1994 College Scholastic Ability Test No.1

Mathematics·Studies (I)

For positive numbers \(a, x\) and \(y\), let
\(A=\log_a \dfrac{x^2}{y^3}\:\) and \(\:B=\log_a\dfrac{y^2}{x^3}\).
Which option is equal to \(3A+2B\)? (※ \(a\ne1\))
  1. \(\log_a\dfrac{1}{x^5}\)
  2. \(\log_a\dfrac{1}{y^5}\)
  3. \(\log_a\dfrac{1}{xy}\)
  4. \(\log_a\dfrac{x^5}{y^5}\)
  5. \(\log_a\dfrac{x^5}{y^7}\)
For a polynomial function \(f(x)\), what is the value of
\(\displaystyle\lim_{n\to\infty} n\left\{f\!\left(\!a+\dfrac{b}{n}\!\right) - f\!\left(\!a-\dfrac{b}{n}\!\right) \right\}\)?
(※ \(b\ne0\))
  1. \(\dfrac{1}{b}f'(a)\)
  2. \(0\)
  3. \(f'(a)\)
  4. \(bf'(a)\)
  5. \(2bf'(a)\)
\(A\) and \(B\) are square matrices of order \(2\) that are invertible. Which option below is incorrect?
  1. \((A^2)^{-1}=(A^{-1})^2\)
  2. \((B^{-1}AB)^2=B^{-1}A^2B\)
  3. If \(A^2=B^2\) then \(A=B\) or \(A=-B\).
  4. \(A^{-1}(A+B)B^{-1}=A^{-1}+B^{-1}\)
  5. If \(A\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}\) then \(x=y=0\).
The Ahmes Papyrus, a mathematical textbook from ancient Egypt(circa B.C. 1650), includes the following problem.
Divide \(120\) loaves of bread among \(5\) people, such that the lot of each person forms an arithmetic progression, and the sum of lots of the person who received the least and the person who received the second least is \(\dfrac{1}{7}\) of the sum of lots of the other \(3\) people.
If we divide breads as shown above, what is the lot of the person who receives the most?
  1. \(52\)
  2. \(50\)
  3. \(48\)
  4. \(46\)
  5. \(44\)

Mathematics·Studies (I)

Let \(a\) and \(b\) be the coefficient of \(x^3\) in two polynomials
\((1+x+x^2+x^3)^3\:\) and \(\:(1+x+x^2+x^3+x^4)^3\)
respectively. what is the value of \(a-b\)?
  1. \(4^3-5^3\)
  2. \(3^3-3^4\)
  3. \(0\)
  4. \(1\)
  5. \(-1\)
Two functions \(f(x)\) and \(g(x)\) satisfy
\(f(x)g(x)=0\)
for all real numbers \(x\). Which option below is correct about the two sets
\(A=\{x\,|\,f(x)=0\}\:\) and \(\:\{x\,|\,g(x)=0\}\)?
  1. \(A\) and \(B\) are both infinite sets.
  2. \(A\) and \(B\) are both finite sets.
  3. If \(A\) is a finite set, then \(B\) is an infinite set.
  4. If \(A\) is an infinite set, then \(B\) is a finite set.
  5. If \(A\) is an infinite set, then \(B\) is an infinite set.
The graph of two functions \(y=f(x)\) and \(y=g(x)\) are as the figures below.
Which option below is an appropriate graph of \(y=(g\circ f)(x)\)?

Mathematics·Studies (I)

A cubic equation \(x^3+ax^2+bx-3=0\) has \(1+\sqrt{2}i\) as a solution. For two real numbers \(a\) and \(b\), what is the product \(ab\)? (※ \(i=\sqrt{-1}\))
  1. \(10\)
  2. \(5\)
  3. \(0\)
  4. \(-15\)
  5. \(-10\)
For an integer \(n\,(n\geq 4)\), let
\(A_n=\{x\,|\,x\) is the length of a diagonal of a regular polygon with \(n\) sides of length \(1\}\),
and let \(a_n\) be the number of elements in the set \(A_n\). For example, \(a_4=1\). What is the value of \(\displaystyle\sum_{n=4}^{25}a_n\)?
  1. \(140\)
  2. \(138\)
  3. \(136\)
  4. \(134\)
  5. \(132\)
On the network of roads in the figure below, a car drives from point \(\mathrm{A}\) to point \(\mathrm{B}\) in the shortest path possible. Let \(a\) and \(b\) be the number of right-turns and left-turns made by the car respectively.
Which option below is always correct?
  1. \(a\) is even.
  2. \(b\) is odd.
  3. if \(a\) is even, then \(b\) is even.
  4. if \(a\) is even, then \(b\) is odd.
  5. if \(a\) is odd, then \(b\) is odd.
A function \(f(x)\) defined on the set of all real numbers satisfies the conditions in the <List> below.
  1. \(f(x)\) is continuous and \(f(x)=f(-x)\).
  2. \(f(x)=0\) when \(|x|>5\).
  3. \(|f(x)|\leq10\) when \(|x|<5\), and there is exactly \(1\) value of \(x\) where \(f(x)=10\).
Which option below is incorrect?
  1. \(f(5)=f(-5)=0\)
  2. \(f(x)\) has a global maximum at \(x=0\).
  3. There are at least \(2\) values of \(x\) where \(f(x)=5\).
  4. There is exactly \(1\) value of \(x\) where \(f(x)\) has a global minimum.
  5. \(f(x+5)f(x-5)=0\) for all real numbers \(x\).

Mathematics·Studies (I)

Set \(P\) is a subset of the set of all real numbers, and has the following properties \((A)\) and \((B)\).
\((A)\:\) For all real numbers \(a\), among the statements
\(a\in P,\: a=0\:\) and \(\:-a\in P\),
at least one, but no more than one, is satisfied.
\((B)\:\) If \(a\in P\) and \(b\in P\), then \(ab\in P\).
The following is a process proving ‘if \(a\in P\), then \(\dfrac{1}{a}\in P\,\)’ using the properties above.
(Proof) \(a\in P\) by the antedecent, so \(a\ne0\) by \((A)\). Therefore the real number \(\dfrac{1}{a}\) is not \(0\), so
\(\dfrac{1}{a}\in P\:\) or \(\:-\dfrac{1}{a}\in P\:\) by the property \(\fbox{\(\;(\alpha)\;\)}\).
Suppose \(-\dfrac{1}{a}\in P\). Then, by the property \(\fbox{\(\;(\beta)\;\)}\) and the antedecent, \(-1=a\times\left(\!-\dfrac{1}{a}\!\right)\in P\). However, if \(-1\in P\:\) then \(\:1=(-1)\times(-1)\in P\) by \((B)\), thus contradicting the property \(\fbox{\(\;(\gamma)\;\)}\).
Therefore, \(\dfrac{1}{a}\in P\:\).
In the proof above, what are appropriate for \((\alpha), (\beta)\) and \((\gamma)\) in this order?
  1. \((A), (B)\) and \((A)\)
  2. \((A), (B)\) and \((B)\)
  3. \((B), (A)\) and \((A)\)
  4. \((A), (A)\) and \((B)\)
  5. \((B), (A)\) and \((B)\)
On the \(xy\)-plane, there is a circle \((x-1)^2+(y-2)^2=r^2\) and a point \(\mathrm{A}(5, 4)\) outside the circle. Two tangent lines to the circle that passes through point \(\mathrm{A}\) are perpendicular to each other. What is the length of the radius \(r\)?
  1. \(\sqrt{10}\)
  2. \(\sqrt{11}\)
  3. \(\sqrt{12}\)
  4. \(\sqrt{13}\)
  5. \(\sqrt{14}\)
On the \(xy\)-plane, there are two points \(\mathrm{A}(x_1,y_1)\) and \(\mathrm{B}(x_2,y_2)\). Let points \(\mathrm{C}(x_3,y_3)\) and \(\mathrm{D}(x_4,y_4)\) respectively be the point internally dividing, and the point externally dividing, the line segment \(\mathrm{AB}\) in the ratio \(4:3\). A square matrix \(X\) of order \(2\) always satisfies
\(X\begin{pmatrix}x_1&y_1\\x_2&y_2\end{pmatrix} =\begin{pmatrix}x_3&y_3\\x_4&y_4\end{pmatrix}\).
Which option is equal to \(X\)?
  1. \(\begin{pmatrix}\dfrac{4}{7}&\dfrac{3}{7}\\-3&4\end{pmatrix}\)
  2. \(\begin{pmatrix}\dfrac{4}{7}&\dfrac{3}{7}\\4&-3\end{pmatrix}\)
  3. \(\begin{pmatrix}\dfrac{3}{7}&-3\\\dfrac{4}{7}&4\end{pmatrix}\)
  4. \(\begin{pmatrix}\dfrac{3}{7}&\dfrac{4}{7}\\3&-4\end{pmatrix}\)
  5. \(\begin{pmatrix}\dfrac{3}{7}&\dfrac{4}{7}\\-3&4\end{pmatrix}\)
As the figure shows, there is a square pyramid where all side lengths are \(1\). For a point \(\mathrm{P}\) moving on the edge \(\mathrm{EC}\), let \(\angle\mathrm{BPD}=\theta\). What is the sum of the maximum value and the minimum value of \(\cos\theta\)?
  1. \(-\dfrac{1}{3}\)
  2. \(-\dfrac{\sqrt{3}}{6}\)
  3. \(0\)
  4. \(\dfrac{\sqrt{3}}{6}\)
  5. \(\dfrac{1}{3}\)

Mathematics·Studies (I)

What is the sum of the \(55\) numbers listed below?
\(1\)
\(2\quad4\)
\(3\quad6\quad9\)
\(4\quad8\quad12\quad16\)
\(5\quad10\quad15\quad20\quad25\)
\(6\quad12\quad18\quad24\quad30\quad36\)
\(7\quad14\quad21\quad28\quad35\quad42\quad49\)
\(8\quad16\quad24\quad32\quad40\quad48\quad56\quad64\)
\(9\quad18\quad27\quad36\quad45\quad54\quad63\quad72\quad81\)
\(10\quad20\quad30\quad40\quad50\quad60\quad70\quad80\quad90\quad100\)
  1. \(1755\)
  2. \(1705\)
  3. \(1655\)
  4. \(1605\)
  5. \(1555\)
On the \(xy\)-plane, consider the region that satisfies three inequalities
\(3x+4y-16<0\), \(\:3x-4y+10>0\:\) and \(\:y>0\)
at the same time. What is the number of points in this region for which the distance between the point and the three line segments forming the boundary of this region, are all positive integers?
  1. \(0\)
  2. \(1\)
  3. \(3\)
  4. \(5\)
  5. \(7\)
Two cars \(\mathrm{A}\) and \(\mathrm{B}\) start from the same position at the same time, and only moves forward on a straight road. The distance \(\mathrm{A}\) and \(\mathrm{B}\) travels for \(t\) seconds are given as two differentiable functions \(f(t)\) and \(g(t)\) respectively, that satisfy the following.
  1. \(f(20)=g(20)\)
  2. \(f'(t)<g'(t)\) for \(10\leq t\leq 30\)
When \(10\leq t\leq 30\), which option below is correct about the positions of \(\mathrm{A}\) and \(\mathrm{B}\)?
  1. \(\mathrm{B}\) is always ahead of \(\mathrm{A}\).
  2. \(\mathrm{B}\) is always ahead of \(\mathrm{A}\).
  3. \(\mathrm{B}\) overtakes \(\mathrm{A}\) once.
  4. \(\mathrm{A}\) overtakes \(\mathrm{B}\) once.
  5. \(\mathrm{A}\) overtakes \(\mathrm{B}\), and then \(\mathrm{B}\) overtakes \(\mathrm{A}\) again.

Mathematics·Studies (I)

Suppose we build a rectangular stadium with a
perimeter of \(800\) meters on a circular ground with a diameter of \(300\) meters. For the stadium with the smallest possible area, what is the difference of its length and width in meters?
  1. \(100\sqrt{3}\)
  2. \(100\sqrt{2}\)
  3. \(50\sqrt{2}\)
  4. \(50\sqrt{3}\)
  5. \(100\)
In \(1993\), the education budget of Korea is \(3.7\%\) of the GNP. Suppose the growth rate of the GNP of Korea is \(7\%\) every year from \(1993\) to \(1998\). For the next \(5\) years, in what rate should the education budget be increased every year, so that in \(1998\) the education budget is \(5\%\) of the GNP?
Table of common logarithms (\(\log\,3.7=0.5682,\; \log\,5=0.6990,\; \log\,7=0.8451\))
  1. About \(10.7\%\)
  2. About \(11.7\%\)
  3. About \(12.7\%\)
  4. About \(13.7\%\)
  5. About \(14.7\%\)