1999 College Scholastic Ability Test

Mathematics·Studies (I)

Arts & Phys. Ed.
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. \(-\dfrac{1}{2}\)
For two functions \(f(x)=2x+1\) and \(g(x)=3x^2-1\), what is the value of \(g(f(0))\)? [2 points]
  1. \(-1\)
  2. \(0\)
  3. \(1\)
  4. \(2\)
  5. \(3\)
What is the number of integers \(x\) that satisfy the following system of inequalities? [3 points]
\(\begin{cases} 2x<x+4\\\\ x^2-4x-5<0 \end{cases}\)
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
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)

Arts & Phys. Ed.
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. \(3\)
If \(2^a=c\) and \(2^b=d\), which option below is equal to \(\left(\!\dfrac{1}{2}\!\right)^{\!2a+b}\)? [3 points]
  1. \(\dfrac{1}{cd}\)
  2. \(\dfrac{1}{2cd}\)
  3. \(\dfrac{1}{c^2d}\)
  4. \(-cd\)
  5. \(-2cd\)
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)

Arts & Phys. Ed.
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. \(\{3,5\}\)
Figure shows a rectangular cuboid with \(\overline{\mathrm{AB}}=2, \overline{\mathrm{BC}}=1\) and \(\overline{\mathrm{BE}}=1\). What is the area of the triangle \(\mathrm{AEC}\)? [3 points]

  1. \(1\)
  2. \(\sqrt{2}\)
  3. \(\dfrac{3}{2}\)
  4. \(\dfrac{3}{2}\sqrt{2}\)
  5. \(2\)
On the \(xy\)-plane, what is the distance between the two points where the circle \((x-1)^2+(y-1)^2=10\) meets the \(y\)-axis? [3 points]
  1. \(2\)
  2. \(4\)
  3. \(6\)
  4. \(8\)
  5. \(10\)

Mathematics·Studies (I)

Arts & Phys. Ed.
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)

Arts & Phys. Ed.
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 tetrahedron \(\mathrm{ABCD}\), let \(\mathrm{P,Q,R}\) and \(\mathrm{S}\) be the midpoint of edges \(\mathrm{BC,CD,DB}\) and \(\mathrm{AD}\) respectively. What is the ratio of the volume of tetrahedrons \(\mathrm{APQR}\) and \(\mathrm{SQDR}\)? [3 points]
  1. \(1:1\)
  2. \(2:1\)
  3. \(3:1\)
  4. \(3:2\)
  5. \(4:1\)
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)

Arts & Phys. Ed.
As the figure shows, there is a point \(\mathrm{A}\) on a circle with radius \(1\), and triangles inscribed in the circle which have point \(\mathrm{A}\) as a vertex with the angle at \(\mathrm{A}\) being \(30°\).
Let us draw as many such triangles as possible without having an overlap. Let \(\mathrm{P}_1, \mathrm{P}_2, \cdots, \mathrm{P}_n, \mathrm{P}_{n+1}\) be the vertices of the triangles except vertex \(\mathrm{A}\), in counter-clockwise order. What is the sum of the lengths of line segments \(\overline{\mathrm{P_1 P_2}},\overline{\mathrm{P_2 P_3}},\cdots,\overline{\mathrm{P_n P_{n+1}}}\)? [3 points]
  1. \(5\)
  2. \(5\sqrt{3}\)
  3. \(\dfrac{5}{2}\sqrt{3}\)
  4. \(4\)
  5. \(4\sqrt{3}\)
Rain intensity is an indicator of how concentrated rainfall is. In some city, the rain intensity \(I\) is a function of the duration \(T\) as follows.
\(I=\dfrac{1}{60}\left(\dfrac{T+6571}{T+41}-1\right)\)
Which option below appropriately depicts the relation between \(I\) and \(T\)? (※ \(T>0\,\)) [2 points]
There are two points \(\mathrm{A}\) and \(\mathrm{B}\) on earth, and two satellites are positioned in points \(\mathrm{A'}\) and \(\mathrm{B'}\) above the two points on earth respectively, with an altitude of \(3600\text{km}\). Given that \(\angle\mathrm{BB'A'}=60°\), what is the distance between the two satellites in \(\text{km}\)?
(※ Suppose the earth is a sphere with a radius of \(6400\text{km}\).) [3 points]
  1. \(5000\)
  2. \(5000\sqrt{2}\)
  3. \(5000\sqrt{3}\)
  4. \(6400\)
  5. \(10000\)
If a radio wave pass through some wall and its strength changes from \(\mathrm{A}\) to \(\mathrm{B}\), then the damping ratio \(\mathrm{F}\) of that wall is defined as
\(\mathrm{F}=10\log\left(\!\dfrac{\mathrm{B}}{\mathrm{A}}\!\right)\) (decibels).
If a radio wave passes through a wall with a damping ratio of \(-7\) (decibels), what is the strength of the radio wave after it passes through the wall, divided by its strength before it passes through the wall?
(※ Suppose \(10^{^3/_{10}}=2\).) [3 points]
  1. \(\dfrac{1}{10}\)
  2. \(\dfrac{1}{5}\)
  3. \(\dfrac{3}{10}\)
  4. \(\dfrac{1}{2}\)
  5. \(\dfrac{7}{10}\)

Mathematics·Studies (I)

Arts & Phys. Ed.
Short Answers (25~30)
Given that the equation \(x^2-ax+b=0\) has a solution \(1+2i\), compute the product \(ab\) of the real numbers \(a\) and \(b\). [3 points]
The following is the process of dividing a cubic equation on \(x\) with a linear equation on \(x\). For five numbers \(a,b,c,d\) and \(e\), compute the sum \(a+b+c+d+e\). [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)

Arts & Phys. Ed.
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, compute the radius of a circle with center \((1,2)\) that is tangent to the line \(3x+4y=1\). [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]