1999 College Scholastic Ability Test

Mathematics·Studies (I)

Nat. Sciences
What is the value of \(\log_2 6-\log_2\dfrac{3}{2}\)? [2 points]
  1. \(0\)
  2. \(-1\)
  3. \(1\)
  4. \(-2\)
  5. \(2\)
If \(\sin x+\cos x=\sqrt{2}\), what is the value of \(\sin x\cos x\)? [2 points]
  1. \(1\)
  2. \(\sqrt{2}\)
  3. \(-\sqrt{2}\)
  4. \(\dfrac{1}{2}\)
  5. \(-\dfrac{1}{2}\)
What is the value of \(\displaystyle\lim_{x\to0}\frac{\ln(1+x)}{2x}\)? [2 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(\dfrac{1}{2}\)
  5. \(\dfrac{1}{3}\)
For \(z=\dfrac{-1+\sqrt{3}i}{2}\), let \(\mathrm{A, B}\) and \(\mathrm{C}\) be three points corresponding to three complex numbers \(1, z\) and \(z^2\) on the complex plane respectively. What is the value of \(\angle\mathrm{ABC}\)? [3 points]
  1. \(30°\)
  2. \(45°\)
  3. \(60°\)
  4. \(90°\)
  5. \(120°\)
Which option contains every pair of functions in the <List> that are exactly the same? [2 points]
  1. \(\begin{cases} y=\log(x-1)(x-2)\\[4pt] y=\log(x-1)+\log(x-2) \end{cases}\)
  2. \(\begin{aligned} \\ \Bigg\{ \end{aligned}\) \(\begin{array}{l} y=\dfrac{x^2-1}{x-1}\\[4pt] y=x+1 \end{array}\)
  3. \(\begin{cases} y=x\\[4pt] y=\sqrt[3]{x^3} \end{cases}\)
  1. a
  2. b
  3. c
  4. b, c
  5. a, c

Mathematics·Studies (I)

Nat. Sciences
The inverse of the fuction \(f(x)=\dfrac{x-1}{x-2}\) is \(f^{-1}(x)=\dfrac{ax+b}{x+c}\). What is the sum \(a+b+c\) of the constants \(a,b\) and \(c\)? [2 points]
  1. \(-1\)
  2. \(0\)
  3. \(1\)
  4. \(2\)
  5. \(3\)
Which option contains every sequence \(\{a_n\}\) in the <List> for which the limit
\(\displaystyle\lim_{n\to\infty}\frac{a_1+a_2+\cdots+a_n}{n}\)
exists? [3 points]
  1. \(a_n=n\)
  2. \(a_n=\dfrac{1}{2^n}\)
  3. \(a_n=(-1)^n\)
  1. a
  2. b
  3. c
  4. b, c
  5. a, b, c
For a positive integer \(n\), let the set \(A_n\) be
\(A_n\!=\{x\,|\,x\) is a positive integer coprime with \(n\}\).
Which option contains every correct statement in the <List>? [3 points]
  1. \(A_2=A_4\)
  2. \(A_3=A_6\)
  3. \(A_6=A_3\cap A_4\)
  1. a
  2. b
  3. c
  4. a, c
  5. a, b, c
What is the image of the function
\(f(x)=[x]+[-x]\)
defined for all real numbers \(x\)? (※ \([x]\) is the greatest integer less than or equal to \(x\).) [3 points]
  1. \(\{0,-1\}\)
  2. \(\{1,-1\}\)
  3. \(\{0,1\}\)
  4. \(\{0,1,-1\}\)
  5. \(\{0\}\)

Mathematics·Studies (I)

Nat. Sciences
Let us depict the union and the intersection of two sets \(A\) and \(B\) as the figure below.
In the figure below, what is appropriate for \((\alpha)\)? [3 points]
  1. \(\{1,2,3,4\}\)
  2. \(\{1,2,3,5\}\)
  3. \(\{2,3,5\}\)
  4. \(\{1,3,5\}\)
  5. \(\{3,5\}\)
Which definite integral below has a value equal to \(\displaystyle\int_a^b\frac{1}{x}dx\)? (※ \(0<a<b\)) [3 points]
  1. \(\displaystyle\int_{a+1}^{b+1}\frac{1}{x}dx\)
  2. \(\displaystyle\int_{2a}^{2b}\frac{1}{x}dx\)
  3. \(\displaystyle\int_{a^2}^{b^2}\frac{1}{x}dx\)
  4. \(\displaystyle\int_{\sqrt{a}}^{\sqrt{b}}\frac{1}{x}dx\)
  5. \(\displaystyle\int_{\frac{1}{a}}^{\frac{1}{b}}\frac{1}{x}dx\)
Let us take \(1\) ball out from a box containing \(2\) white balls and \(2\) black balls. Suppose we throw a coin \(3\) times if the ball taken out is white, and throw a coin \(4\) times if the ball taken out is black. What is the probability that the number of times the coin lands on heads is \(3\)? (※ The coin lands on heads or tails with the same probability.) [3 points]
  1. \(\dfrac{3}{16}\)
  2. \(\dfrac{5}{16}\)
  3. \(\dfrac{7}{16}\)
  4. \(\dfrac{9}{16}\)
  5. \(\dfrac{11}{16}\)

Mathematics·Studies (I)

Nat. Sciences
For two real numbers \(x\) and \(y\), let us use \(x*y\) to denote
\(x*y=\begin{cases} x&(x\geq y)\\ y&(x\leq y). \end{cases}\)
For example, \(2*1=2\). A set \(A=\{a,b,c,d\}\) of \(4\) distinct real numbers satisfies the following.
  1. \(x*a=x\) for all elements \(x\) in \(A\).
  2. \(c*d<c*b\)
Which option below is correct? [3 points]
  1. \(b<c<a\)
  2. \(b<d<a\)
  3. \(d<b<c\)
  4. \(a<b<c\)
  5. \(a<c<b\)
On the \(xy\)-plane, a point \(\mathrm{P}(x,y)\) moves or does not move according to the following rules. \(\mathrm{P}\) started from point \(\mathrm{A}(6,5)\), and stopped moving after reaching point \(\mathrm{B}\). What is the number of times it moves from \(\mathrm{A}\) until it reaches \(\mathrm{B}\)? [2 points]
  1. If \(y=2x\), do not move.
  2. If \(y<2x\), move \(-1\) units horizontally.
  3. If \(y>2x\), move \(-1\) units vertically.
  1. \(4\)
  2. \(5\)
  3. \(6\)
  4. \(7\)
  5. \(8\)
The following is a proof of the statement ‘If \(n\) is not divisible by any primes that are \(\sqrt{n}\) or less, then \(n\) is prime.’ where \(n\) is an integer greater than \(1\).
<Proof>
Let us assume the conclusion is false and \(n\) is not prime. Then there are integers \(l\) and \(m\) greater than \(1\) such that \(n=lm\). Let \(p\) be a prime factor of \(l\), and \(q\) be a prime factor of \(m\). Then \(pq\) is a divisor of \(lm\), so \(pq\leq n\). If \(p>\sqrt{n}\) and \(q>\sqrt{n}\), then \(pq>\sqrt{n}\sqrt{n}=n\) which is a contradiction. Therefore it should be that \(\fbox{\(\qquad\quad(\alpha)\quad\qquad\)}\).
This means there is a prime factor of \(n\) that is \(\sqrt{n}\) or less. However, this contradicts the premise. Therefore \(n\) is prime.
In the proof above, what is appropriate for \((\alpha)\)? [2 points]
  1. \(p\leq\sqrt{n}\) or \(q\leq\sqrt{n}\)
  2. \(p\leq\sqrt{n}\) and \(q\leq\sqrt{n}\)
  3. \(p\leq\sqrt{n}\) or \(q\geq\sqrt{n}\)
  4. \(p\leq\sqrt{n}\) and \(q\geq\sqrt{n}\)
  5. \(p\geq\sqrt{n}\) or \(q\geq\sqrt{n}\)
What is the equation of the tangent line to the circle \(x^2+y^2=5\) at point \((1,2)\)? [2 points]
  1. \(x+y=3\)
  2. \(2x-y=0\)
  3. \(x-2y=-3\)
  4. \(2x+y=4\)
  5. \(x+2y=5\)

Mathematics·Studies (I)

Nat. Sciences
There is a square with side lengths of \(1\). Let us divide the square into four rectangles with two arbitrary lines that are perpendicular. Let the area of each region be \(\mathrm{A,B,C}\) and \(\mathrm{D}\) respectively as the figure shows. Which option contains every statement in the <List> that is always true? [3 points]
  1. If \(\mathrm{A}>\dfrac{1}{4}\), then \(\:\mathrm{C}<\dfrac{1}{4}\).
  2. If \(\mathrm{A}<\dfrac{1}{4}\), then \(\:\mathrm{D}>\dfrac{1}{4}\).
  3. If \(\mathrm{A}>\dfrac{1}{4}\), then \(\:\mathrm{D}<\dfrac{1}{4}\).
  1. a
  2. b
  3. c
  4. a, c
  5. b, c
For all positive numbers \(x\), let \(f(x)\) be the number of primes no larger than \(x\). For example, \(f\!\left(\!\dfrac{5}{2}\!\right)=1\) and \(f(5)=3\). Which option contains every correct statement in the <List>? [3 points]
  1. \(f(10)=4\)
  2. \(f(x)<x\) for all positive numbers \(x\).
  3. \(f(x+1)=f(x)\) for all positive numbers \(x\).
  1. a
  2. a, b
  3. a, c
  4. b, c
  5. a, b, c
For a \(3\)-dimensional vector \(\overrightarrow{\mathrm{OP}}=(1,-1,1)\), let its projection onto the \(xy\)-plane, the \(yz\)-plane, and the \(zx\)-plane be \(\overrightarrow{\mathrm{OA}}\), \(\overrightarrow{\mathrm{OB}}\), and \(\overrightarrow{\mathrm{OC}}\) respectively. Given that \(\overrightarrow{\mathrm{OP}}=a\,\overrightarrow{\mathrm{OA}} +b\,\overrightarrow{\mathrm{OB}}+c\,\overrightarrow{\mathrm{OC}}\) for three real numbers \(a, b\) and \(c\), what is the sum \(a+b+c\)? [3 points]
  1. \(-\dfrac{3}{2}\)
  2. \(-1\)
  3. \(0\)
  4. \(1\)
  5. \(\dfrac{3}{2}\)
On the \(xy\)-plane, suppose a point \((x,y)\) is moving in the region satisfying the inequality \(-x\leq y\leq 2-x^2\). What is the maximum value of \(x+y\)? [3 points]
  1. \(\dfrac{5}{4}\)
  2. \(\dfrac{7}{4}\)
  3. \(\dfrac{9}{4}\)
  4. \(\dfrac{11}{4}\)
  5. \(\dfrac{13}{4}\)

Mathematics·Studies (I)

Nat. Sciences
There is a circle whose diameter \(\mathrm{AB}\) has a length of \(10\). Two points \(\mathrm{P}\) and \(\mathrm{Q}\) on the circle satisfy \(\overline{\mathrm{AP}}=8\) and \(\angle\mathrm{QAB}=2\angle\mathrm{PAB}\). What is the length of the line segment \(\overline{\mathrm{AQ}}\)? [3 points]
  1. \(\dfrac{10}{5}\)
  2. \(\dfrac{11}{5}\)
  3. \(\dfrac{12}{5}\)
  4. \(\dfrac{13}{5}\)
  5. \(\dfrac{14}{5}\)
In the \(12\)th grade of some high school, the number of male students is \(1.5\) times the number of female students. According to the statistics of a mock test of College Scholastic Ability Test, out of \(400\) points, male students scored \(225\) points on average, and female students scored \(235\) points on average. What is the average score of all \(12\)th grade students? [2 points]
  1. \(229\)
  2. \(230\)
  3. \(231\)
  4. \(232\)
  5. \(233\)
In the \(12\)-tone system in Western music, the frequencies of notes form a geometric progression. As the note goes up every semitone, the frequency is multiplied by a certain ratio, so that the frequency is doubled after going up \(12\) semitones. In the piano keys below, what is the ratio of integers closest to the ratio of frequencies \(a_1:a_5:a_8\) of the notes Do, Mi and Sol?
(※ Approximate \(2^{^1/_3}=\dfrac{5}{4}, 2^{^5/_{12}}=\dfrac{4}{3}\) and \(2^{^7/_{12}}=\dfrac{3}{2}\).) [3 points]
  1. \(2:3:4\)
  2. \(3:4:5\)
  3. \(4:5:6\)
  4. \(5:6:8\)
  5. \(6:8:9\)
Suppose cars with speed \(v\,(\text{m/s})\) are driving on a highway without changing lanes. Considering the braking distance, the minimum distance between cars has to be
\(f(v)=\dfrac{1}{20}v^2+\dfrac{1}{2}v+5\,(\text{m})\).
During \(60\) seconds, what is the maximum number of cars that can pass through one point on a lane?
(※ Ignore the length of cars.) [3 points]
  1. \(16\)
  2. \(40\)
  3. \(60\)
  4. \(90\)
  5. \(225\)

Mathematics·Studies (I)

Nat. Sciences
Short Answers (25~30)
In \(3\)-dimensional space, compute the radius of a sphere with center \((1,1,1)\) that is tangent to the plane \(x+2y-2z=31\). [3 points]
The two inequalities \(\dfrac{1}{x-3}\leq\dfrac{1}{x-2}\) and \(x^2-ax+b<0\) have identical solutions for some real numbers \(a\) and \(b\). Compute \(a+b\). [2 points]
There is a sphere with a radius of \(30\). Let us fix one end of a string with a length of \(5\pi\) at a point \(\mathrm{N}\) on the sphere. Let us revolve the other end of the string once around the sphere while pulling the string tight along the surface of the sphere. Let \(l\) be the length of the locus of the end of the string on the surface of the sphere. Compute \(\dfrac{l}{\pi}\). [3 points]

Mathematics·Studies (I)

Nat. Sciences
Compute the positive integer \(n\) that satisfies all of the following. [3 points]
  1. \(n\) is a divisor of \(60\).
  2. \(n\) is the sum of two positive integers with a ratio of \(3:7\).
  3. \(n\) has \(6\) divisors.
On the \(xy\)-plane, the rotation \(f\) and reflection \(g\) are represented by matrices \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) and \(\begin{pmatrix}-1&0\\0&1\end{pmatrix}\) respectively. What is the number of points that point \(\mathrm{P}\left(\!\dfrac{\sqrt{3}}{2},\dfrac{1}{2}\!\right)\) can be mapped to, including point \(\mathrm{P}\) itself, by a linear map formed by a composition of finite numbers of maps \(f\) and \(g\)? [3 points]
Suppose a deposit product of a bank is presented with its annual interest rate. For a compound deposit that calculates interest \(n\) times a year, the rate of interest calculated each time is \(\dfrac{\text{Annual interest rate}}{n}\), and the effective rate of return is defined as
\(\dfrac{\text{Total interest after }1\text{ year}}{\text{Principal}}\times 100(\%)\).
For a compound deposit product with an annual interest rate of \(10\%\) that calculates interest every \(6\) months, compute its effective rate of return\((\%)\) to the second place after the decimal. [3 points]