)\) \((\beta)\)
① \(\mathrm{P}(A_n \cap B_n)\) \(^1/_{10}\)
② \(\mathrm{P}(B_n)\) \(^1/_{10}\)
③ \(\mathrm{P}(B_n)\) \(^1/_9\)
④ \(\mathrm{P}(B_n | A_n)\) \(^9/_{10}\)
⑤ \(\mathrm{P}(B_n | A_n)\) \(^1/_{10}\)
Let \(g(x)\) be the inverse of the function \(f(x) = \dfrac{x^2}{4}+a \,(x\geq 0)\). What is the range of values of \(a\) for which the equation \(f(x)=g(x)\) has two distinct nonnegative solutions? [1.5 points]
  1. \(0 \leq a < 1\)
  2. \(a \geq 0\)
  3. \(a < 1\)
  4. \(0 < a < 2\)
  5. \(a < 2\)
As the figure shows, suppose we endlessly repeat the process of attaching right isosceles triangles and squares alternately to a square. Let \(S_1, S_2, S_3, \cdots\) be the squares and \(T_1, T_2, T_3, \cdots\) be the triangles. Given that the side length of \(S_1\) is \(2\), what is the total area of these triangles and squares? [1.5 points]
  1. \(10\)
  2. \(11\)
  3. \(12\)
  4. \(13\)
  5. \(14\)

Mathematics·Studies (I)

Hum. & Arts
Consider three points \(\mathrm{A}(0,2)\), \(\mathrm{B}(-1,0)\) and \(\mathrm{C}(1,0)\) on the \(xy\)-plane. Let \(a, b\) and \(c\) be the distance from a point \(\mathrm{P}\) in the inside or boundary of \(\triangle \mathrm{ABC}\), to edges \(\mathrm{AB}, \mathrm{BC}\) and \(\mathrm{CA}\) respectively. Given that \(4b=5(a+c)^2\), what is the shape created by the locus of point \(\mathrm{P}\)? [2 points]
  1. A point
  2. A line segment parallel to the \(x\)-axis
  3. A line segment parallel to the \(y\)-axis
  4. A curve that is part of a parabola
  5. A curve that is part of a circle
Consider the region on the \(xy\)-plane created by the following system of inequalities.
\(\begin{cases} |x|+|y| \leq 4 \\\\ \log_2 (x+y)^4 - \log_2 (x+y)^2 \geq 2 \end{cases}\)
If a circle with center \(\left(\!\dfrac{1}{2}, -1\!\right)\) and radius \(r\) meets this region, what is the minimum and maximum value of \(r\)? [2 points]
[Minimum value] [Maximum value]
① \(\dfrac{\sqrt{2}}{4}\) \(\dfrac{\sqrt{85}}{2}\)
② \(\dfrac{5\sqrt{2}}{4}\) \(\dfrac{\sqrt{101}}{2}\)
③ \(\dfrac{3\sqrt{2}}{4}\) \(\dfrac{\sqrt{85}}{2}\)
④ \(\dfrac{3\sqrt{2}}{4}\) \(\dfrac{\sqrt{101}}{2}\)
⑤ \(\dfrac{\sqrt{2}}{4}\) \(\dfrac{\sqrt{101}}{2}\)
Cars are driving on a \(4\)-lane highway in one direction, with a speed of \(100\,\text{km/hour}\) or less, while maintaining a distance of \(100\,\text{m}\) or greater between each car. Suppose we count the number of all cars that pass through a point on the 4 lanes of the highway in one direction for one hour. What is the maximum number of cars that can pass through?
(※ Ignore the length of the cars.) [1.5 points]
  1. \(2000\)
  2. \(4000\)
  3. \(6000\)
  4. \(8000\)
  5. \(10000\)
The Mathematics·Studies (I) exam of the College Scholastic Ability Test contains \(30\) questions, and the total score is \(40\) points. The score type of each question can be either \(1\) point, \(1.5\) points, or \(2\) points. Given that there should be at least one question of each score type, what is the minimum number of \(1\) point questions? [1.5 points]
  1. \(8\)
  2. \(9\)
  3. \(10\)
  4. \(11\)
  5. \(12\)

Mathematics·Studies (I)

Hum. & Arts
Consider a rectangle with width \(10\) and height \(6\) as the figure shows. Suppose a circle with radius \(1\) moves inside this rectangle, and the region it passes through is colored with fluorescent paint. Given that the center of the circle moved from \(\mathrm{A}\) to \(\mathrm{B}\) along the path denoted with arrows in the figure, what is the area of the region inside the rectangle that is not colored with fluorescent paint? (※ All line segments in the path are either parallel to or perpendicular to edges of the rectangle.) [2 points]
  1. \(0\)
  2. \(10 - \dfrac{5}{2}\pi\)
  3. \(8 - 2\pi\)
  4. \(6 - \dfrac{3}{2}\pi\)
  5. \(4 - pi\)
Figure shows the network of roads of some city. The squares formed by main roads all have a distance of \(1\) between the crossroads, and the ring road of the city is a portion of a circle with center \(\mathrm{O}\). Suppose a distribution company that owns four retail stores \(\mathrm{A, B, C}\) and \(\mathrm{D}\) wants to build a warehouse in one position among points \(\alpha, \beta, \gamma, \delta\) and \(\varepsilon\) on the ring road. Let \(a, b, c\) and \(d\) be the minimum distance along the road from the warehouse to retail stores \(\mathrm{A, B, C}\) and \(\mathrm{D}\) respectively. What is the position of the warehouse that minimizes \(a+b+c+d\)? [2 points]
  1. \(\alpha\)
  2. \(\beta\)
  3. \(\gamma\)
  4. \(\delta\)
  5. \(\varepsilon\)