2018 College Scholastic Ability Test

Mathematics (Type Ga)

Multiple Choice Questions
For vectors \(\vec{a}=(3,-1)\) and \(\vec{b}=(1,2)\), what is the sum of all components of the vector \(\vec{a}+\vec{b}\)? [2 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
What is the value of \(\displaystyle\lim_{x\;\!\to\;\!0} \dfrac{\ln(1+5x)}{e^{2x}-1}\)? [2 points]
  1. \(1\)
  2. \(\dfrac{3}{2}\)
  3. \(2\)
  4. \(\dfrac{5}{2}\)
  5. \(3\)
For points \(\mathrm{A}(1, 6, 4)\) and \(\mathrm{B}(a, 2, -4)\) in
\(3\)-dimensional space, the point internally dividing the line segment \(\mathrm{AB}\) in the ratio \(1:3\) has coordinates \((2,5,2)\). What is the value of \(a\)? [2 points]
  1. \(1\)
  2. \(3\)
  3. \(5\)
  4. \(7\)
  5. \(9\)
Consider events \(A\) and \(B\) that are independent. Given that
\(\mathrm{P}(A) = \dfrac{2}{3}\, \) and \(\, \mathrm{P}(A \cup B) = \dfrac{5}{6}\),
what is the value of \(\mathrm{P}(B)\)? [3 points]
  1. \(\dfrac{1}{3}\)
  2. \(\dfrac{5}{12}\)
  3. \(\dfrac{1}{2}\)
  4. \(\dfrac{7}{12}\)
  5. \(\dfrac{2}{3}\)

Mathematics (Type Ga)

What is the maximum value of the function \(f(x)=1+\left(\!\dfrac{1}{3}\!\right)^{\!x-1}\) on the closed interval \([1,3]\)? [3 points]
  1. \(\dfrac{5}{3}\)
  2. \(2\)
  3. \(\dfrac{7}{3}\)
  4. \(\dfrac{8}{3}\)
  5. \(3\)
In the expansion of \(\left(\!x+\dfrac{2}{x}\!\right)^{\!8}\), what is the coefficient of \(x^4\)? [3 points]
  1. \(108\)
  2. \(112\)
  3. \(116\)
  4. \(120\)
  5. \(124\)
For \(0\leq x<2\pi\), what is the sum of all solutions to the following equation? [3 points]
\(\cos^2x=\sin^2x-\sin x\)
  1. \(2\pi\)
  2. \(\dfrac{5}{2}\pi\)
  3. \(3\pi\)
  4. \(\dfrac{7}{2}\pi\)
  5. \(4\pi\)

Mathematics (Type Ga)

Given that the ellipse \(\dfrac{(x-2)^2}{a}+\dfrac{(y-2)^2}{4}=1\) has two foci with coordinates \((6,b)\) and \((-2,b)\), what is the value of \(ab\)? (※ \(a\) is a positive number.) [3 points]
  1. \(40\)
  2. \(42\)
  3. \(44\)
  4. \(46\)
  5. \(48\)
For a function \(f(x)\) differentiable on the set of all real numbers, let us define the function \(g(x)\) as
\(g(x)=\dfrac{f(x)}{e^{x-2}}\).
Given that \(\displaystyle\lim_{x\;\!\to\;\!2}\frac{f(x)-3}{x-2}=5\), what is the value of \(g'(2)\)? [3 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
The weight of a cosmetic product manufactured in some company follows a normal distribution with a mean of \(201.5\text{g}\) and a standard deviation of \(1.8\text{g}\). What is the probability that \(9\) products randomly
sampled from the cosmetic products manufactured in this company, have a sample mean of \(200\text{g}\) or more, computed with 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. \(0.7745\)
  2. \(0.8413\)
  3. \(0.9332\)
  4. \(0.9772\)
  5. \(0.9938\)

Mathematics (Type Ga)

Consider two functions \(f(x)\) \(g(x)\) differentiable on the set of all real numbers. \(f(x)\) is an inverse of \(g(x)\), and \(f(1)=2\) and \(f'(1)=3\). For the function \(h(x)=xg(x)\), what is the value of \(h'(2)\)? [3 points]
  1. \(1\)
  2. \(\dfrac{4}{3}\)
  3. \(\dfrac{5}{3}\)
  4. \(2\)
  5. \(\dfrac{7}{3}\)
Let \(A\) be the region enclosed by the curve \(y=e^{2x}\), the \(y\)-axis, and the line \(y=-2x+a\). Let \(B\) be the region enclosed by the curve \(y=e^{2x}\) and two lines \(y=-2x+a\) and \(x=1\). If the areas of \(A\) and \(B\) are the same, what is the value of the constant \(a\)?
(※ \(1<a<e^2\)) [3 points]
  1. \(\dfrac{e^2+1}{2}\)
  2. \(\dfrac{2e^2+1}{4}\)
  3. \(\dfrac{e^2}{2}\)
  4. \(\dfrac{2e^2-1}{4}\)
  5. \(\dfrac{e^2-1}{2}\)

Mathematics (Type Ga)

Suppose we throw a die twice. Given that it never landed on \(6\), what is the probability that the sum of the two numbers it landed on is a multiple of \(4\)? [3 points]
  1. \(\dfrac{4}{25}\)
  2. \(\dfrac{1}{5}\)
  3. \(\dfrac{6}{25}\)
  4. \(\dfrac{7}{25}\)
  5. \(\dfrac{8}{25}\)
As the figure shows, for a triangle \(\mathrm{ABC}\) with \(\overline{\mathrm{AB}}=5\) and \(\overline{\mathrm{AC}}=2\sqrt{5}\), let \(\mathrm{D}\) be the perpendicular foot from point \(\mathrm{A}\) to the line segment \(\mathrm{BC}\). A point \(\mathrm{E}\) internally dividing the line segment \(\mathrm{AD}\) in the ratio \(3:1\) satisfies \(\overline{\mathrm{EC}}=\sqrt{5}\). Given that \(\angle\mathrm{ABD}=\alpha\) and \(\angle\mathrm{DCE}=\beta\), what is the value of \(\cos(\alpha-\beta)\)? [4 points]
  1. \(\dfrac{\sqrt{5}}{5}\)
  2. \(\dfrac{\sqrt{5}}{4}\)
  3. \(\dfrac{3\sqrt{5}}{10}\)
  4. \(\dfrac{7\sqrt{5}}{20}\)
  5. \(\dfrac{2\sqrt{5}}{5}\)

Mathematics (Type Ga)

For the function
\(f(x)=\displaystyle\int_0^x\frac{1}{1+e^{-t}}dt\),
what is the value of the real number \(a\) satisfying \((f\circ f)(a)=\ln5\)? [4 points]
  1. \(\ln11\)
  2. \(\ln13\)
  3. \(\ln15\)
  4. \(\ln17\)
  5. \(\ln19\)
Suppose a point \(\mathrm{P}\) is moving on the \(xy\)-plane and its position \(\mathrm{P}(x,y)\) at time \(t\,(0< t<\pi)\) is
\(x=\sqrt{3}\sin t\:\) and \(\:y=2\cos t-5\).
Given that the velocity \(\overrightarrow{v}\) of point \(\mathrm{P}\) is parallel to \(\overrightarrow{\mathrm{OP}}\) at time \(t=\alpha\,(0<\alpha<\pi)\), what is the value of \(\cos\alpha\)? (※ \(\mathrm{O}\) is the origin.) [4 points]
  1. \(\dfrac{1}{10}\)
  2. \(\dfrac{1}{5}\)
  3. \(\dfrac{3}{10}\)
  4. \(\dfrac{2}{5}\)
  5. \(\dfrac{1}{2}\)

Mathematics (Type Ga)

As the figure shows, there is a rhombus \(\mathrm{ABCD}\) with side lengths of \(1\). Let \(\mathrm{E}\) be the perpendicular foot from point \(\mathrm{C}\) to line \(\mathrm{AB}\), let \(\mathrm{F}\) be the perpendicular foot from point \(\mathrm{E}\) to line \(\mathrm{AC}\), and let \(\mathrm{G}\) be the intersection of lines \(\mathrm{EF}\) and \(\mathrm{BC}\). For \(\angle\mathrm{DAB}=\theta\), let \(S(\theta)\) be the area of the triangle \(\mathrm{CFG}\).
What is the value of \(\displaystyle\lim_{\theta\;\!\to\;\!0+}\frac{S(\theta)}{\theta^5}\)? (※ \(0<\theta<\dfrac{\pi}{2}\)) [4 points]
  1. \(\dfrac{1}{24}\)
  2. \(\dfrac{1}{20}\)
  3. \(\dfrac{1}{16}\)
  4. \(\dfrac{1}{12}\)
  5. \(\dfrac{1}{8}\)
Suppose we put \(4\) distinct balls into \(4\) distinct boxes without leaving any ball outside the boxes. What is the number of cases where a box with exactly \(1\) ball exists? (※ There can be a box without any balls.) [4 points]
  1. \(220\)
  2. \(216\)
  3. \(212\)
  4. \(208\)
  5. \(204\)

Mathematics (Type Ga)

There are \(6\) coins that weigh \(1\), \(3\) coins that weigh \(2\), and an empty sack. Let us perform the following trial using a die. (※ The unit of weight is \(\text{g}\).)
Throw the die once. If it lands on \(2\) or less, put a coin with weight \(1\) in the sack. If it lands on \(3\) or more, put a coin with weight \(2\) in the sack.
Let us repeat the trial above until the total weight in the sack becomes \(6\) or more for the first time. Let the random variable \(X\) be the number of coins in the sack at this point. The following is a process computing \(\mathrm{P}(X=x)\,(x=3,4,5,6)\).
  1. The event \(X=3\) happens if the sack has \(3\) coins with weight \(2\). Therefore
    \(\mathrm{P}(X=3)=\fbox{\(\;(\alpha)\;\)}\)
  2. The event \(X=4\) happens
    if the total weight after the \(3\)rd trial is \(4\) and a coin with weight \(2\) is put in the \(4\)th trial,
    or if the total weight after the \(3\)rd trial is \(5\). Therefore
    \(\mathrm{P}(X=4)=\fbox{\(\;(\beta)\;\)}+\,_3\mathrm{C}_1\left(\!\dfrac{1}{3}\!\right)^{\!\!1}\!\left(\!\dfrac{2}{3}\!\right)^{\!\!2}\)
  3. The event \(X=5\) happens
    if the total weight after the \(4\)th trial is \(4\) and a coin with weight \(2\) is put in the \(5\)th trial,
    or if the total weight after the \(4\)th trial is \(5\). Therefore
    \(\mathrm{P}(X=5)=\,_4\mathrm{C}_4\left(\!\dfrac{1}{3}\!\right)^{\!\!4}\!\left(\!\dfrac{2}{3}\!\right)^{\!\!0}\times\dfrac{2}{3}+\fbox{\(\;(\gamma)\;\)}\)
  4. The event \(X=6\) happens if the total weight after the \(5\)th trial is \(5\). Therefore
    \(\mathrm{P}(X=6)=\left(\!\dfrac{1}{3}\!\right)^{\!\!5}\)
Let \(a\), \(b\) and \(c\) be the correct numbers for \((\alpha)\), \((\beta)\) and \((\gamma)\) respectively. What is the value of \(\dfrac{ab}{c}\)? [4 points]
  1. \(\dfrac{4}{9}\)
  2. \(\dfrac{7}{9}\)
  3. \(\dfrac{10}{9}\)
  4. \(\dfrac{13}{9}\)
  5. \(\dfrac{16}{9}\)
† Original question used weights instead of coins, which was confusing in English.
In \(3\)-dimensional space, consider three points \(\mathrm{A,B}\) and \(\mathrm{C}\) that are not on a line. For a plane \(\alpha\) that satisfies the following, let \(d(\alpha)\) be the smallest among the three distances from points \(\mathrm{A,B}\) and \(\mathrm{C}\) to plane \(\alpha\) respectively.
  1. Plane \(\alpha\) meets line segments \(\mathrm{AC}\) and \(\mathrm{BC}\).
  2. Plane \(\alpha\) does not meet line segment \(\mathrm{AB}\).
Among all planes \(\alpha\) that satisfy the above, let \(\beta\) be the plane with the greatest value of \(d(\alpha)\). What is the list of correct statements in the <List>? [4 points]
  1. Plane \(\beta\) is perpendicular to the plane that passes through points \(\mathrm{A, B}\) and \(\mathrm{C}\).
  2. Plane \(\beta\) passes through either the midpoint of the line segment \(\mathrm{AC}\) or the midpoint of the line segment \(\mathrm{BC}\) (or both).
  3. For points \(\mathrm{A}(2,3,0), \mathrm{B}(0,1,0)\) and \(\mathrm{C}(2,-1,0)\), \(d(\beta)\) is equal to the distance between point \(\mathrm{B}\) and plane \(\beta\).
  1. a
  2. c
  3. a, b
  4. b, c
  5. a, b, c

Mathematics (Type Ga)

For positive numbers \(t\), let us define a function \(f(x)\) on the interval \([1,\infty)\) as
\(f(x)=\begin{cases} \ln x &\; (1\leq x<e)\\\\ -t+\ln x &\; (x\geq e), \end{cases}\)
and let \(h(t)\) be the minimum value of the slope of the line \(y=g(x)\) where \(g(x)\) is a linear function that satisfies the following.
\((x-e)\{g(x)-f(x)\}\geq 0\) for all real numbers \(x\) greater than or equal to \(1\).
\(h(t)\) is a differentiable function, and a positive number \(a\) satisfies \(h(a)=\dfrac{1}{e+2}\). What is the value of \(h'\!\left(\!\dfrac{1}{2e}\!\right)\times h'(a)\)? [4 points]
  1. \(\dfrac{1}{(e+1)^2}\)
  2. \(\dfrac{1}{e(e+1)}\)
  3. \(\dfrac{1}{e^2}\)
  4. \(\dfrac{1}{(e-1)(e+1)}\)
  5. \(\dfrac{1}{e(e-1)}\)
Short Answer Questions
Compute \(_5\mathrm{C}_3\). [3 points]
For the function \(f(x)=\ln(x^2+1)\), compute \(f'(1)\). [3 points]

Mathematics (Type Ga)

Compute the slope of the tangent line to the curve \(2x+x^2y-y^3=2\) at point \((1,1)\). [3 points]
On the \(xy\)-plane, a line passes through the point \((4,1)\) and is perpendicular to the vector \(\overrightarrow{n}=(1,2)\). The coordinates of points where this line meets the
\(x\)-axis and the \(y\)-axis are \((a,0)\) and \((0,b)\) respectively. Compute \(a+b\). [3 points]
A random variable \(X\) follows a normal distribution with a mean of \(m\) and a standard deviation of \(\sigma\), and satisfies
\(\mathrm{P}(X\leq3)=\mathrm{P}(3\leq X\leq80)=0.3\).
Compute \(m+\sigma\).
(※ For a random variable \(Z\) that follows the standard normal distribution, suppose \(\mathrm{P}(0\leq Z\leq 0.25) = 0.1\) and \(\mathrm{P}(0\leq Z\leq 0.52) = 0.2\).) [4 points]

Mathematics (Type Ga)

As the figure shows, a point \(\mathrm{P}\) is on the hyperbola \(\dfrac{x^2}{8}-\dfrac{y^2}{17}=1\) with two foci \(\mathrm{F}\) and \(\mathrm{F'}\), and a circle \(C\) whose center is on the \(y\)-axis is tangent to line \(\mathrm{FP}\) and line \(\mathrm{F'P}\) at the same time. Let \(\mathrm{Q}\) be the point of tangency between line \(\mathrm{F'P}\) and circle \(C\).
Given that \(\overline{\mathrm{F'Q}}=5\sqrt{2}\), compute \(\overline{\mathrm{FP}}^2+\overline{\mathrm{F'P}}^2\).
(※ \(\overline{\mathrm{F'P}}<\overline{\mathrm{FP}}\)) [4 points]
Let us randomly select one out of all \(3\)-tuples \((x,y,z)\) where \(x,y\) and \(z\) are nonnegative integers satisfying \(x+y+z=10\). The probability that the selected \(3\)-tuple \((x,y,z)\) satisfies \((x-y)(y-z)(z-x)\ne0\) is \(\dfrac{q}{p}\). Compute \(p+q\).
(※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]

Mathematics (Type Ga)

In \(3\)-dimensional space, consider a circle \(C\) which is the intersection of the sphere \(x^2+y^2+z^2=6\) and the plane \(x+2z-5=0\). Let \(\mathrm{P}\) be the point on circle \(C\) with the smallest \(y\)-coordinate, and let \(\mathrm{Q}\) be the perpendicular foot from point \(\mathrm{P}\) to the \(xy\)-plane.
For a point \(\mathrm{X}\) moving on the circle \(C\), the maximum value of \(\big|\overrightarrow{\mathrm{PX}}+\overrightarrow{\mathrm{QX}}\big|^2\) is \(a+b\sqrt{30}\).
Compute \(10(a+b)\). (※ \(a\) and \(b\) are rational numbers.) [4 points]
For real numbers \(t\), let \(f(x)\) be
\(f(x)=\begin{cases} 1-|x-t| & (|x-t|\leq1)\\\\ \qquad 0 & (|x-t|>1). \end{cases}\)
For some odd number \(k\), the function
\(g(t)=\displaystyle\int_k^{k+8}\!f(x)\cos(\pi x)dx\)
satisfies the following.
Let us list all \(\alpha\) for which the function \(g(t)\) has a local minimum at \(t=\alpha\) and \(g(\alpha)<0\).
Let \(\alpha_1,\alpha_2,\cdots,\alpha_m\) (\(m\) is an integer) be this list in ascending order. Then \(\displaystyle\sum_{i\;\!=\;\!1}^m\alpha_i=45\).
Compute \(k-\pi^2\displaystyle\sum_{i\;\!=\;\!1}^mg(\alpha_i)\). [4 points]