2015 College Scholastic Ability Test

Mathematics (Type B)

Multiple Choice Questions
For two matrices \(A=\begin{pmatrix} 1&1 \\ 0&2 \end{pmatrix}\) and \(B=\begin{pmatrix} 1&1 \\ 3&0 \end{pmatrix}\), what is the sum of all elements in the matrix \(A+B\)? [2 points]
  1. \(5\)
  2. \(6\)
  3. \(7\)
  4. \(8\)
  5. \(9\)
What is the value of \(\displaystyle\lim_{x\to0} \dfrac{\ln(1+x)}{3x}\)? [2 points]
  1. \(1\)
  2. \(\dfrac{1}{2}\)
  3. \(\dfrac{1}{3}\)
  4. \(\dfrac{1}{4}\)
  5. \(\dfrac{1}{5}\)
What is the maximum value of the function \(f(x)=\sin x+\sqrt{7}\cos x-\sqrt{2}\)? [2 points]
  1. \(\sqrt{2}\)
  2. \(\sqrt{3}\)
  3. \(2\)
  4. \(\sqrt{5}\)
  5. \(\sqrt{6}\)
What is the value of \(\displaystyle\int_{0}^{1}3\sqrt{x}\:dx\)? [3 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)

Mathematics (Type B)

For points \(\mathrm{A}(2,a,-2)\) and \(\mathrm{B}(5,-3,b)\) in \(3\)-dimensional space, a point internally dividing the line segment \(\mathrm{AB}\) in the ratio \(2:1\) is on the \(x\)-axis. What is the value of \(a+b\)? [3 points]
  1. \(10\)
  2. \(9\)
  3. \(8\)
  4. \(7\)
  5. \(6\)
Two linear maps \(f\) and \(g\) are represented with matrices \(\begin{pmatrix} 2&1 \\ 4&2 \end{pmatrix}\) and \(\begin{pmatrix} 2&0 \\ 1&-1 \end{pmatrix}\) respectively. The composite map \(f\circ g\) maps the point \((1,2)\) to point \((a,6)\). What is the value of \(a\)? [3 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
A geometric progression \(\{a_n\}\) satisfies \(a_1=3\) and \(a_2=1\). What is the value of \(\displaystyle\sum_{n=1}^{\infty}(a_n)^2\)? [3 points]
  1. \(\dfrac{81}{8}\)
  2. \(\dfrac{83}{8}\)
  3. \(\dfrac{85}{8}\)
  4. \(\dfrac{87}{8}\)
  5. \(\dfrac{89}{8}\)

Mathematics (Type B)

Consider two events \(A\) and \(B\) where \(A^C\) and \(B\) are mutually exclusive. Given that
\(\mathrm{P}(A)=2 \mathrm{P}(B)=\dfrac{3}{5}\),
what is the value of \(\mathrm{P}(A\cap B^C)\)?
(※ \(A^C\) is the complement of \(A\).) [3 points]
  1. \(\dfrac{7}{20}\)
  2. \(\dfrac{3}{10}\)
  3. \(\dfrac{1}{4}\)
  4. \(\dfrac{1}{5}\)
  5. \(\dfrac{3}{20}\)
For the function \(f(x)=\dfrac{1}{x}\), what is the value of \(\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n} f\!\left(\!1+\dfrac{2k}{n}\!\right) \dfrac{2}{n}\)? [3 points]
  1. \(\ln 2\)
  2. \(\ln 3\)
  3. \(2\ln 2\)
  4. \(\ln 5\)
  5. \(\ln 6\)
As the figure shows, let \(\mathrm{A}\) and \(\mathrm{B}\) be two points where a line passing through the focus \(\mathrm{F}\) of the parabola \(y^2=12x\) meets the parabola. Let \(\mathrm{C}\) and \(\mathrm{D}\) be perpendicular foots from points \(\mathrm{A}\) and \(\mathrm{B}\) to the directrix \(l\) respectively. Given that \(\overline{\mathrm{AC}}=4\), what is the length of the line segment \(\mathrm{BD}\)? [3 points]
  1. \(12\)
  2. \(\dfrac{25}{2}\)
  3. \(13\)
  4. \(\dfrac{27}{2}\)
  5. \(14\)

Mathematics (Type B)

The weight of a bag of chips manufactured in some company follows a normal distribution with a mean of \(75\text{g}\) and a standard deviation of \(2\text{g}\). What is the probability that a randomly sampled bag of chips
manufactured in this company has a weight between \(76\text{g}\) and \(78\text{g}\), computed using the standard normal table to the right? [3 points]
\(z\)\(\mathrm{P}(0\!\leq\! Z \!\leq\!z)\)
\(0.5\)\(0.1915\)
\(1.0\)\(0.3413\)
\(1.5\)\(0.4332\)
\(2.0\)\(0.4772\)
  1. \(0.0440\)
  2. \(0.0919\)
  3. \(0.1359\)
  4. \(0.1498\)
  5. \(0.2417\)
Let \(l\) be a line passing through two distinct points \(\mathrm{A}\) and \(\mathrm{B}\) on a plane \(\alpha\). Let \(\mathrm{P}\) be a point not on plane \(\alpha\), and let \(\mathrm{H}\) be the perpendicular foot from point \(\mathrm{P}\) to plane \(\alpha\). Given that \(\overline{\mathrm{AB}}=\overline{\mathrm{PA}}=\overline{\mathrm{PB}}=6\) and \(\overline{\mathrm{PH}}=4\), what is the distance between the point \(\mathrm{H}\) and line \(l\)? [3 points]
  1. \(\sqrt{11}\)
  2. \(2\sqrt{3}\)
  3. \(\sqrt{13}\)
  4. \(\sqrt{14}\)
  5. \(\sqrt{15}\)

Mathematics (Type B)

[13~14]  For a constant \(\boldsymbol{a>3}\), two curves \(\boldsymbol{y=a^x-1}\) and \(\boldsymbol{y=3^x}\) meet at point \(\boldsymbol{\mathrm{P}}\).
Let \(\boldsymbol{k}\) be the \(\boldsymbol{x}\)-coordinate of point \(\boldsymbol{\mathrm{P}}\).
Answer the questions 13 and 14.
What is the value of \(\displaystyle\lim_{n\to\infty} \frac{\left(\!\dfrac{a}{3}\!\right)^{n+k}}{\left(\!\dfrac{a}{3}\!\right)^{n+1}+1}\)? [3 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
Let \(\mathrm{A}\) be the point where a tangent line to the curve \(y=3^x\) at point \(\mathrm{P}\) meets the \(x\)-axis, and let \(\mathrm{B}\) be the point where a tangent line to the curve \(y=a^{x-1}\) at point \(\mathrm{P}\) meets the \(x\)-axis. If the point \(\mathrm{H}(k,0)\) satisfies \(\overline{\mathrm{AH}}=2 \overline{\mathrm{BH}}\), what is the value of \(a\)? [4 points]
  1. \(6\)
  2. \(7\)
  3. \(8\)
  4. \(9\)
  5. \(10\)

Mathematics (Type B)

A survey was conducted on all of the \(320\) students of some school about whether they are a member of the math club. The results showed that \(60\%\) of male students and \(50\%\) of female students were a member of the math club. Let \(p_1\) be the probability that a randomly selected student among the members of the math club is male, and let \(p_2\) be the probability that a randomly selected student among the members of the math club is female. Given that \(p_1=2p_2\), what is the number of male students in this school? [4 points]
  1. \(170\)
  2. \(180\)
  3. \(190\)
  4. \(200\)
  5. \(210\)
Two square matrices \(A\) and \(B\) of order \(2\) satisfy
\(A^2-AB=3E\:\) and \(\:A^2B-B^2A=A+B\).
Which option only contains every correct statement in the <List>? (※ \(E\) is the identity matrix.) [4 points]
  1. \(A\) is invertible.
  2. \(AB = BA\)
  3. \((A+2B)^2=24E\)
  1. a
  2. c
  3. a, b
  4. b, c
  5. a, b, c

Mathematics (Type B)

For a sequence \(\{a_n\}\) with \(a_1=1\), and \(S_n=\displaystyle\sum_{k=1}^{n}a_k\),
\(a_{n+1}=(n+1)S_n+n!\quad(n\geq 1)\)
is satisfied. The following is a process computing the general term \(a_n\).
\(a_{n+1}=S_{n+1}-S_n\) for all positive integers \(n\), so the given equation becomes
\(S_{n+1}=(n+2)S_n+n!\quad(n\geq 1)\).
Dividing both sides with \((n+2)!\) gives
\(\dfrac{S_{n+1}}{(n+2)!}=\dfrac{S_n}{(n+1)!}+\dfrac{1}{(n+1)(n+2)}\).
Let \(b_n=\dfrac{S_n}{(n+1)!}\). Then \(b_1=\dfrac{1}{2}\), and
\(b_{n+1}=b_n+ \dfrac{1}{(n+1)(n+2)}\).
The general term of the sequence \(\{b_n\}\) is
\(b_n=\dfrac{\fbox{\(\;(\alpha)\;\)}}{n+1}\).
Therefore
\(S_n=\fbox{\(\;(\alpha)\;\)}\times n!\)
and
\(a_n=\fbox{\(\;(\beta)\;\)}\times (n-1)!\quad(n\geq1)\).
Let \(f(n)\) and \(g(n)\) be the correct expression for \((\alpha)\) and \((\beta)\) respectively. What is the value of \(f(7)+g(6)\)? [4 points]
  1. \(44\)
  2. \(41\)
  3. \(38\)
  4. \(35\)
  5. \(32\)
There is a sack containing \(1\) ball marked with number \(1\), \(2\) balls marked with number \(2\), and \(5\) balls marked with number \(3\). Let us randomly take out a ball from this sack, check the number marked on it, and put it back in. After repeating such trial \(2\) times, let \(\overline{X}\) be the mean of the numbers marked on the balls taken out. What is the value of \(\mathrm{P}(\overline{X}=2)\)? [4 points]
  1. \(\dfrac{5}{32}\)
  2. \(\dfrac{11}{64}\)
  3. \(\dfrac{3}{16}\)
  4. \(\dfrac{13}{64}\)
  5. \(\dfrac{7}{32}\)

Mathematics (Type B)

In \(3\)-dimensional space, the line \(l: \dfrac{x}{2}=6-y=z-6\) and a plane \(\alpha\) meets at point \(\mathrm{P}(2,5,7)\) perpendicular to each other. A point \(\mathrm{A}(a,b,c)\) on line \(l\) and a point \(\mathrm{Q}\) on plane \(\alpha\) satisfy \(\overrightarrow{\mathrm{AP}}\cdot \overrightarrow{\mathrm{AQ}}=6\). What is the value of \(a+b+c\)?
(※ \(a>0\)) [4 points]
  1. \(15\)
  2. \(16\)
  3. \(17\)
  4. \(18\)
  5. \(19\)
As the figure shows, there is an isosceles triangle \(\mathrm{ABC}\) with an inscribed circle with radius \(1\), where \(\angle \mathrm{CAB}=\angle \mathrm{BCA}=\theta\). Let \(\mathrm{D}\) be a point on line \(\mathrm{AB}\) that is not point \(\mathrm{A}\), such that \(\angle \mathrm{DCB}=\theta\). Let \(S(\theta)\) be the area of the triangle \(\mathrm{BDC}\). What is the value of \(\displaystyle\lim_{\theta\to+0}\{\theta\times S(\theta)\}\)? (※ \(0< \theta < \dfrac{\pi}{4}\)) [4 points]
  1. \(\dfrac{2}{3}\)
  2. \(\dfrac{8}{9}\)
  3. \(\dfrac{10}{9}\)
  4. \(\dfrac{4}{3}\)
  5. \(\dfrac{14}{9}\)

Mathematics (Type B)

For a positive integer \(n\), let \(a_n\) be the smallest positive integer \(m\) that satisfies the following. What is the value of \(\displaystyle\sum_{n=1}^{10}a_n\)? [4 points]
  1. Point \(\mathrm{A}\) has coordinates \((2^n,0)\).
  2. Among the points on a line passing through two points \(\mathrm{B}(1,0)\) and \(\mathrm{C}(2^m,m)\), let \(\mathrm{D}\) be the point with an \(x\)-coordinate of \(2^n\). Then the area of the triangle \(\mathrm{ABD}\) is \(\dfrac{m}{2}\) or less.
  1. \(109\)
  2. \(111\)
  3. \(113\)
  4. \(115\)
  5. \(117\)
Short Answer Questions
Compute the solution to the equation \(\log_2(x+6)=5\). [3 points]
For the function \(f(x)=\cos x+4e^{2x}\), compute \(f'(0)\). [3 points]

Mathematics (Type B)

Let \(k\) be the product of all real solutions to the equation \(x^2-6x-\sqrt{x^2-6x-1}=3\). Compute \(k^2\). [3 points]
Upon compressing a digital image, let \(P\) be the peak signal-to-noise ratio, which is an indicator of the difference between the original and compressed image, and let \(E\) be the mean square error between the original and compressed image. Then the following relation is known.
\(P=20\log_{10} 255-10\log_{10} E\quad(E>0)\)
Upon compressing two original images \(A\) and \(B\), let \(P_A\) and \(P_B\) be their peak signal-to-noise ratio respectively, and let \(E_A\) and \(E_B\) (\(E_A>0, E_B>0\)) be their mean square error respectively. Given that \(E_B=100E_A\), compute \(P_A-P_B\). [3 points]
Compute the number of all \(3\)-tuples \((a,b,c)\) where \(a, b\) and \(c\) are positive integers that satisfy the following. [4 points]
  1. \(a\times b\times c\) is odd.
  2. \(a\leq b\leq c\leq 20\)

Mathematics (Type B)

Among the two foci of the ellipse \(\dfrac{x^2}{9}+ \dfrac{y^2}{4}=1\), let \(\mathrm{F}\) be the point with a positive \(x\)-coordinate, and \(\mathrm{F'}\) be the point with a negative \(x\)-coordinate. Let \(\mathrm{P}\) be a point on this ellipse in the \(1\)st quadrant such that \(\angle \mathrm{FPF'}=\dfrac{\pi}{2}\), and let \(\mathrm{Q}\) be a point on line \(\mathrm{FP}\) with a positive \(y\)-coordinate such that \(\overline{\mathrm{FQ}}=6\). Compute the area of the triangle \(\mathrm{QF'F}\). [4 points]
Consider a positive number \(a\) for which the function \(f(x)=\displaystyle\int_{0}^{x}(a-t)e^t dt\) has a global maximum value of \(32\). Compute the area of the region enclosed by the curve \(y=3e^x\) and two lines \(x=a\) and \(y=3\). [4 points]

Mathematics (Type B)

Consider the sphere \(S: x^2+y^2+z^2=50\) and the point \(\mathrm{P}(0,5,5)\) in \(3\)-dimensional space. Among all circles \(C\) that satisfy the following, the maximum area of the projection of \(C\) onto the \(xy\)-plane is \(\dfrac{q}{p}\pi\). Compute \(p+q\). (※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]
  1. Circle \(C\) is a cross section of the sphere \(S\) and a plane passing through point \(\mathrm{P}\).
  2. Circle \(C\) has a radius of \(1\).
For the function \(f(x)=e^{x+1}-1\) and for a positive integer \(n\), let us define a function \(g(x)\) as
\(g(x)=100\,|\,f(x)\,|- \displaystyle\sum_{k=1}^{n}|\,f(x^k)\,|\).
Compute the sum of all positive integers \(n\) for which \(g(x)\) is differentiable on the set of all real numbers. [4 points]