2017 College Scholastic Ability Test

Mathematics (Type Ga)

Multiple Choice Questions
For vectors \(\vec{a}=(1,3)\) and \(\vec{b}=(5,-6)\), 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 \(\displaystyle\lim_{x\to\,0} \dfrac{e^{6x}-1}{\ln(1+3x)}\)? [2 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
What is the value of \(\displaystyle\int_{0}^{^\pi/_2}2\sin x\,dx\)? [2 points]
  1. \(0\)
  2. \(\dfrac{1}{2}\)
  3. \(1\)
  4. \(\dfrac{3}{2}\)
  5. \(2\)
Consider two events \(A\) and \(B\) that are independent. Given that
\(\mathrm{P}(B^C) = \dfrac{1}{3}\, \) and \(\, \mathrm{P}(A | B) = \dfrac{1}{2}\),
what is the value of \(\mathrm{P}(A)\mathrm{P}(B)\)?
(※ \(B^C\) is the complement of \(B\).) [3 points]
  1. \(\dfrac{5}{6}\)
  2. \(\dfrac{2}{3}\)
  3. \(\dfrac{1}{2}\)
  4. \(\dfrac{1}{3}\)
  5. \(\dfrac{1}{6}\)

Mathematics (Type Ga)

Let us select four numbers among numbers \(1,2,3,4\) and \(5\), where each number may be selected multiple times, and arrange them in a line to make a \(4\)-digit integer. What is the number of all \(4\)-digit integers that are a multiple of \(5\)? [3 points]
  1. \(115\)
  2. \(120\)
  3. \(125\)
  4. \(130\)
  5. \(135\)
For the function \(f(x)=x^3+x+1\) and its inverse \(g(x)\), what is the value of \(g'(1)\)? [3 points]
  1. \(\dfrac{1}{3}\)
  2. \(\dfrac{2}{5}\)
  3. \(\dfrac{2}{3}\)
  4. \(\dfrac{4}{5}\)
  5. \(1\)
Suppose we throw a die \(3\) times. What is the probability that it lands on \(4\) exactly once? [3 points]
  1. \(\dfrac{25}{72}\)
  2. \(\dfrac{13}{36}\)
  3. \(\dfrac{3}{8}\)
  4. \(\dfrac{7}{18}\)
  5. \(\dfrac{29}{72}\)

Mathematics (Type Ga)

For points \(\mathrm{A}(1,a,-6)\) and \(\mathrm{B}(-3,2,b)\) in \(3\)-dimensional space, the point externally dividing the line segment \(\mathrm{AB}\) in the ratio \(3:2\) is on the \(x\)-axis. What is the value of \(a+b\)? [3 points]
  1. \(-1\)
  2. \(-2\)
  3. \(-3\)
  4. \(-4\)
  5. \(-5\)
What is the value of \(\displaystyle\int_{1}^{e}\ln \dfrac{x}{e}dx\)? [3 points]
  1. \(\dfrac{1}{e}-1\)
  2. \(2-e\)
  3. \(\dfrac{1}{e}-2\)
  4. \(1-e\)
  5. \(\dfrac{1}{2}-e\)
Suppose a point \(\mathrm{P}\) is moving on the \(xy\)-plane and its position \((x,y)\) at time \(t\,(t>0)\) is
\(x=t- \dfrac{2}{t}\:\) and \(\:y=2t+ \dfrac{1}{t}\).
What is the speed of point \(\mathrm{P}\) at time \(t=1\)? [3 points]
  1. \(2\sqrt{2}\)
  2. \(3\)
  3. \(\sqrt{10}\)
  4. \(\sqrt{11}\)
  5. \(2\sqrt{3}\)

Mathematics (Type Ga)

As the figure shows, there is a \(3\)-dimensional solid where one of its faces is the region enclosed by the curve \(y=\sqrt{x}+1\), the \(x\)-axis, the \(y\)-axis, and the line \(x=1\). Suppose a cross section of this solid and any plane perpendicular to the \(x\)-axis is always a square. What is the volume of this solid? [3 points]
  1. \(\dfrac{7}{3}\)
  2. \(\dfrac{5}{2}\)
  3. \(\dfrac{8}{3}\)
  4. \(\dfrac{17}{6}\)
  5. \(3\)
In \(3\)-dimensional space, let \(\theta\) be the acute angle between the plane \(2x+2y-z+5=0\) and the
\(xy\)-plane. What is the value of \(\cos\theta\)? [3 points]
  1. \(\dfrac{1}{12}\)
  2. \(\dfrac{1}{6}\)
  3. \(\dfrac{1}{4}\)
  4. \(\dfrac{1}{3}\)
  5. \(\dfrac{5}{12}\)

Mathematics (Type Ga)

Let \(\overline{X}\) be the sample mean of a random sample of size \(9\) from a population following the normal distribution \(\mathrm{N}(0,4^2)\). Let \(\overline{Y}\) be the sample mean of a random sample of size \(16\) from a population following the normal distribution \(\mathrm{N}(3,2^2)\). What is the value of the constant \(a\) that satisfies \(\mathrm{P}\big(\overline{X}\geq 1\big)=\mathrm{P}\big(\overline{Y}\leq a\big)\)? [3 points]
  1. \(\dfrac{19}{8}\)
  2. \(\dfrac{5}{2}\)
  3. \(\dfrac{21}{8}\)
  4. \(\dfrac{11}{4}\)
  5. \(\dfrac{23}{8}\)
As the figure shows, there is a sector \(\mathrm{OAB}\) with radius \(1\) and central angle \(\dfrac{\pi}{2}\). For a point \(\mathrm{P}\) on arc \(\mathrm{AB}\), let \(\mathrm{H}\) be the perpendicular foot from point \(\mathrm{P}\) to line \(\mathrm{OA}\), and let \(\mathrm{Q}\) be the intersection of lines \(\mathrm{PH}\) and \(\mathrm{AB}\). For \(\angle \mathrm{POH}=\theta\), let \(S(\theta)\) be the area of the triangle \(\mathrm{AQH}\). What is the value of \(\displaystyle\lim_{\theta\to\,0+}\!\frac{S(\theta)}{\theta^4}\)?
(※ \(0<\theta<\dfrac{\pi}{2}\)) [4 points]
  1. \(\dfrac{1}{8}\)
  2. \(\dfrac{1}{4}\)
  3. \(\dfrac{3}{8}\)
  4. \(\dfrac{1}{2}\)
  5. \(\dfrac{5}{8}\)

Mathematics (Type Ga)

For a point \(\mathrm{P}(t,2e^{-t})\) on the curve \(y=2e^{-x}\), let \(\mathrm{A}\) be the perpendicular foot from point \(\mathrm{P}\) to the \(y\)-axis, and let \(\mathrm{B}\) be the point where the tangent line at point \(\mathrm{P}\) meets the \(y\)-axis. What is the value of \(t\) for which the area of the triangle \(\mathrm{APB}\) is the greatest? [4 points]
  1. \(1\)
  2. \(\dfrac{e}{2}\)
  3. \(\sqrt{2}\)
  4. \(2\)
  5. \(e\)
In \(3\)-dimensional space, the position of three points \(\mathrm{A,B}\) and \(\mathrm{C}\) are represented by vectors \(\vec{a}, \vec{b}\) and \(\vec{c}\) respectively. The inner products between these vectors are shown in the following table.
\(\cdot\) \(\vec{a}\) \(\vec{b}\) \(\vec{c}\)
\(\vec{a}\) \(2\) \(1\) \(-\sqrt{2}\)
\(\vec{b}\) \(1\) \(2\) \(0\)
\(\vec{c}\) \(-\sqrt{2}\) \(0\) \(2\)
For example, \(\vec{a}\cdot\vec{c}=-\sqrt{2}\). For points \(\mathrm{A, B}\) and \(\mathrm{C}\), which option below is correct about the distances between each pair of points? [4 points]
  1. \(\overline{\mathrm{AB}}<\overline{\mathrm{AC}}<\overline{\mathrm{BC}}\)
  2. \(\overline{\mathrm{AB}}<\overline{\mathrm{BC}}<\overline{\mathrm{AC}}\)
  3. \(\overline{\mathrm{AC}}<\overline{\mathrm{AB}}<\overline{\mathrm{BC}}\)
  4. \(\overline{\mathrm{BC}}<\overline{\mathrm{AB}}<\overline{\mathrm{AC}}\)
  5. \(\overline{\mathrm{BC}}<\overline{\mathrm{AC}}<\overline{\mathrm{AB}}\)

Mathematics (Type Ga)

On the \(xy\)-plane, a jump is defined as moving from a point \((x,y)\) to one of three points \((x+1,y)\), \((x,y+1)\), or \((x+1,y+1)\).
Suppose we move from point \((0,0)\) to point \((4,3)\) by performing jumps repeatedly. Among all possible cases of doing such, let the random variable \(X\) be the number of jumps in a randomly selected case. The following is a process computing \(\mathrm{E}(X)\). (※ The probability of selecting each case is the same.)
Let \(N\) be the number of all cases of moving from point \((0,0)\) to point \((4,3)\) by performing jumps repeatedly. Let \(k\) be the smallest value that the random variable \(X\) can have. Then \(k=\fbox{\(\;(\alpha)\;\)}\), and the largest value that can be taken is \(k+3\).
\(\mathrm{P}(X=k)=\dfrac{1}{N}\times \dfrac{4!}{3!}=\dfrac{4}{N}\)
\(\mathrm{P}(X=k+1)=\dfrac{1}{N}\times \dfrac{5!}{2!2!}=\dfrac{30}{N}\) \(\mathrm{P}(X=k+2)=\dfrac{1}{N}\times \fbox{\(\;(\beta)\;\)}\) \(\mathrm{P}(X=k+3)=\dfrac{1}{N}\times \dfrac{7!}{3!4!}=\dfrac{35}{N}\)
and
\(\displaystyle\sum_{i\,=\,k}^{k+3}\mathrm{P}(X=i)=1\),
therefore \(N=\fbox{\(\;(\gamma)\;\)}\).
Therefore the mean of the random variable \(X\) is
\(\mathrm{E}(X)=\displaystyle\sum_{i\,=\,k}^{k+3}\{i\times \mathrm{P}(X=i)\}=\dfrac{257}{43}\).
Let \(a, b\) and \(c\) be the correct number for \((\alpha), (\beta)\) and \((\gamma)\) respectively. What is the value of \(a+b+c\)? [4 points]
  1. \(190\)
  2. \(193\)
  3. \(196\)
  4. \(199\)
  5. \(202\)
A random variable \(X\) follows a normal distribution with a mean of \(m\) and a standard deviation of \(5\), and its probability density function \(f(x)\) satisfies the following.
  1. \(f(10)>f(20)\)
  2. \(f(4)<f(22)\)
Given that \(m\) is an integer, what is the value of \(\mathrm{P}(17\leq X\leq 18)\), computed using the standard normal table to the right? [4 points]
\(z\)\(\mathrm{P}(0\!\leq\! Z \!\leq\!z)\)
\(0.6\)\(0.226\)
\(0.8\)\(0.288\)
\(1.0\)\(0.341\)
\(1.2\)\(0.385\)
\(1.4\)\(0.419\)
  1. \(0.044\)
  2. \(0.053\)
  3. \(0.062\)
  4. \(0.078\)
  5. \(0.097\)

Mathematics (Type Ga)

For positive numbers \(k\) and \(p\), consider two tangent lines to the parabola \(y^2=4px\) that passes through point \(\mathrm{A}(-k,0)\). Let \(\mathrm{F}\) and \(\mathrm{F'}\) be the points where the two lines meet the \(y\)-axis respectively, and let \(\mathrm{P}\) and \(\mathrm{Q}\) be the points where the two lines meet the parabola respectively. It is given that \(\angle \mathrm{PAQ}=\dfrac{\pi}{3}\), and an ellipse with points \(\mathrm{F}\) and \(\mathrm{F'}\) as the foci which passes through points \(\mathrm{P}\) and \(\mathrm{Q}\) has a major axis with a length of \(4\sqrt{3}+12\). What is the value of \(k+p\)? [4 points]
  1. \(8\)
  2. \(10\)
  3. \(12\)
  4. \(14\)
  5. \(16\)
For the function \(f(x)=e^{-x}\!\displaystyle\int_{0}^{x}\!\sin(t^2)dt\), which option only contains every correct statement in the <List>? [4 points]
  1. \(f(\sqrt{\pi})>0\)
  2. There exists a number \(a\) in the open interval \((0, \sqrt{\pi})\) that satisfies \(f'(a)>0\).
  3. There exists a number \(b\) in the open interval \((0, \sqrt{\pi})\) that satisfies \(f'(b)=0\).
  1. a
  2. c
  3. a, b
  4. b, c
  5. a, b, c

Mathematics (Type Ga)

A continuous function \(f(x)\) strictly increases on the closed interval \([0,1]\) and satisfies
\(\displaystyle\int_{0}^{1}\!f(x)dx=2\:\) and \(\:\displaystyle\int_{0}^{1}\!\big|f(x)\big|dx=2\sqrt{2}\).
For the function \(F(x)\) defined as
\(F(x)=\displaystyle\int_{0}^{x}\big|f(t)\big|dt\quad(0\leq x\leq1)\),
What is the value of \(\displaystyle\int_{0}^{1}f(x)F(x)dx\)? [4 points]
  1. \(4-\sqrt{2}\)
  2. \(2+\sqrt{2}\)
  3. \(5-\sqrt{2}\)
  4. \(1+2\sqrt{2}\)
  5. \(2+2\sqrt{2}\)
Short Answer Questions
Compute \(_4\mathrm{H}_2\). [3 points]
Compute the sum of all positive integers \(x\) that satisfy \(\left(\!\dfrac{1}{2}\!\right)^{\!x-5}\geq4\). [3 points]

Mathematics (Type Ga)

In \(3\)-dimensional space, compute the sum of all real numbers \(k\) for which the plane \(x+8y-4z+k=0\) is tangent to the sphere \(x^2+y^2+z^2+2y-3=0\). [3 points]
For \(0<x<2\pi\), the sum of all real solutions to the equation \(\cos^2 x-\sin x=1\) is \(\dfrac{q}{p}\pi\). Compute \(p+q\).
(※ \(p\) and \(q\) are positive integers that are coprime.) [3 points]
There are two sacks \(\mathrm{A}\) and \(\mathrm{B}\) each containing \(4\) cards marked with numbers \(1,2,3\) and \(4\) respectively. Suppose James randomly takes two cards from sack \(\mathrm{A}\), and Mary randomly takes two cards from sack \(\mathrm{B}\). The probability that the sum of James's two cards is equal to the sum of Mary's two cards is \(\dfrac{q}{p}\).
Compute \(p+q\). (※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]

Mathematics (Type Ga)

Compute the number of all \(3\)-tuples \((a,b,c)\) where \(a,b\) and \(c\) are nonnegative integers that satisfy the following. [4 points]
  1. \(a+b+c=7\)
  2. \(2^a\times 4^b\) is a multiple of \(8\).
A hyperbola whose asymptotes are lines \(y=\pm \dfrac{4}{3}x\), and whose foci are \(\mathrm{F}(c,0)\) and \(\mathrm{F'}(-c,0)\;(c>0)\), satisfies the following.
  1. A point \(\mathrm{P}\) on the hyperbola satisfies \(\overline{\mathrm{PF'}}=30\) and \(16\leq\overline{\mathrm{PF}}\leq 20\).
  2. For the vertex \(\mathrm{A}\) with a positive
    \(x\)-coordinate, the length of the line segment \(\mathrm{AF}\) is an integer.
Compute the length of the major axis of this hyperbola. [4 points]

Mathematics (Type Ga)

For a regular tetrahedron \(\mathrm{ABCD}\) with side lengths of \(4\), let \(\mathrm{O}\) be the centroid of the triangle \(\mathrm{ABC}\), and let \(\mathrm{P}\) be the midpoint of the line segment \(\mathrm{AD}\). Consider a point \(\mathrm{Q}\) on face \(\mathrm{BCD}\) of the tetrahedron \(\mathrm{ABCD}\). Given that the vectors \(\overrightarrow{\mathrm{OQ}}\) and \(\overrightarrow{\mathrm{OP}}\) are perpendicular, the maximum value of \(\big|\overrightarrow{\mathrm{PQ}}\big|\) is \(\dfrac{q}{p}\). Compute \(p+q\). (※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]
A function \(f(x)\) defined on \(x>a\), and a quartic function \(g(x)\) with a leading coefficient of \(-1\), satisfy the following. (※ \(a\) is a constant.)
  1. \((x-a)f(x)=g(x)\) for all \(x>a\).
  2. For some real numbers \(\alpha\) and \(\beta\) that are distinct, the function \(f(x)\) has a local maximum of \(M\) at both \(x=\alpha\) and \(x=\beta\).
    (※ \(M>0\))
  3. The number of values \(x\) where the function \(f(x)\) has a local extremum is greater than the number of values \(x\) where the function \(g(x)\) has a local extremum.
Given that \(\beta-\alpha=6\sqrt{3}\), compute the minimum value of \(M\). [4 points]