2014 College Scholastic Ability Test

Mathematics (Type A)

Multiple Choice Questions
What is the value of \(8^{^2/_3}\times 9^{^1/_2}\)? [2 points]
  1. \(12\)
  2. \(10\)
  3. \(8\)
  4. \(6\)
  5. \(4\)
For two matrices \(A=\begin{pmatrix} 2&0 \\ 1&1 \end{pmatrix}\) and \(B=\begin{pmatrix} 1&1 \\ 0&1 \end{pmatrix}\), what is the sum of all elements in the matrix \(A+2B\)? [2 points]
  1. \(8\)
  2. \(9\)
  3. \(10\)
  4. \(11\)
  5. \(12\)
What is the value of \(\displaystyle\lim_{n\to\infty}\frac{2\times 3^{n+1}+5}{3^n}\)? [2 points]
  1. \(10\)
  2. \(9\)
  3. \(8\)
  4. \(7\)
  5. \(6\)
In the following graph and its adjacency matrix, what is the value of \(a+b+c+d+e\)? [3 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)

Mathematics (Type A)

If the function \(f(x)=2x^2+ax\) satisfies \(\displaystyle\lim_{h\to\,0}\dfrac{f(1+h)-f(1)}{h}=6\), what is the value of the constant \(a\)? [3 points]
  1. \(-4\)
  2. \(-2\)
  3. \(0\)
  4. \(2\)
  5. \(4\)
For an arithmetic progression \(\{a_n\}\) with an initial term of \(6\) and a common difference of \(d\), let \(S_n\) be the sum of its first \(n\) terms. Given that
\(\dfrac{a_8-a_6}{S_8-S_6}=2\),
what is the value of \(d\)? [3 points]
  1. \(-1\)
  2. \(-2\)
  3. \(-3\)
  4. \(-4\)
  5. \(-5\)
If two events \(A\) and \(B\) are independent, and \(\mathrm{P}(A)=\dfrac{1}{3}\) and \(\mathrm{P}(B)=\dfrac{1}{3}\), what is the value of \(\mathrm{P}(A\cap B^{\,C})\)? (※ \(B^{\,C}\) is the complement of \(B\)). [3 points]
  1. \(\dfrac{5}{27}\)
  2. \(\dfrac{2}{9}\)
  3. \(\dfrac{7}{27}\)
  4. \(\dfrac{8}{27}\)
  5. \(\dfrac{1}{3}\)

Mathematics (Type A)

What is the area of the region enclosed by the curve \(y=x^2-4x+3\) and the line \(y=3\)? [3 points]
  1. \(10\)
  2. \(\dfrac{31}{3}\)
  3. \(\dfrac{32}{3}\)
  4. \(11\)
  5. \(\dfrac{34}{3}\)
A random variable \(X\) following the binomial distribution \(\mathrm{B}(9,p)\) satisfies \(\{\mathrm{E}(X)\}^2=\mathrm{V}(X)\). What is the value of \(p\)? (※ \(0< p< 1\)) [3 points]
  1. \(\dfrac{1}{13}\)
  2. \(\dfrac{1}{12}\)
  3. \(\dfrac{1}{11}\)
  4. \(\dfrac{1}{10}\)
  5. \(\dfrac{1}{9}\)
Suppose water completely fills and flows in a cylindrical pipe, whose cross section has a radius of \(R\,(R<1)\). Let \(v_c\) be the speed of water at the center of a cross section of this pipe, and let \(v\) be the speed of water at a point \(x\,(0<x \leq R)\) units apart from the walls of the pipe towards the center. Then the following relation is known.
\(\dfrac{v_c}{v}=1-k\log_{10} \dfrac{x}{R}\)
(※ \(k\) is a positive constant. Then units of length and speed are \(\text{m}\) and \(\text{m/s}\) respectively.)
For this pipe satisfying \(R< 1\), the speed of water at a point \(R^{\small{\dfrac{27}{23}}}\) units apart from the walls towards the center is \(\dfrac{1}{2}\) times the speed of water at the center, and the speed of water at a point \(R^a\) units apart from the walls towards the center is \(\dfrac{1}{3}\) times the speed of water at the center. What is the value of \(a\)? [3 points]
  1. \(\dfrac{39}{23}\)
  2. \(\dfrac{37}{23}\)
  3. \(\dfrac{35}{23}\)
  4. \(\dfrac{33}{23}\)
  5. \(\dfrac{31}{23}\)

Mathematics (Type A)

Figure below is the graph of the function \(y=f(x)\).
What is the value of \(\displaystyle\lim_{x\to\,-1-0}f(x)+ \lim_{x\to+0}\!f(x)\)? [3 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
The volume of one medicine bottle produced by some pharmaceutical company follows a normal distribution with a mean of \(m\) and a standard deviation of \(10\). The probability that a random sample of \(25\) bottles from the bottles produced by
this company has a mean volume of \(2000\) or more, is \(0.9772\). What is the value of \(m\) computed using the standard normal table to the right?
(※ The unit of volume is \(\text{mL}\).) [3 points]
\(z\)\(\mathrm{P}(0\!\leq\! Z \!\leq\!z)\)
\(1.5\)\(0.4332\)
\(2.0\)\(0.4772\)
\(2.5\)\(0.4938\)
\(3.0\)\(0.4987\)
  1. \(2003\)
  2. \(2004\)
  3. \(2005\)
  4. \(2006\)
  5. \(2007\)

Mathematics (Type A)

[13~14]  For positive integers \(\boldsymbol{n}\), \(\boldsymbol{f(n)}\) is as follows.
\(\boldsymbol{f(n)=\begin{cases} \boldsymbol{\log_3 n} &\; \boldsymbol{(n \textbf{ is odd})} \\ \boldsymbol{\log_2 n} &\; \boldsymbol{(n \textbf{ is even})} \end{cases}}\)
Answer the questions 13 and 14.
For a sequence \(\{a_n\}\) satisfying \(a_n=f(6^n)-f(3^n)\), what is the value of \(\displaystyle\sum_{n=1}^{15}a_n\)? [3 points]
  1. \(120(\log_2 3 - 1)\)
  2. \(105 \log_3 2\)
  3. \(105 \log_2 3\)
  4. \(120 \log_2 3\)
  5. \(120(\log_3 2 + 1)\)
For two integers \(m\) and \(n\) less than or equal to \(20\), what is the number of pairs \((m,n)\) that satisfy \(f(mn)=f(m)+f(n)\)? [4 points]
  1. \(220\)
  2. \(230\)
  3. \(240\)
  4. \(250\)
  5. \(260\)

Mathematics (Type A)

There are \(2\) white balls and \(3\) black balls in sack \(\mathrm{A}\), and \(1\) white ball and \(3\) black balls in sack \(\mathrm{B}\). Suppose we randomly take out \(1\) ball from sack \(\mathrm{A}\). If the ball is white, let us put \(2\) white balls in sack \(\mathrm{B}\), and if the ball is black, let us put \(2\) black balls in sack \(\mathrm{B}\). After this, suppose we randomly take out a ball from sack \(\mathrm{B}\). What is the probability that this ball taken out is white? [4 points]
  1. \(\dfrac{1}{6}\)
  2. \(\dfrac{1}{5}\)
  3. \(\dfrac{7}{30}\)
  4. \(\dfrac{4}{15}\)
  5. \(\dfrac{3}{10}\)
A sequence \(\{a_n\}\) whose terms are all positive, satisfies \(a_1=10\) and
\(\big(a_{n+1}\big)^n=10\big(a_n\big)^{n+1}\quad(n\geq 1)\).
The following is a process computing the general term \(a_n\).
Taking the common logarithm to both sides of the equation gives
\(n\log_{10} a_{n+1}=(n+1)\log_{10}a_n+1\).
Dividing both sides with \(n(n+1)\) gives
\(\dfrac{\log_{10} a_{n+1}}{n+1}=\dfrac{\log_{10}a_n}{n}+ \fbox{\(\;(\alpha)\;\)}\).
Let \(b_n=\dfrac{\log_{10}a_n}{n}\). 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
\(\log_{10}a_n=n\times \fbox{\(\;(\beta)\;\)}\).
Therefore \(a_n=10^{n\times \fbox{\(\;(\beta)\;\)}}\).
Let \(f(n)\) and \(g(n)\) be the correct expression for \(\alpha\) and \(\beta\) respectively. What is the value of \(\dfrac{g(10)}{f(4)}\)? [3 points]
  1. \(38\)
  2. \(40\)
  3. \(42\)
  4. \(44\)
  5. \(46\)

Mathematics (Type A)

Consider a rectangle \(\mathrm{A_1B_1C_1D_1}\) with \(\overline{\mathrm{A_1B_1}}=1\) and \(\overline{\mathrm{A_1D_1}}=2\). As the figure shows, let \(\mathrm{M_1}\) and \(\mathrm{N_1}\) be the midpoints of line segments \(\mathrm{A_1D_1}\) and \(\mathrm{B_1C_1}\) respectively. Let us draw a sector \(\mathrm{N_1M_1B_1}\) with center \(\mathrm{N_1}\), radius \(\overline{\mathrm{B_1N_1}}\), and centeral angle \(\dfrac{\pi}{2}\). Let us draw a sector \(\mathrm{D_1M_1C_1}\) with center \(\mathrm{D_1}\), radius \(\overline{\mathrm{C_1D_1}}\), and centeral angle \(\dfrac{\pi}{2}\). Figure \(R_1\) is obtained by coloring the shape consisting of the region enclosed by arc \(\mathrm{M_1B_1}\) of sector \(\mathrm{N_1M_1B_1}\) and line \(\mathrm{M_1B_1}\), and the region enclosed by arc \(\mathrm{M_1C_1}\) of sector \(\mathrm{D_1M_1C_1}\) and line \(\mathrm{M_1C_1}\).
Starting from figure \(R_1\), let us draw a rectangle \(\mathrm{A_2B_2C_2D_2}\) where \(\mathrm{A_2}\) is on line \(\mathrm{M_1B_1}\), \(\mathrm{D_2}\) is on arc \(\mathrm{M_1C_1}\), \(\mathrm{B_2}\) and \(\mathrm{C_2}\) are on the edge \(\mathrm{B_1C_1}\), and \(\overline{\mathrm{A_2B_2}}:\overline{\mathrm{A_2D_2}}=1:2\). With the same method used to obtain figure \(R_1\), figure \(R_2\) is obtained by drawing and coloring the shape inside rectangle \(\mathrm{A_2B_2C_2D_2}\).
Continue this process, and let \(S_n\) be the area of the colored region in \(R_n\), the \(n\)th obtained figure.
What is the value of \(\displaystyle\lim_{n\to\infty} S_n\)? [4 points]
  1. \(\dfrac{25}{19}\!\left(\!\dfrac{\pi}{2}-1\!\right)\)
  2. \(\dfrac{5}{4}\!\left(\!\dfrac{\pi}{2}-1\!\right)\)
  3. \(\dfrac{25}{21}\!\left(\!\dfrac{\pi}{2}-1\!\right)\)
  4. \(\dfrac{25}{22}\!\left(\!\dfrac{\pi}{2}-1\!\right)\)
  5. \(\dfrac{25}{23}\!\left(\!\dfrac{\pi}{2}-1\!\right)\)
There are \(8\) white ping-pong balls and \(7\) orange ping-pong balls. Suppose we distribute all of the ping-pong balls to \(3\) students. What is the number of cases where each student receives at least \(1\) white ping-pong ball and \(1\) orange ping-pong ball? [4 points]
  1. \(295\)
  2. \(300\)
  3. \(305\)
  4. \(310\)
  5. \(315\)

Mathematics (Type A)

Two square matrices \(A\) and \(B\) of order \(2\) satisfy
\(AB+A^2B=E\:\) and \(\:(A-E)^2+B^2=O\).
Which option only contains every correct statement in the <List>? (※ \(E\) is the identity matrix and \(O\) is the zero matrix.) [4 points]
  1. \(B\) is invertible.
  2. \(AB=BA\)
  3. \((A^3-A)^2+E=O\)
  1. b
  2. c
  3. a, b
  4. a, c
  5. a, b, c
For a positive real number \(x\), let \(f(x)\) and \(g(x)\) be the characteristic and mantissa of \(\log_{10}x\) respectively. For a positive integer \(n\), let \(a_n\) be the product of all values of \(x\) that satisfies \(f(x)-(n+1)g(x)=n\). What is the value of \(\displaystyle\lim_{n\to\infty}\dfrac{\log_{10} a_n}{n^2}\)? [4 points]
  1. \(1\)
  2. \(\dfrac{3}{2}\)
  3. \(2\)
  4. \(\dfrac{5}{2}\)
  5. \(3\)

Mathematics (Type A)

On the \(xy\)-plane, for the cubic function \(f(x)=x^3+ax^2+bx\) and a positive number \(t\), let \(\mathrm{P}\) be the point where a tangent line to the curve \(y=f(x)\) at point \((t,f(t))\) meets the \(y\)-axis. Let \(g(t)\) be the distance between the origin and point \(\mathrm{P}\). Functions \(f(x)\) and \(g(t)\) satisfy the following.
  1. \(f(1)=2\)
  2. Function \(g(t)\) is differentiable on the set of all real numbers.
What is the value of \(f(3)\)? (※ \(a\) and \(b\) are constants.) [4 points]
  1. \(21\)
  2. \(24\)
  3. \(27\)
  4. \(30\)
  5. \(33\)
Short Answer Questions
Compute \(\displaystyle\lim_{x\to\,0} \sqrt{2x+9}\). [3 points]
For a real number \(a\) satisfying \(\displaystyle\int_{-a}^{a}(3x^2+2x)dx=\dfrac{1}{4}\), compute \(50a\). [3 points]

Mathematics (Type A)

A sequence \(\{a_n\}\) satisfies the following.
  1. \(a_1=a_2+3\)
  2. \(a_{n+1}=-2 a_n\,(n\geq 1)\)
Compute \(a_9\). [3 points]
A function \(f(x)=2x^3-12x^2+ax-4\) has a local maximum value of \(M\) at \(x=1\). Compute \(a+M\).
(※ \(a\) is a constant.) [3 points]
Compute the sum of all real numbers \(a\) for which the following system of linear equations has a solution other than \(x=0, y=0\). [4 points]
\(\begin{pmatrix} 5&a \\ a&3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} =\begin{pmatrix} x+5y \\ 6x+y \end{pmatrix}\)

Mathematics (Type A)

There are \(5\) drawers marked with integers from \(1\) to \(5\) respectively. Suppse \(2\) of the \(5\) drawers are randomly assigned for Mary. Let the random variable \(X\) be the smaller of the two integers marked on the drawers assigned to Mary. Compute \(\mathrm{E}(10X)\). [4 points]
For the function
\(f(x)=\begin{cases} x+1 &\; (x \leq 0) \\\\ -\dfrac{1}{2}x+7 &\; (x>0), \end{cases}\)
compute the sum of all real numbers \(a\) for which the function \(f(x)f(x-a)\) is continuous at \(x=a\). [4 points]

Mathematics (Type A)

A function \(f(x)=3x^2-ax\) satisfies
\(\displaystyle\lim_{n\to\infty}\dfrac{1}{n} \sum_{k=1}^{n}f\! \left(\!\dfrac{3k}{n}\!\right)=f(1)\).
Compute the value of the constant \(a\). [4 points]
On the \(xy\)-plane, for an integer \(a>1\), consider the region enclosed by two curves \(y=4^x\) and \(y=a^{-x+4}\) and the line \(y=1\). Consider the points inside this region or on the boundary of this region, whose \(x\) and \(y\)-coordinates are both integers. Compute the number of values of \(a\) for which the number of such points is between \(20\) and \(40\), inclusive. [4 points]