2016 College Scholastic Ability Test

Mathematics (Type A)

Multiple Choice Questions
Consider matrices \(A=\begin{pmatrix} 2&1 \\ 5&0 \end{pmatrix}\) and \(B=\begin{pmatrix} 1&0 \\ 1&1 \end{pmatrix}\). What is the sum of all elements in the matrix \(A-B\)? [2 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
What is the value of \(8^{\,\begin{array}{c}1 \\\hline 3\end{array}} + 27^{\,\begin{array}{c}2 \\\hline 3\end{array}}\)? [2 points]
  1. \(8\)
  2. \(9\)
  3. \(10\)
  4. \(11\)
  5. \(12\)
What is the value of \(\displaystyle\lim_{x\;\!\to\;\!-2}\frac{(x+2)(x^2+5)}{x+2}\)? [2 points]
  1. \(7\)
  2. \(8\)
  3. \(9\)
  4. \(10\)
  5. \(11\)
In the adjacency matrix of the following graph, what is the number of elements with a value of \(1\)? [3 points]
  1. \(10\)
  2. \(14\)
  3. \(18\)
  4. \(22\)
  5. \(26\)

Mathematics (Type A)

For the function \(f(x)=x^3+7x+3\), what is the value of \(f'(1)\)? [3 points]
  1. \(4\)
  2. \(6\)
  3. \(8\)
  4. \(10\)
  5. \(12\)
Let \(A\) and \(B\) be events such that
\(\mathrm{P}(A)=\dfrac{2}{5}\:\) and \(\:\mathrm{P}(B|A)=\dfrac{5}{6}\).
What is the value of \(\mathrm{P}(A\cap B)\)? [3 points]
  1. \(\dfrac{1}{3}\)
  2. \(\dfrac{4}{15}\)
  3. \(\dfrac{1}{5}\)
  4. \(\dfrac{2}{15}\)
  5. \(\dfrac{1}{15}\)
A geometric progression \({a_n}\) whose initial term is not \(0\) satisfies
\(a_3=4a_1\:\) and \(\:a_7=\big(a_6\big)^2\).
What is the value of the initial term \(a_1\)? [3 points]
  1. \(\dfrac{1}{16}\)
  2. \(\dfrac{1}{8}\)
  3. \(\dfrac{3}{16}\)
  4. \(\dfrac{1}{4}\)
  5. \(\dfrac{5}{16}\)

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. \(5\)
Let \(\overline{X}\) be the sample mean of a random sample of size \(n\) from a population with a standard deviation of \(14\). Given that \(\sigma(\overline{X})=2\), what is the value of \(n\)? [3 points]
  1. \(9\)
  2. \(16\)
  3. \(25\)
  4. \(36\)
  5. \(49\)
Let \(\{a_n\}\) be a sequence such that the curve \(y=x^2-(n+1)x+a_n\) meets the \(x\)-axis, and the curve \(y=x^2-nx+a_n\) does not meet the \(x\)-axis. What is the value of \(\displaystyle\lim_{n\;\!\to\;\!\infty} \dfrac{a_n}{n^2}\)? [3 points]
  1. \(\dfrac{1}{20}\)
  2. \(\dfrac{1}{10}\)
  3. \(\dfrac{3}{20}\)
  4. \(\dfrac{1}{5}\)
  5. \(\dfrac{1}{4}\)

Mathematics (Type A)

Consider the inequality
\(\log_5 (x-1) \leq \log_5 \left(\!\dfrac{1}{2}x+k\!\right)\).
If the number of all integers \(x\) satisfying this inequality is \(3\), what is the value of the positive integer \(k\)? [3 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
The weight of rice donated by a student participating in some rice collection event follows a normal distribution with a mean of \(1.5\text{kg}\) and a standard deviation of \(0.2\text{kg}\). Suppose we randomly select one of the students participating in this event.
What is the probability that the weight of rice donated by this student is between \(1.3\text{kg}\) and \(1.8\text{kg}\), computed with the standard normal table to the right? [3 points]
\(z\) \(\mathrm{P}(0\!\leq\! Z \!\leq\!z)\)
\(1.00\) \(0.3413\)
\(1.25\) \(0.3944\)
\(1.50\) \(0.4332\)
\(1.75\) \(0.4599\)
  1. \(0.8543\)
  2. \(0.8012\)
  3. \(0.7745\)
  4. \(0.7357\)
  5. \(0.6826\)

Mathematics (Type A)

[13~14]  For positive integers \(\boldsymbol{n}\), let \(\boldsymbol{\mathrm{P}}\) be a point with coordinates \(\boldsymbol{(0,2n+1)}\), and let \(\boldsymbol{\mathrm{Q}}\) be a point on the graph of the function \(\boldsymbol{f(x)=nx^2}\) in the \(\boldsymbol{1}\)st quadrant with a \(\boldsymbol{y}\)-coordinate of \(\boldsymbol{1}\).
Answer the questions 13 and 14.
Given that \(n=1\), what is the area of the region enclosed by the line segment \(\mathrm{PQ}\), the curve \(y=f(x)\) and the \(y\)-axis? [3 points]
  1. \(\dfrac{3}{2}\)
  2. \(\dfrac{19}{12}\)
  3. \(\dfrac{5}{3}\)
  4. \(\dfrac{7}{4}\)
  5. \(\dfrac{11}{6}\)
For the point \(\mathrm{R}(0,1)\), let \(S_n\) be the area of the triangle \(\mathrm{PRQ}\), and let \(l_n\) be the length of the line segment \(\mathrm{PQ}\). What is the value of \(\displaystyle\lim_{n\;\!\to\;\!\infty}\dfrac{S_n^2}{l_n}\)? [4 points]
  1. \(\dfrac{3}{2}\)
  2. \(\dfrac{5}{4}\)
  3. \(1\)
  4. \(\dfrac{3}{4}\)
  5. \(\dfrac{1}{2}\)

Mathematics (Type A)

As the figure shows, there is a square \(\mathrm{ABCD}\) with side lengths of \(5\), and points \(\mathrm{P_1,P_2,P_3}\) and \(\mathrm{P_4}\) divide its diagonal \(\mathrm{BD}\) into \(5\) equal parts. Obtain figure \(R_1\) by drawing three squares with line segments \(\mathrm{BP_1}\), \(\mathrm{P_2P_3}\) and \(\mathrm{P_4D}\) respectively as diagonals, and two circles with line segments \(\mathrm{P_1P_2}\) and \(\mathrm{P_3P_4}\) respectively as diameters, and then coloring the shape.
Starting from figure \(R_1\), consider the square with the line segment \(\mathrm{P_2P_3}\) as a diagonal. Let \(\mathrm{Q_1}\) be the vertex closest to point \(\mathrm{A}\), and let \(\mathrm{Q_2}\) be the vertex closest to point \(\mathrm{C}\). Draw two squares with line segments \(\mathrm{AQ_1}\) and \(\mathrm{CQ_2}\) as a diagonal respectively. Obtain figure \(R_2\) by drawing and coloring two shapes, by applying the process of obtaining figure \(R_1\) to the newly drawn two squares respectively.
Starting from figure \(R_2\), draw and color four shapes on two squares with line segments \(\mathrm{AQ_1}\) and \(\mathrm{CQ_2}\) as a diagonal respectively, using the same process as obtaining figure \(R_2\) from figure \(R_1\).
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. \(\dfrac{24}{17}(\pi+3)\)
  2. \(\dfrac{25}{17}(\pi+3)\)
  3. \(\dfrac{26}{17}(\pi+3)\)
  4. \(\dfrac{24}{17}(2\pi+1)\)
  5. \(\dfrac{25}{17}(2\pi+1)\)
Suppose an initial asset of \(W_0\) is invested to some financial product. Then, the expected asset \(W\) after \(t\) years is given as follows.
\(W=\dfrac{W_0}{2} 10^{at}(1+10^{at})\)
(※ \(W_0 > 0\:\) and \(\:t \geq 0.\:\) \(a\) is a constant.)
The expected asset \(15\) years after investing the initial asset of \(w_0\) to this financial product, is \(3\) times the initial asset. Given this, the expected asset \(30\) years after investing the initial asset of \(w_0\) to this financial product, is \(k\) times the initial asset. What is the value of the real number \(k\)? (※ \(w_0 > 0\)) [4 points]
  1. \(9\)
  2. \(10\)
  3. \(11\)
  4. \(12\)
  5. \(13\)

Mathematics (Type A)

What is the number of all \(5\)-tuples \((a,b,c,d,e)\) where \(a,b,c,d\) and \(e\) are nonnegative integers that satisfy the following? [4 points]
  1. Exactly \(2\) numbers among \(a,b,c,d\) and \(e\) have a value of \(0\).
  2. \(a+b+c+d+e=10\)
  1. \(240\)
  2. \(280\)
  3. \(320\)
  4. \(360\)
  5. \(400\)
Let \(A\) and \(B\) be square matrices of order \(2\) such that
\(A+B=(BA)^2\:\) and \(\:ABA=B+E\).
What is the list of correct statements in the <List>? (※ \(E\) is the identity matrix.) [4 points]
  1. \(A=B^2\)
  2. \(B^{-1} = A^2+E\)
  3. \(A^5-B^5=A+B\)
  1. a
  2. b
  3. a, c
  4. b, c
  5. a, b, c

Mathematics (Type A)

A sequence \(\{a_n\}\) whose terms are all positive, satisfy \(a_1=a_2=1\) and
\(a_{n+1} = \dfrac{S_n^2}{S_{n-1}} + (2n-1)S_n \quad (n\geq 2)\)
where \(S_n= \displaystyle\sum_{k\;\!=\;\!1}^{n}a_k\). The following is a process computing the general term \(a_n\).
Since \(a_{n+1}=S_{n+1}-S_n\), we have
\(S_{n+1} = \dfrac{S_n^2}{S_{n-1}} + 2nS_n \quad (n\geq 2)\).
Dividing both sides with \(S_n\), we have
\(\dfrac{S_{n+1}}{S_n} = \dfrac{S_n}{S_{n-1}}+2n\).
Let \(b_n = \dfrac{S_{n+1}}{S_n}\). Then \(b_1=2\), and
\(b_n=b_{n-1}+2n \quad (n\geq 2)\).
The general term of the sequence \(\{b_n\}\) is
\(b_n = \fbox{\(\;(\alpha)\;\)} \times (n+1) \quad (n\geq 1)\),
therefore
\(S_n =\fbox{\(\;(\alpha)\;\)} \times \{(n-1)!\}^2 \quad (n\geq 1)\).
Therefore \(a_1=1\), and
\(\begin{align} a_n &= S_n-S_{n-1} \\ &= \fbox{\(\;(\beta)\;\)} \times \{(n-2)!\}^2 \end{align}\)
for \(n\geq 2\).
Let \(f(n)\) and \(g(n)\) be the correct expression for \(\alpha\) and \(\beta\) respectively. What is the value of \(f(10)+g(6)\)? [4 points]
  1. \(110\)
  2. \(125\)
  3. \(140\)
  4. \(155\)
  5. \(170\)
Functions \(f(x)\) and \(g(x)\) satisfy
\(f(-x)=-f(x)\:\) and \(\:g(-x)=g(x)\)
for all real numbers \(x\). Given that the function \(h(x)=f(x)g(x)\) satisfies
\(\displaystyle\int_{-3}^{3}(x+5)h'(x)dx = 10\),
what is the value of \(h(3)\)? [4 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)

Mathematics (Type A)

Among all cubic functions \(f(x)\) that satisfy the following, let \(M\) and \(m\) be the maximum value and minimum value of \(\dfrac{f'(0)}{f(0)}\), respectively.
What is the value of \(Mm\)? [4 points]
  1. The function \(|f(x)|\) is not differentiable only at \(x=-1\).
  2. The equation \(f(x)=0\) has at least one real solution on the closed interval \([3, 5]\).
  1. \(\dfrac{1}{15}\)
  2. \(\dfrac{1}{10}\)
  3. \(\dfrac{2}{15}\)
  4. \(\dfrac{1}{6}\)
  5. \(\dfrac{1}{5}\)
Short Answer Questions
An arithmetic progression \(\{a_n\}\) satisfies \(a_8-a_4=28.\) Compute the common difference of the sequence \(\{a_n\}\). [3 points]
Compute \(\displaystyle\lim_{n\;\!\to\;\!\,\infty} \dfrac{3\times 9^n - 13}{9^n}\). [3 points]

Mathematics (Type A)

Compute the value of the constant \(a\) for which the following system of linear equations has a solution other than \(x=0, y=0\). [3 points]
\(\begin{pmatrix} 1&a-2 \\ 2&-1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} =3 \begin{pmatrix} x \\ y \end{pmatrix}\)
The probability distribution of a discrete random variable \(X\) is as follows.
\(X\) \(-5\) \(0\) \(5\) Total
\(\mathrm{P}(X=x)\) \(\dfrac{1}{5}\) \(\dfrac{1}{5}\) \(\dfrac{3}{5}\) \(1\)
Compute \(\mathrm{E}(4X+3)\). [3 points]
A company has \(60\) employees in total, and each employee is either in department \(\mathrm{A}\) or department \(\mathrm{B}\). Department \(\mathrm{A}\) consists of \(20\) employees and department \(\mathrm{B}\) consists of \(40\) employees. \(50\%\) of employees in department \(\mathrm{A}\) are female. \(60\%\) of female employees of this company are in department \(\mathrm{B}\). Suppose one of the \(60\) employees of this company is chosen at random. Given that this employee is in department \(\mathrm{B}\), the probability that this employee is female is \(p\). Compute \(80p\). [4 points]

Mathematics (Type A)

For functions
\(f(x)=\begin{cases} x+3 &\; (x \leq a) \\ x^2-x &\; (x > a) \end{cases}\:\)
and \(\:g(x)=x-(2a+7)\),
compute the product of all real numbers \(a\) for which the function \(f(x)g(x)\) is continuous on the set of all real numbers. [4 points]
Let \(f(x)\) and \(g(x)\) be polynomial functions that satisfy the following.
  1. \(g(x)=x^3f(x)-7\)
  2. \(\displaystyle\lim_{x\;\!\to\;\!2} \dfrac{f(x)-g(x)}{x-2} = 2\)
The equation of the tangent line to the curve \(y=g(x)\) at point \((2, g(2))\) is \(ax+b\). compute \(a^2+b^2\). (※ \(a\) and \(b\) are constants.) [4 points]

Mathematics (Type A)

A quadratic function \(f(x)\) satisfies \(f(0)=0\) and the following.
  1. \(\displaystyle\int_{0}^{2}|f(x)|dx = -\int_{0}^{2}f(x)dx = 4\)
  2. \(\displaystyle\int_{2}^{3}|f(x)|dx = \int_{2}^{3}f(x)dx\)
Compute \(f(5)\). [4 points]
For real numbers \(x \geq \dfrac{1}{100}\), let \(f(x)\) be the mantissa of \(\log_{10}x\). Let \(R\) be the region created by points \((a,b)\) on the \(xy\)-plane, where \(a\) and \(b\) are real numbers that satisfy the following.
  1. \(a<0\) and \(b>10\).
  2. The graph of the function \(y=9f(x)\) and the line \(ax+b\) meet at only one point.
Among all points \((a,b)\) in the region \(R\), the minimum value of \((a+20)^2+b^2\) is \(100\times \dfrac{q}{p}\). Compute \(p+q\).
(※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]