2020 College Scholastic Ability Test

Mathematics (Type Ga)

Multiple Choice Questions

For two vectors \(\vec{a}=(3,1)\) and \(\vec{b}=(-2,4)\), what is the sum of all components of the vector \(\vec{a}+\dfrac{1}{2}\vec{b}\)? [2 points]

\(1\)
\(2\)
\(3\)
\(4\)
\(5\)

What is the value of \(\displaystyle\lim_{x\to0} \dfrac{6x}{e^{4x}-e^{2x}}\)? [2 points]

\(1\)
\(2\)
\(3\)
\(4\)
\(5\)

A point on the \(y\)-axis whose distance to points \(\mathrm{A}(2,0,1)\) and \(\mathrm{B}(3,2,0)\) are equal, has coordinates of \((0,a,0)\). What is the value of \(a\)? [2 points]

\(1\)
\(2\)
\(3\)
\(4\)
\(5\)

In the expansion of \(\left(\!2x+\dfrac{1}{x^2}\!\right)^{\!4}\), what is the coefficient of \(x\)? [3 points]

\(16\)
\(20\)
\(24\)
\(28\)
\(32\)

Mathematics (Type Ga)

What is the slope of the tangent line to the curve \(x^2-3xy+y^2=x\) at point \((1,0)\) on the curve? [3 points]

\(\dfrac{1}{12}\)
\(\dfrac{1}{6}\)
\(\dfrac{1}{4}\)
\(\dfrac{1}{3}\)
\(\dfrac{5}{12}\)

There is a sack containing \(3\) white balls and \(4\) black balls. Suppose four balls are randomly taken out from this sack at the same time. What is the probability that \(2\) white balls and \(2\) black balls are taken out? [3 points]

\(\dfrac{2}{5}\)
\(\dfrac{16}{35}\)
\(\dfrac{18}{35}\)
\(\dfrac{4}{7}\)
\(\dfrac{22}{35}\)

Given that \(0<x<2\pi\), what is the sum of all values of \(x\) that satisfy the equation \(4\cos^2 x-1=0\) and the inequality \(\sin x\cos x<0\) at the same time? [3 points]

\(2\pi\)
\(\dfrac{7}{3}\pi\)
\(\dfrac{8}{3}\pi\)
\(3\pi\)
\(\dfrac{10}{3}\pi\)

Mathematics (Type Ga)

What is the value of \(\displaystyle\int_e^{e^2}\frac{\ln x-1}{x^2}dx\)? [3 points]

\(\dfrac{e+2}{e^2}\)
\(\dfrac{e+1}{e^2}\)
\(\dfrac{1}{e}\)
\(\dfrac{e-1}{e^2}\)
\(\dfrac{e-2}{e^2}\)

Suppose a point \(\mathrm{P}\) is moving on the \(xy\)-plane and its position \((x,y)\) at time \(t\,\left(\!0< t<\dfrac{\pi}{2}\!\right)\) is

\(x=t+\sin t\cos t\:\) and \(\:y=\tan t\).

What is the minimum value of the speed of point \(\mathrm{P}\) in \(0< t<\dfrac{\pi}{2}\)? [3 points]

\(1\)
\(\sqrt{3}\)
\(2\)
\(2\sqrt{2}\)
\(2\sqrt{3}\)

Consider an isosceles triangle \(\mathrm{ABC}\) where \(\overline{\mathrm{AB}}=\overline{\mathrm{AC}}\), and let \(\angle\mathrm{A}=\alpha\) and \(\angle\mathrm{B}=\beta\). If \(\tan(\alpha+\beta)=-\dfrac{3}{2}\), what is the value of \(\tan\alpha\)? [3 points]

\(\dfrac{21}{10}\)
\(\dfrac{11}{5}\)
\(\dfrac{23}{10}\)
\(\dfrac{12}{5}\)
\(\dfrac{5}{2}\)

Mathematics (Type Ga)

What is the number of integers \(a\) for which the curve \(y=ax^2-2\sin2x\) has an inflection point? [3 points]

\(4\)
\(5\)
\(6\)
\(7\)
\(8\)

As the figure shows, for a positive number \(k\), there is a \(3\)-dimensional solid where one of its faces is the region enclosed by the curve \(y=\sqrt{\dfrac{e^x}{e^x+1}}\), the
\(x\)-axis, the \(y\)-axis, and the line \(x=k\). Suppose a cross section of this solid and any plane perpendicular to the \(x\)-axis is always a square. If the volume of this solid is \(\ln 7\), what is the value of \(k\)? [3 points]

\(\ln11\)
\(\ln13\)
\(\ln15\)
\(\ln17\)
\(\ln19\)

Mathematics (Type Ga)

As the figure shows, let \(\mathrm{A}\) be the point where the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{25}=1\) with two foci \(\mathrm{F}(0,c)\) and \(\mathrm{F'}(0,-c)\), meets the \(x\)-axis with a positive
\(x\)-coordinate. Let \(\mathrm{B}\) be the point where the line \(y=c\) meets the line \(\mathrm{AF'}\), and let \(\mathrm{P}\) be the point where the line \(y=c\) meets the ellipse with a positive
\(x\)-coordinate. Given that the perimeter of triangle \(\mathrm{BPF'}\) and the perimeter of triangle \(\mathrm{BFA}\) have a difference of \(4\), what is the area of triangle \(\mathrm{AFF'}\)?
(※ \(0<a<5\) and \(c>0\).) [3 points]

\(5\sqrt{6}\)
\(\dfrac{9\sqrt{6}}{2}\)
\(4\sqrt{6}\)
\(\dfrac{7\sqrt{6}}{2}\)
\(3\sqrt{6}\)

There is a sack containing \(10\) balls marked with the number \(1\), \(20\) balls marked with the number \(2\), and \(30\) balls marked with the number \(3\). Let us randomly take out a ball from this sack, check the number marked on it, and put it back in. Let the random variable \(Y\) be the sum of \(10\) numbers checked by repeating this trial \(10\) times. The following is a process computing the mean \(\mathrm{E}(Y)\) and variance \(\mathrm{V}(Y)\) of the random variable \(Y\).

Consider the \(60\) balls in the sack as a population. Let the random variable \(X\) be the number marked on a ball randomly taken out from this population. The distribution of \(X\), which is the distribution of the population, is as the table below.

\(X\)	\(1\)	\(2\)	\(3\)	Total
\(\mathrm{P}(X=x)\)	\(\dfrac{1}{6}\)	\(\dfrac{1}{3}\)	\(\dfrac{1}{2}\)	\(1\)

Therefore the mean \(m\) and variance \(\sigma^2\) of the population are

\(m=\mathrm{E}(X)=\dfrac{7}{3}\:\) and \(\:\sigma^2=\mathrm{V}(X)=\fbox{\(\;(\alpha)\;\)}\).

Let \(\overline{X}\) be the mean of a random sample of size \(10\) from this population. Then,

\(\mathrm{E}(\overline{X})=\dfrac{7}{3}\:\) and \(\:\mathrm{V}(\overline{X})=\fbox{\(\;(\beta)\;\)}\).

Let \(X_n\) be the number marked on the \(n\)th ball taken out from the sack. Then,

\(\displaystyle Y=\sum_{n=1}^{10}X_n=10\overline{X}\)

therefore

\(\mathrm{E}(Y)=\dfrac{70}{3}\:\) and \(\:\mathrm{V}(Y)=\fbox{\(\;(\gamma)\;\)}\).

Let \(p, q\) and \(r\) be the correct number for \((\alpha), (\beta)\) and \((\gamma)\) respectively. What is the value of \(p+q+r\)? [4 points]

\(\dfrac{31}{6}\)
\(\dfrac{11}{2}\)
\(\dfrac{35}{6}\)
\(\dfrac{37}{6}\)
\(\dfrac{13}{2}\)

Mathematics (Type Ga)

Let \(\mathrm{A}\) be the point where the graph of the exponential function \(y=a^x\,(a>1)\) meets the line \(y=\sqrt{3}\). For point \(\mathrm{B}(4,0)\), what is the product of all values of \(a\) for which the lines \(\mathrm{OA}\) and \(\mathrm{AB}\) are perpendicular? (※ \(\mathrm{O}\) is the origin.) [4 points]

\(3^{^1/_3}\)
\(3^{^2/_3}\)
\(3\)
\(3^{^4/_3}\)
\(3^{^5/_3}\)

What is the number of \(4\)-tuples \((a,b,c,d)\) where \(a,b,c\) and \(d\) are nonnegative integers that satisfy the following? [4 points]

\(a+b+c-d=9\)
\(d\leq4\) and \(c\geq d\).

\(265\)
\(270\)
\(275\)
\(280\)
\(285\)

Mathematics (Type Ga)

There is an equilateral triangle \(\mathrm{ABC}\) with side lengths of \(10\) on a plane. Let \(\mathrm{P}\) be a point satisfying \(\overline{\mathrm{PB}}-\overline{\mathrm{PC}}=2\), such that the line segment \(\mathrm{PA}\) has the smallest possible length. What is the area of the triangle \(\mathrm{PBC}\)? [4 points]

\(20\sqrt{3}\)
\(21\sqrt{3}\)
\(22\sqrt{3}\)
\(23\sqrt{3}\)
\(24\sqrt{3}\)

A random variable \(X\) follows the normal distribution \(\mathrm{N}(10,2^2)\), and a random variable \(Y\) follows the normal distribution \(\mathrm{N}(m,2^2)\). The probability density functions of \(X\) and \(Y\) are \(f(x)\) and \(g(x)\) respectively.

For values of \(m\) that satisfy

\(f(12)\leq g(20)\),

what is the maximum value of \(\mathrm{P}(21\leq Y\leq 24)\) computed using the standard normal table to the right? [4 points]

\(z\)	\(\mathrm{P}(0\!\leq\! Z \!\leq\!z)\)
\(0.5\)	\(0.1915\)
\(1.0\)	\(0.3413\)
\(1.5\)	\(0.4332\)
\(2.0\)	\(0.4772\)

\(0.5328\)
\(0.6247\)
\(0.7745\)
\(0.8185\)
\(0.9104\)

Mathematics (Type Ga)

Four distinct points \(\mathrm{A,B,C}\) and \(\mathrm{D}\) on a circle satisfies the following. What is the value of \(\big|\overrightarrow{\mathrm{AD}}\big|^2\)? [4 points]

\(\big|\overrightarrow{\mathrm{AB}}\big|=8\:\) and \(\:\overrightarrow{\mathrm{AC}}\cdot\overrightarrow{\mathrm{BC}}=0\).
\(\overrightarrow{\mathrm{AD}}= \dfrac{1}{2}\overrightarrow{\mathrm{AB}}-2\overrightarrow{\mathrm{BC}}\)

\(32\)
\(34\)
\(36\)
\(38\)
\(40\)

Let us throw a coin \(7\) times. What is the probability that the following is satisfied? [4 points]

The coin lands on heads at least \(3\) times.
At some point the coin lands on heads consecutively.

\(\dfrac{11}{16}\)
\(\dfrac{23}{32}\)
\(\dfrac{3}{4}\)
\(\dfrac{25}{32}\)
\(\dfrac{13}{16}\)

Mathematics (Type Ga)

For a real number \(t\), let \(y=f(x)\) be the equation of the tangent line to the curve \(y=e^x\) at point \(\big(t,e^t\big)\). Let \(g(t)\) be the minimum value of a real number \(k\) for which the function \(y=|f(x)+k-\ln x|\) is differentiable on the set of all positive numbers. For two real numbers \(a\) and \(b\,(a<b)\), let \(\displaystyle\int_a^b g(t)dt=m\). Which option only contains every correct statement in the <List>? [4 points]

There exist numbers \(a\) and \(b\,(a<b)\) for which \(m<0\).
If \(g(c)=0\) for a real number \(c\), then \(g(-c)=0\).
If the value of \(m\) is the smallest when \(a=\alpha\) and \(b=\beta\,(\alpha<\beta)\), then \(\dfrac{1+g'(\beta)}{1+g'(\alpha)}<-e^2\).

a
b
a, b
a, c
a, b, c

Short Answer Questions

For the function \(f(x)=x^3\ln x\), compute \(\dfrac{f'(e)}{e^2}\). [3 points]

A random variable \(X\) following the binominal distribution \(\mathrm{B}(80,p)\) satisfies \(\mathrm{E}(X)=20\). Compute \(\mathrm{V}(X)\). [3 points]

Mathematics (Type Ga)

On the \(xy\)-plane, consider a point \(\mathrm{P}(t,\sin t)\) on the curve \(y=\sin x\,(0<t<\pi)\). Let \(C\) be a circle with center \(\mathrm{P}\) that is tangent to the \(x\)-axis. Let \(\mathrm{Q}\) and \(\mathrm{R}\) be points where the circle \(C\) meets the \(x\)-axis and the line segment \(\mathrm{OP}\) respectively. Given that \(\displaystyle\lim_{t\to0+}\frac{\overline{\mathrm{OQ}}}{\overline{\mathrm{OR}}}=a+b\sqrt{2}\), compute \(a+b\).
(※ \(\mathrm{O}\) is the origin. \(a\) and \(b\) are integers.) [3 points]

Let us throw a die \(5\) times, and let \(a\) be the number of times it lands on an odd number. Let us throw a coin \(4\) times, and let \(b\) be the number of times it lands on heads. The probability that \(a-b\) equals \(3\) is \(\dfrac{q}{p}\). Compute \(p+q\). (※ \(p\) and \(q\) are positive integers that are coprime.) [3 points]

For the function \(f(x)=(x^2+2)e^{-x}\), a differentiable function \(g(x)\) satisfies

\(g\!\left(\!\dfrac{x+8}{10}\!\right)=f^{-1}(x)\:\) and \(\:g(1)=0\).

Compute \(\big|g'(1)\big|\). [4 points]

Mathematics (Type Ga)

As the figure shows, there is a piece of paper in the shape of a rhombus \(\mathrm{ABCD}\) with side lengths of \(4\) and \(\angle\mathrm{BAD}=\dfrac{\pi}{3}\). Let \(\mathrm{M}\) and \(\mathrm{N}\) be the midpoints of edges \(\mathrm{BC}\) and \(\mathrm{CD}\) respectively. Suppose we fold the paper along the line segments \(\mathrm{AM}, \mathrm{AN}\) and \(\mathrm{MN}\) to create the tetrahedron \(\mathrm{PAMN}\). The projection of the triangle \(\mathrm{AMN}\) onto plane \(\mathrm{PAM}\) has an area of \(\dfrac{q}{p}\sqrt{3}\). Compute \(p+q\). (※ Ignore the thickness of the paper. Point \(\mathrm{P}\) is where three points \(\mathrm{B, C}\) and \(\mathrm{D}\) meet when the paper is folded. \(p\) and \(q\) are positive integers that are coprime.) [4 points]

Let us select five numbers among numbers \(1,2,3,4,5\) and \(6\), and arrange them in a line to make a \(5\)-digit integer. Each number may be selected multiple times, and the following should be satisfied. Compute the number of all \(5\)-digit integers that can be made. [4 points]

Each odd number is either not selected or selected exactly once.
Each odd number is either not selected or selected exactly twice.

Mathematics (Type Ga)

In \(3\)-dimensional space, for two points \(\mathrm{A}(3,-3,3)\) and \(\mathrm{B}(-2,7,-2)\), let \(\alpha\) and \(\beta\) be the two planes that contain the line \(\mathrm{AB}\) and is tangent to the sphere \(x^2+y^2+z^2=1\). Let \(\mathrm{C}\) and \(\mathrm{D}\) be the point of tangency between the sphere \(x^2+y^2+z^2=1\) and two planes \(\alpha\) and \(\beta\) respectively. The tetrahedron \(\mathrm{ABCD}\) has a volume of \(\dfrac{q}{p}\sqrt{3}\). Compute \(p+q\). (※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]

For a positive real number \(t\), let \(f(t)\) be the value of the real number \(a\) for which the curve \(y=t^3\ln(x-t)\) and the curve \(y=2e^{x-a}\) meet at exactly one point. Compute \(\left\{\!f'\!\left(\!\dfrac{1}{3}\!\right)\!\right\}^{\!2}\). [4 points]