2016 College Scholastic Ability Test

Mathematics (Type B)

Multiple Choice Questions
Consider matrices \(A=\begin{pmatrix} a&3 \\ 0&1 \end{pmatrix}\) and \(B=\begin{pmatrix*}[r] 4&1 \\ -1&0 \end{pmatrix*}\).
If the sum of all elements in the matrix \(A+B\) is \(9\), what is the value of \(a\)? [2 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
What is the value of \(\displaystyle\lim_{x\;\!\to\;\!0} \dfrac{\ln(1+5x)}{\sin3x}\)? [2 points]
  1. \(1\)
  2. \(\dfrac{4}{3}\)
  3. \(\dfrac{5}{3}\)
  4. \(2\)
  5. \(\dfrac{7}{3}\)
In \(3\)-dimensional space, a triangle with three points \(\mathrm{A}(a,0,5), \mathrm{B}(1,b,-3)\) and \(\mathrm{C}(1,1,1)\) as vertices has a centroid with coordinates \((2,2,1)\). What is the value of \(a+b\)? [2 points]
  1. \(6\)
  2. \(7\)
  3. \(8\)
  4. \(9\)
  5. \(10\)
What is the value of \(\displaystyle\int_{0}^{e}\dfrac{5}{x+e}dx\)? [3 points]
  1. \(\ln2\)
  2. \(2\ln2\)
  3. \(3\ln2\)
  4. \(4\ln2\)
  5. \(5\ln2\)

Mathematics (Type B)

Let \(A\) and \(B\) be independent events such that
\(\mathrm{P}(A^C)=\dfrac{1}{4}\:\) and \(\:\mathrm{P}(A\cap B)=\dfrac{1}{2}\).
What is the value of \(\mathrm{P}(B|A^C)\)?
(※ \(A^C\) is the complement of \(A\).) [3 points]
  1. \(\dfrac{5}{12}\)
  2. \(\dfrac{1}{2}\)
  3. \(\dfrac{7}{12}\)
  4. \(\dfrac{2}{3}\)
  5. \(\dfrac{3}{4}\)
A linear map represented with the matrix \(\begin{pmatrix*}[r] 1&2 \\ -2&1 \end{pmatrix*}\) maps the point \(\mathrm{P}(2,-1)\) to point \(\mathrm{Q}\). What is the slope of the line \(\mathrm{PQ}\)? [3 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
A tangent line to the curve \(y=3e^{x-1}\) at point \(\mathrm{A}\), passes through the origin \(\mathrm{O}\). What is the length of the line segment \(\mathrm{OA}\)? [3 points]
  1. \(\sqrt{6}\)
  2. \(\sqrt{7}\)
  3. \(2\sqrt{2}\)
  4. \(3\)
  5. \(\sqrt{10}\)

Mathematics (Type B)

Suppose we throw a coin \(5\) times. What is the probability that the number of times it lands on heads and the number of times it lands on tails has a product of \(6\)? [3 points]
  1. \(\dfrac{5}{8}\)
  2. \(\dfrac{9}{16}\)
  3. \(\dfrac{1}{2}\)
  4. \(\dfrac{7}{16}\)
  5. \(\dfrac{3}{8}\)
Let \(l\) be the tangent line to the parabola \(y^2=4x\) at point \(\mathrm{A}(4,4)\). Let \(\mathrm{B}\) be the intersection of line \(l\) and the directrix of the parabola, let \(\mathrm{C}\) be the intersection of line \(l\) and the \(x\)-axis, and let \(\mathrm{D}\) be the intersection of the directrix of the parabola and the \(x\)-axis. What is the area of the triangle \(\mathrm{BCD}\)? [3 points]
  1. \(\dfrac{7}{4}\)
  2. \(2\)
  3. \(\dfrac{9}{4}\)
  4. \(\dfrac{5}{2}\)
  5. \(\dfrac{11}{4}\)
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 B)

[11~12]  The graph of the function
\(\boldsymbol{f(x)=\begin{cases} \boldsymbol{|5x(x+2)|} &\; \boldsymbol{(x < 0)} \\\\ \boldsymbol{|5x(x-2)|} &\; \boldsymbol{(x \geq 0)} \end{cases}}\)
is as follows. Answer the questions 11 and 12.
Consider the region enclosed by the graph of the function \(y=f(x)\) on the closed interval \([0,1]\), the
\(x\)-axis, and the line \(x=1\). What is the volume of the solid formed by rotating this region about the \(x\)-axis? [3 points]
  1. \(\dfrac{65}{6}\pi\)
  2. \(\dfrac{35}{3}\pi\)
  3. \(\dfrac{25}{2}\pi\)
  4. \(\dfrac{40}{3}\pi\)
  5. \(\dfrac{85}{6}\pi\)
What is the number of distinct real solutions to the equation \(\sqrt{4-f(x)}=1-x\)? [3 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)

Mathematics (Type B)

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 \(\displaystyle\lim_{n\;\!\to\;\!\infty} S_n\)? [3 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. \(\dfrac{25}{17}(2\pi+1)\)
For three integers \(a, b\) and \(c\) that satisfy
\(1 \leq |a| \leq |b| \leq |c| \leq 5\),
What is the number of all \(3\)-tuples \((a,b,c)\)? [4 points]
  1. \(360\)
  2. \(320\)
  3. \(280\)
  4. \(240\)
  5. \(200\)

Mathematics (Type B)

On the \(xy\)-plane, there is a point \(\mathrm{A}\) with coordinates \((1,0)\), and a point \(\mathrm{B}\) with coordinates \((\cos\theta, \sin\theta)\) for \(0<\theta<\dfrac{\pi}{2}\). Let \(\mathrm{C}\) be a point in the \(1\)st quadrant such that the quadrilateral \(\mathrm{OACB}\) is a parallelogram. Let \(f(\theta)\) be the area of the quadrilateral \(\mathrm{OACB}\), and let \(g(\theta)\) be the square of the length of the line segment \(\mathrm{OC}\). What is the maximum value of \(f(\theta)+g(\theta)\)? (※ \(\mathrm{O}\) is the origin.) [4 points]
  1. \(2+\sqrt{5}\)
  2. \(2+\sqrt{6}\)
  3. \(2+\sqrt{7}\)
  4. \(2+2\sqrt{2}\)
  5. \(5\)
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 B)

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. \(170\)
Let \(\overline{X}\) be the mean of a random sample of size \(16\) from a population following the normal distribution \(\mathrm{N}(50, 8^2)\), and let \(\overline{Y}\) be the mean of a random sample of size \(25\) from a population following the normal
distribution \(\mathrm{N}(75, \sigma^2)\). Given that \(\mathrm{P}(\overline{X} \!\leq\! 53) \!+\! \mathrm{P}(\overline{Y} \!\leq\! 69) \!=\! 1\), what is the value of \(\mathrm{P}(\overline{Y} \geq 71)\), computed with the standard normal table to the right? [4 points]
\(z\) \(\mathrm{P}(0\!\leq\! Z \!\leq\!z)\)
\(1.0\) \(0.3413\)
\(1.2\) \(0.3849\)
\(1.4\) \(0.4192\)
\(1.6\) \(0.4452\)
  1. \(0.8413\)
  2. \(0.8644\)
  3. \(0.8849\)
  4. \(0.9192\)
  5. \(0.9452\)

Mathematics (Type B)

In \(3\)-dimensional space, there is a point \(\mathrm{A}(2,2,1)\) and a plane \(\alpha:\, x+2y+2z-14=0\). Consider the shape created by points \(\mathrm{P}\) on plane \(\alpha\) that satisfy \(\overline{\mathrm{AP}} \leq 3\). What is the area of the projection of this shape onto the \(xy\)-plane? [4 points]
  1. \(\dfrac{14}{3}\pi\)
  2. \(\dfrac{13}{3}\pi\)
  3. \(4\pi\)
  4. \(\dfrac{11}{3}\pi\)
  5. \(\dfrac{10}{3}\pi\)
For positive numbers \(x\), let \(f(x)\) be the characteristic of \(\log_{10}x\). What is the number of positive integers \(n\) that satisfy
\(f(n+10)=f(n)+1\),
if \(n\) is less than or equal to \(100\)? [4 points]
  1. \(11\)
  2. \(13\)
  3. \(15\)
  4. \(17\)
  5. \(19\)

Mathematics (Type B)

For real numbers \(t\) that satisfy \(0<t<41\), consider three points where the curve \(y=x^3+2x^2-15x+5\) meets the line \(y=t\). Let \((f(t),t)\) be the point with the greatest \(x\)-coordinate, and \((g(t),t)\) be the point with the smallest \(x\)-coordinate among the three points. Let \(h(t)=t \times \{f(t)-g(t)\}\). What is the value of \(h'(5)\)? [4 points]
  1. \(\dfrac{79}{12}\)
  2. \(\dfrac{85}{12}\)
  3. \(\dfrac{91}{12}\)
  4. \(\dfrac{97}{12}\)
  5. \(\dfrac{103}{12}\)
Short Answer Questions
An arithmetic progression \(\{a_n\}\) with an initial term of \(2\) satisfies
\(2(a_2+a_3)=a_9\).
Compute the common difference of the sequence \(\{a_n\}\). [3 points]
For the function \(f(x)=4\sin 7x\), compute \(f'(2\pi)\). [3 points]

Mathematics (Type B)

An absolutely continuous random variable \(X\) can take all values in the interval \([0, 1]\), and its probability density function is
\(f(x)=kx(1-x^3)\quad(0 \leq x \leq 1)\).
Compute \(24k\). (※ \(k\) is a constant.) [3 points]
For a geometric progression \(\{a_n\}\) with an initial term of \(1\) and a common ratio of \(r\,(r>1)\), let \(S_n= \displaystyle\sum_{k\;\!=\;\!1}^{n}a_k\). Given that \(\displaystyle\lim_{n\;\!\to\;\!\infty} \dfrac{a_n}{S_n} = \dfrac{3}{4}\), compute \(r\). [3 points]
As the figure shows, there is an ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) with two foci \(\mathrm{F}(c,0)\) and \(\mathrm{F'}(-c,0)\). For a point \(\mathrm{P}\) on the ellipse in the \(2\)nd quadrant, let \(\mathrm{Q}\) be the midpoint of the line segment \(\mathrm{PF'}\), and let \(\mathrm{R}\) be the point internally dividing the line segment \(\mathrm{PF}\) in the ratio \(1:3\). Given that \(\angle \mathrm{PQR}=\dfrac{\pi}{2}, \overline{\mathrm{QR}}=\sqrt{5}\) and \(\overline{\mathrm{RF}}=9\), compute \(a^2+b^2\).
(※ \(a, b\) and \(c\) are positive numbers.) [4 points]

Mathematics (Type B)

In \(3\)-dimensional space, consider two planes \(\alpha\) and \(\beta\) that are perpendicular. There are two points \(\mathrm{A}\) and \(\mathrm{B}\) on plane \(\alpha\) such that \(\overline{\mathrm{AB}}=3\sqrt{5}\) and line \(\mathrm{AB}\) is parallel to plane \(\beta\). The distance from point \(\mathrm{A}\) to plane \(\beta\) is \(2\), and the distance from a point \(\mathrm{P}\) on plane \(\beta\) to plane \(\alpha\) is \(4\). Compute the area of the triangle \(\mathrm{PAB}\). [4 points]
As the figure shows, on the \(xy\)-plane, let \(\mathrm{A}\) be the point in the \(1\)st quadrant where the circle \(x^2+y^2=1\) meets the curve \(y=\ln(x+1)\). For the point \(\mathrm{B}(1,0)\) and a point \(\mathrm{P}\) on arc \(\mathrm{AB}\), let \(\mathrm{H}\) be the perpendicular foot from point \(\mathrm{P}\) to the \(y\)-axis, and let \(\mathrm{Q}\) be the intersection of line \(\mathrm{PH}\) and the curve \(y=\ln(x+1)\). For \(\angle \mathrm{POB}=\theta\), let \(S(\theta)\) be the area of the triangle \(\mathrm{OPQ}\), and let \(L(\theta)\) be the length of the line segment \(\mathrm{HQ}\). Given that \(\displaystyle\lim_{\theta\;\!\to\;\!+0} \dfrac{S(\theta)}{L(\theta)} = k\),
compute \(60k\). (※ \(0<\theta< \dfrac{\pi}{6}\). \(\mathrm{O}\) is the origin.) [4 points]

Mathematics (Type B)

For points \(\mathrm{A}(2,\sqrt{2},\sqrt{3})\) and \(\mathrm{B}(1,-\sqrt{2},2\sqrt{3})\) in \(3\)-dimensional space, there is a point \(\mathrm{P}\) that satisfies the following.
  1. \(\big|\overrightarrow{\mathrm{AP}}\big|=1\)
  2. The angle between \(\overrightarrow{\mathrm{AP}}\) and \(\overrightarrow{\mathrm{AB}}\) is \(\dfrac{\pi}{6}\).
For a point \(\mathrm{Q}\) on a sphere with radius \(1\) and the origin as the center, the maximum value of \(\overrightarrow{\mathrm{AP}}\cdot \overrightarrow{\mathrm{AQ}}\) is \(a+b\sqrt{33}\). Compute \(16(a^2+b^2)\).
(※ \(a\) and \(b\) are rational numbers.) [4 points]
A function \(f(x)\) continuous on the set of all real numbers satisfies the following.
  1. \(f(x)=a(x-b)^2+c\:\) for \(\:x\leq b\).
    (※ \(a, b\) and \(c\) are constants.)
  2. \(f(x)=\displaystyle\int_{0}^{x}\!\sqrt{4-2f(t)}\,dt\:\) for all real numbers \(x\).
Given that \(\displaystyle\int_{0}^{6}\!f(x)dx = \dfrac{q}{p}\), compute \(p+q\).
(※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]