2011 College Scholastic Ability Test

Mathematics (Type Na)

What is the value of \(4^{\frac{3}{2}}\times\log_3\sqrt{3}\)? [2 points]
  1. \(5\)
  2. \(4\)
  3. \(3\)
  4. \(2\)
  5. \(1\)
For matrices \(A=\begin{pmatrix*}[r] 1 & -1 \\ 1 & 1 \end{pmatrix*}\) and \(B=\begin{pmatrix*}[r] 1 & 1 \\ -1 & 1 \end{pmatrix*}\), what is the sum of all of the elements in the matrix \(A(A+B)\)? [2 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
Given that \(\displaystyle\lim_{n\to\infty}\!\dfrac{a\times6^{n+1}-5^n}{6^n+5^n}=4\), what is the value of the constant \(a\)?
  1. \(\dfrac{1}{3}\)
  2. \(\dfrac{1}{2}\)
  3. \(\dfrac{2}{3}\)
  4. \(\dfrac{4}{3}\)
  5. \(\dfrac{3}{2}\)
What is the sum of all positive integers \(x\) for which \((3^x-5)(3^x-100)<0\) is satisfied? [2 points]
  1. \(5\)
  2. \(7\)
  3. \(9\)
  4. \(11\)
  5. \(13\)
Consider two events \(A\) and \(B\) that are independent. Given that
\({\mathrm{P}(A) = \dfrac{2}{3}}\, \) and \(\, {\mathrm{P}(A \cap B) = \mathrm{P}(A)-\mathrm{P}(B)}\),
what is the value of \(\mathrm{P}(B)\)? [3 points]
  1. \({\dfrac{1}{10}}\)
  2. \({\dfrac{1}{5}}\)
  3. \({\dfrac{3}{10}}\)
  4. \({\dfrac{2}{5}}\)
  5. \({\dfrac{1}{2}}\)

Mathematics (Type Na)

At some event hall, there are \(5\) places where one banner can be installed. There are three types of banners \(\mathrm{A}\), \(\mathrm{B}\) and \(\mathrm{C}\), and we have \(1\) banner of type \(\mathrm{A}\), \(4\) of type \(\mathrm{B}\), and \(2\) of type \(\mathrm{C}\). If we install \(5\) of our banners in the \(5\) places such that the following is satisfied, how many cases are possible as the result? (※ Banners of the same type are not considered distinct.) [3 points]
  1. Type \(\mathrm{A}\) should be installed.
  2. Type \(\mathrm{B}\) is installed in at least \(2\) places.
  1. \(55\)
  2. \(65\)
  3. \(75\)
  4. \(85\)
  5. \(95\)
Cheolsu participates in a design contest. Participants receive scores in two categories, and \(3\) kinds of scores can be earned in each category as shown in the table. In each category, the probability that Cheolsu will receive score \(\mathrm{A}\), score \(\mathrm{B}\) and score \(\mathrm{C}\) are \(\dfrac{1}{2}\), \(\dfrac{1}{3}\) and \(\dfrac{1}{6}\) respectively. If the event of receiving the audience vote score and the event of receiving the judge panel score are independent, what is the probability that the \(2\) scores Cheolsu receive add up to \(70\)? [3 points]
Score
Category
 Score \(\mathrm{A}\) Score \(\mathrm{B}\) Score \(\mathrm{C}\)
Audience vote \(40\) \(30\) \(20\)
Judge panel \(50\) \(40\) \(30\)
  1. \(\dfrac{1}{3}\)
  2. \(\dfrac{11}{36}\)
  3. \(\dfrac{5}{18}\)
  4. \(\dfrac{1}{4}\)
  5. \(\dfrac{2}{9}\)
The probability distribution table of a random variable \(X\) is as follows.
\(X\) \(-1\) \(0\) \(1\) \(2\) Total
\(\mathrm{P}(X=x)\) \(\dfrac{3-a}{8}\) \(\dfrac{1}{8}\) \(\dfrac{3+a}{8}\) \(\dfrac{1}{8}\) \(1\)
Given that \(\mathrm{P}(0\leq X \leq 2)=\dfrac{7}{8}\), what is \(\mathrm{E}(X)\), the mean of the random variable \(X\)? [3 points]
  1. \(\dfrac{1}{4}\)
  2. \(\dfrac{3}{8}\)
  3. \(\dfrac{1}{2}\)
  4. \(\dfrac{5}{8}\)
  5. \(\dfrac{3}{4}\)
The relative density of the ground can be measured by inserting a probe into the ground. Let \(S\) be the effective vertical stress of the ground, and let \(R\) be the resistance to the probe while inserting to the ground. The relative density of the ground \(D(\%)\) can be calculated as follows.
\(D=-98+66\log\dfrac{R}{\sqrt{S}}\)
(※ The units of \(S\) and \(R\) are \(\text{metric ton/m}^2\).)
The effective vertical stress of ground \(\mathrm{A}\) is \(2\) times the effective vertical stress of ground \(\mathrm{B}\), and the resistance to the probe while inserting to ground \(\mathrm{A}\) is \(1.5\) times the resistance to the probe while inserting to ground \(\mathrm{B}\). If the relative density of ground \(\mathrm{B}\) is \(65(\%)\), what is the relative density of ground \(\mathrm{A}\)?
(※ Suppose \(\log2=0.3\).) [3 points]
  1. \(81.5\)
  2. \(78.2\)
  3. \(74.9\)
  4. \(71.6\)
  5. \(68.3\)

Mathematics (Type Na)

There is a rectangle \(\mathrm{A_1B_1C_1D_1}\) where \(\overline{\mathrm{A_1B_1}}=1\) and \(\overline{\mathrm{B_1C_1}}=2\). As the figure shows, let \(\mathrm{M_1}\) be the midpoint of the line segment \(\mathrm{B_1C_1}\), and let \(\mathrm{B_2}\) and \(\mathrm{C_2}\) be two points on the line segment \(\mathrm{A_1D_1}\) such that \(\angle\mathrm{A_1M_1B_2}=\angle\mathrm{C_2M_1D_1}=15°\) and \(\angle\mathrm{B_2M_1C_2}=60°\). Let \(S_1\) be the sum of the area of triangles \(\mathrm{A_1M_1B_2}\) and \(\mathrm{C_2M_1D_1}\).
As the figure shows, Let \(\mathrm{A_2}\) and \(\mathrm{D_2}\) be points such that the quadrilateral \(\mathrm{A_2B_2C_2D_2}\) is a rectangle where \(\overline{\mathrm{B_2C_2}}=2\overline{\mathrm{A_2B_2}}\). Let \(\mathrm{M_2}\) be the midpoint of the line segment \(\mathrm{B_2C_2}\), and let \(\mathrm{B_3}\) and \(\mathrm{C_3}\) be two points on the line segment \(\mathrm{A_2D_2}\) such that \(\angle\mathrm{A_2M_2B_3}=\angle\mathrm{C_3M_2D_2}=15°\) and \(\angle\mathrm{B_3M_2C_3}=60°\). Let \(S_2\) be the sum of the area of triangles \(\mathrm{A_2M_2B_3}\) and \(\mathrm{C_3M_2D_2}\).
For all \(S_n\) obtained by repeating this process, what is the value of \(\displaystyle\sum_{n=1}^\infty S_n\)? [4 points]
  1. \(\dfrac{2+\sqrt{3}}{6}\)
  2. \(\dfrac{3-\sqrt{3}}{2}\)
  3. \(\dfrac{4+\sqrt{3}}{9}\)
  4. \(\dfrac{5-\sqrt{3}}{5}\)
  5. \(\dfrac{7-\sqrt{3}}{8}\)
On the \(xy\)-plane, there is a graph obtained by reflecting the graph of the function \(y=a^x\) about the
\(y\)-axis, and then translating that graph \(3\) units horizontally and \(2\) units vertically. Given that this graph passes through the point \((1,4)\), what is the value of the positive number \(a\)?
  1. \(\sqrt{2}\)
  2. \(2\)
  3. \(2\sqrt{2}\)
  4. \(4\)
  5. \(4\sqrt{2}\)
A set \(S\) whose elements are \(1\times2\) matrices and a set \(T\) whose elements are \(2\times1\) matrices are as follows.
\(S=\{(a \;\: b)|a+b\ne 0\}\), \(T=\left\{\!\begin{pmatrix} p \\ q \end{pmatrix} \middle| \:pq\ne 0\right\}\)
For an element \(A\) in \(S\), Which option only contains every correct statement in the <List>? [4 points]
  1. For all elements \(P\) in set \(T\), \(PA\) does not have an inverse matrix.
  2. For an element \(B\) in set \(S\) and an element \(P\) in set \(T\), if \(PA=PB\,\) then \(A=B\).
  3. There is an element \(P\) in set \(T\) that satisfies \(PA\begin{pmatrix}1\\1\end{pmatrix}=\begin{pmatrix}1\\1\end{pmatrix}\).
  1. a
  2. c
  3. a, b
  4. b, c
  5. a, b, c

Mathematics (Type Na)

For customers of some traditional market, the distance from home to market follows a normal distribution with a mean of \(1740\text{m}\) and a standard deviation of \(500\text{m}\). \(15\%\) of customers whose distance from home to market is not less than \(2000\text{m}\), and \(5\%\) of customers whose distance from home to market is less than \(2000\text{m}\), drives their car to the market. If we randomly choose a customer who drove their car to the market, what is the probability that the distance from home to market of this customer is less than \(2000\text{m}\)?
(※ For a random variable \(Z\) that follows the standard normal distribution, suppose \(\mathrm{P}(0 \leq Z \leq 0.52) = 0.2\).) [4 points]
  1. \(\dfrac{3}{8}\)
  2. \(\dfrac{7}{16}\)
  3. \(\dfrac{1}{2}\)
  4. \(\dfrac{9}{16}\)
  5. \(\dfrac{5}{8}\)
On the \(xy\)-plane, let \(\mathrm{A}_n\) be the point where the line \(y=\dfrac{1}{n}x\) meets the line \(x=n\), and let \(\mathrm{B}_n\) be the point where the line \(x=n\) meets the \(x\)-axis. Let \(\mathrm{C}_n\) be the center of the inscribed circle of the triangle \(\mathrm{A}_n\mathrm{OB}_n\), and let \(S_n\) be the area of the triangle \(\mathrm{A}_n\mathrm{OC}_n\). What is the value of \(\displaystyle\lim_{n\to\infty}\dfrac{S_n}{n}\)? [4 points]
  1. \(\dfrac{1}{12}\)
  2. \(\dfrac{1}{6}\)
  3. \(\dfrac{1}{4}\)
  4. \(\dfrac{1}{3}\)
  5. \(\dfrac{5}{12}\)
A sequence \(\{a_n\}\) satisfies \(a_1=1\) and
\(a_{n+1}=n+1+\dfrac{(n-1)!}{a_1a_2\cdots a_n} \;(n\geq 1)\).
The following is a part of the process of computing the general term \(a_n\).
The following is true for all positive integers \(n\).
\(a_1a_2\cdots a_n a_{n+1}\)
\(= a_1a_2\cdots a_n\times(n+1)+(n-1)!\)
Let \(b_n=\dfrac{a_1a_2\cdots a_n}{n!}\). Then, \(b_1=1\) and
\(b_{n+1}=b_n + \fbox{\(\;(\alpha)\;\)}\).
The general term of the sequence \(\{b_n\}\) is
\(b_n=\fbox{\(\;(\beta)\;\)}\), therefore \(\dfrac{a_1a_2\cdots a_n}{n!}=\fbox{\(\;(\beta)\;\)}\).
\(\vdots\)
Therefore \(a_1=1\), and \(a_n=\dfrac{(n-1)(2n-1)}{2n-3} \: (n\geq 2)\).
Let \(f(n)\) be the correct expression for \((\alpha)\), and \(g(n)\) be the correct expression for \((\beta)\). What is the value of \(f(13)\times g(7)\)? [4 points]
  1. \(\dfrac{1}{70}\)
  2. \(\dfrac{1}{77}\)
  3. \(\dfrac{1}{84}\)
  4. \(\dfrac{1}{91}\)
  5. \(\dfrac{1}{98}\)

Mathematics (Type Na)

On the \(xy\)-plane, let \(\mathrm{P}(x_1,y_1)\) and \(\mathrm{Q(x_2,y_2)}\) be points where two curves \(y=|\log_2 x|\) and \(y=\!\left(\!\dfrac{1}{2}\!\right)^{\!\!x}\) meet \((x_1<x_2)\). Let \(\mathrm{R}(x_3,y_3)\) be the point where two curves \(y=|\log_2 x|\) and \(y=2^x\) meet. Which option only contains every correct statement in the <List>? [4 points]
  1. \(\dfrac{1}{2}<x<1\)
  2. \(x_2y_2-x_3y_3=0\)
  3. \(x_2(x_1-1)>y_1(y_2-1)\)
  1. a
  2. c
  3. a, b
  4. b, c
  5. a, b, c
There are \(2\) Korean, Chinese, and Japanese students each. Suppose these \(6\) students randomly select and sit on each of the \(6\) seats marked with numbers as the figure below. What is the probability that each pair of students from the same country sit on seats marked with numbers with a difference of \(1\) or \(10\)? [4 points]
\(11\) \(12\) \(13\)
\(21\) \(22\) \(23\)
  1. \(\dfrac{1}{20}\)
  2. \(\dfrac{1}{10}\)
  3. \(\dfrac{3}{20}\)
  4. \(\dfrac{1}{5}\)
  5. \(\dfrac{1}{4}\)
Short Answer Questions
Compute the value of the positive integer \(n\) that satisfies the equation \(2\times_n\!\mathrm{C}_3=3\times_n\!\mathrm{P}_2\). [3 points]
Let \(\alpha\) be the solution to the equation \(\log_3(x-4)=\log_9(5x+4)\). compute \(\alpha\). [3 points]

Mathematics (Type Na)

Suppose \(6\) distinct balls are put into baskets \(\mathrm{A}\) and \(\mathrm{B}\), \(3\) balls each. Compute the number of cases possible as the result. [3 points]
Suppose the trial of throwing \(2\) coins at the same time is repeated \(10\) times. Let a random variable \(X\) be the number of times both coins land on heads. Compute \(\mathrm{V}(4X+1)\), the variance of the random variable \(4X+1\). [3 points]
Consider an arithmetic progression \(\{a_n\}\) whose common differece is not \(0\), and whose three terms \(a_2, a_4\) and \(a_9\), in this order, form a geometric progression with a common ratio of \(r\). Compute \(6r\). [4 points]
For integers \(n\geq2\), let \(S\) be the set of all products of two distinct elements in the set
\(\big\{3^{2k-1} \,\big|\, k\) is an integer, \(1\leq k\leq n\big\}\).
Let \(f(n)\) be the number of elements in \(S\). For example, \(f(4)=5\). Compute \(\displaystyle\sum_{n=2}^{11} f(n)\). [4 points]
For a positive integer \(A\), let \(n\) and \(\alpha\) be the characteristic and mantissa of \(\log_{10}A\) respectively. Compute the number of integers \(A\) for which \(n\leq2\alpha\). (※ \(3.1<\sqrt{10}<3.2\)) [4 points]

Mathematics (Type Na)

For a positive integer \(m\), consider cube-shaped blocks of the same size that are stacked \(1\) block high in column \(1\), \(2\) blocks high in column \(2\), \(3\) blocks high in column \(3\), \(\cdots\), \(m\) blocks high in column \(m\). Let us perform the following trial repeatedly until there are no columns with an even number of blocks.
For each column with an even number of blocks, remove \(\dfrac{1}{2}\) of the blocks in that column.
After all trials are done, let \(f(m)\) be the sum of the number of blocks from column \(1\) to column \(m\). For example, \(f(2)=2, f(3)=5,\) and \(f(4)=6\). Given that
\(\displaystyle\lim_{n\to\infty}\dfrac{f(2^{n+1})-f(2^n)}{f(2^{n+2})}=\dfrac{q}{p}\),
compute \(p+q\). (※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]
Multiple Choice Questions
Consider a sequence \(\{a_n\}\) that satisfies
\(2a_{n+1}=a_n+a_{n+2}\)
for all positive integers \(n\). Given that \(a_2=-1\) and \(a_3=2\), what is the sum of the first \(10\) terms of the sequence \(\{a_n\}\)? [3 points]
  1. \(95\)
  2. \(90\)
  3. \(85\)
  4. \(80\)
  5. \(75\)
The time spent on \(1\) use of public bicycles in some city follows a normal distribution with a mean of \(60\) minutes and a standard deviation of \(10\) minutes. Suppose we examine \(25\) uses of public bicycles that are randomly sampled.
What is the probability that the time spent on those \(25\) uses add up to \(1450\) minutes or more, computed using the standard normal table to the right? [3 points]
\(z\)\(\mathrm{P}(0\!\leq\! Z \!\leq\!z)\)
\(1.0\)\(0.3413\)
\(1.5\)\(0.4332\)
\(2.0\)\(0.4772\)
\(2.5\)\(0.4938\)
  1. \(0.8351\)
  2. \(0.8413\)
  3. \(0.9332\)
  4. \(0.9772\)
  5. \(0.9938\)

Mathematics (Type Na)

A company assesses applicants based on scores converted from the raw scores of the inferential abilty test and the spatial intelligence test. Given that the raw score of the inferential ability test is \(x\) and the raw score of the spatial intelligence test is \(y\), the two converted scores \(p\) and \(q\) are as follows.
\(\begin{pmatrix}p\\q\end{pmatrix}=\begin{pmatrix}3&2\\2&3\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}\)
Suppose the converted scores of applicants \(\mathrm{A, B}\) and \(\mathrm{C}\) is as shown in the table. Let \(a, b\) and \(c\) be the raw scores of the inferential ability test for applicants \(\mathrm{A, B}\) and \(\mathrm{C}\) respectively. Which option correctly shows the relation between \(a, b\) and \(c\)? [4 points]
Applicants
Converted score
 \(\mathrm{A}\) \(\mathrm{B}\) \(\mathrm{C}\)
\(p\) \(45\) \(50\) \(45\)
\(q\) \(40\) \(50\) \(50\)
  1. \(a>b>c\)
  2. \(a>c>b\)
  3. \(b>a>c\)
  4. \(b>c>a\)
  5. \(c>b>a\)
Consider a square matrix \(A\) of order \(2\) whose \((i,j)\) entry, \(a_{ij}\), is equal to
\(a_{ij}=i-j\;(i=1,2,\,j=1,2)\).
What is the \((2,1)\) entry of the matrix \(A+A^2+A^3+\cdots+A^{2010}\)? [4 points]
  1. \(-2010\)
  2. \(-1\)
  3. \(0\)
  4. \(1\)
  5. \(2010\)
Short Answer Questions
A sequence \(\{a_n\}\) satisfies
\(\displaystyle\sum_{k=1}^n a_k=\log_{10}\dfrac{(n+1)(n+2)}{2}\)
for all positive integers \(n\). Given that \(\displaystyle\sum_{k=1}^{20}a_{2k}=p\), compute \(10^p\). [4 points]