2021 College Scholastic Ability Test

Mathematics (Type Na)

Multiple Choice Questions
What is the value of \(3^0 \times 8^{^2/_3} \)? [2 points]
  1. \(1\)
  2. \(2\)
  3. \(3\)
  4. \(4\)
  5. \(5\)
A geometric progression \(\{a_n\}\) with an initial term of \(\dfrac{1}{8}\) satisfies \(\dfrac{a_3}{a_2}=2\). What is the value of \(a_5\)? [2 points]
  1. \(\dfrac{1}{4}\)
  2. \(\dfrac{1}{2}\)
  3. \(1\)
  4. \(2\)
  5. \(4\)
What is the value of \(\displaystyle\lim_{x\to2}\! \dfrac{x^2+2x-8}{x-2} \)? [2 points]
  1. \(2\)
  2. \(4\)
  3. \(6\)
  4. \(8\)
  5. \(10\)
What is the global maximum value of the function \(f(x)=4\cos x+3\)? [3 points]
  1. \(6\)
  2. \(7\)
  3. \(8\)
  4. \(9\)
  5. \(10\)

Mathematics (Type Na)

Two events \(A\) and \(B\) that are independent satisfy
\(\mathrm{P}(A|B)=\mathrm{P}(B)\:\) and \(\mathrm{P}(A\cap B)=\dfrac{1}{9}\).
What is the value of \(\mathrm{P}(A)\)? [3 points]
  1. \(\dfrac{7}{18}\)
  2. \(\dfrac{1}{3}\)
  3. \(\dfrac{5}{18}\)
  4. \(\dfrac{2}{9}\)
  5. \(\dfrac{1}{6}\)
For the function \(f(x)=x^4+3x-2\), what is the value of \(f'(2)\)? [3 points]
  1. \(35\)
  2. \(37\)
  3. \(39\)
  4. \(41\)
  5. \(43\)
What is the number of positive integers \(x\) that satisfy the inequality \(\left(\!\dfrac{1}{9}\!\right)^{\!x}<3^{21-4x}\)? [3 points]
  1. \(6\)
  2. \(7\)
  3. \(8\)
  4. \(9\)
  5. \(10\)

Mathematics (Type Na)

Let us throw a die three times, and let \(a, b\) and \(c\) be the number it lands on in this order. What is the probability that \(a\times b\times c=4\)? [3 points]
  1. \({\dfrac{1}{54}}\)
  2. \({\dfrac{1}{36}}\)
  3. \({\dfrac{1}{27}}\)
  4. \({\dfrac{5}{108}}\)
  5. \({\dfrac{1}{18}}\)
Consider the tangent line to the curve \(y=x^3-3x^2+2x+2\) at point \(\mathrm{A}(0,2)\). What is the
\(x\)-intercept of a line that passes through point \(\mathrm{A}\) perpendicular to this tangent line? [3 points]
  1. \(4\)
  2. \(6\)
  3. \(8\)
  4. \(10\)
  5. \(12\)
Two sequences \(\{a_n\}\) and \(\{b_n\}\) satisfy
\(\displaystyle\sum_{k=1}^5 a_k=8\) and \(\displaystyle\sum_{k=1}^5 b_k=9\).
What is the value of \(\displaystyle\sum_{k=1}^5 (2a_k-b_k+4)\)? [3 points]
  1. \(19\)
  2. \(21\)
  3. \(23\)
  4. \(25\)
  5. \(27\)

Mathematics (Type Na)

Let \(\overline{X}\) be the mean of a sample of size \(16\) randomly sampled from a population following the normal distribution \(\mathrm{N}(20,5^2)\). What is the value of \(\mathrm{E}(\overline{X})+\sigma(\overline{X})\)? [3 points]
  1. \(\dfrac{91}{4}\)
  2. \(\dfrac{89}{4}\)
  3. \(\dfrac{87}{4}\)
  4. \(\dfrac{85}{4}\)
  5. \(\dfrac{83}{4}\)
A sequence \(\{a_n\}\) satisfies \(a_1=1\), and
\(\displaystyle\sum_{k=1}^n (a_k-a_{k+1})=-n^2+n\)
for all positive integers \(n\). What is the value of \(a_{11}\)? [3 points]
  1. \(88\)
  2. \(91\)
  3. \(94\)
  4. \(97\)
  5. \(100\)

Mathematics (Type Na)

For the set \(X=\{1,2,3,4\}\), what is the number of functions \(f:X\to X\) that satisfy the following? [3 points]
\(f(2)\leq f(3)\leq f(4)\)
  1. \(64\)
  2. \(68\)
  3. \(72\)
  4. \(76\)
  5. \(80\)
Suppose a point \(\mathrm{P}\) is moving on the number line, and its position \(v(t)\) at time \(t\) \((t \geq 0)\) is equal to
\(v(t)=2t-6\).
If the distance point \(\mathrm{P}\) travels from time \(t=3\) to \(t=k\,(k>3)\) is equal to \(25\), what is the value of the constant \(k\)? [4 points]
  1. \(6\)
  2. \(7\)
  3. \(8\)
  4. \(9\)
  5. \(10\)

Mathematics (Type Na)

There are \(6\) students including three students \(\mathrm{A, B}\) and \(\mathrm{C}\). Compute the number of ways for these \(6\) students to sit around a circular table in equal distances while satisfying the following.
(※ If rotating one case results in another case, the two are not considered distinct.) [4 points]
  1. \(\mathrm{A}\) and \(\mathrm{B}\) are adjacent.
  2. \(\mathrm{B}\) and \(\mathrm{C}\) are not adjacent.
  1. \(32\)
  2. \(34\)
  3. \(36\)
  4. \(38\)
  5. \(40\)
What is the sum of all solutions to the equation
\(4\sin^2 x-4\cos\!\left(\!\dfrac{\pi}{2}+x\!\right)-3=0\)
where \(0\leq x <4\pi\)? [4 points]
  1. \(5\pi\)
  2. \(6\pi\)
  3. \(7\pi\)
  4. \(8\pi\)
  5. \(9\pi\)

Mathematics (Type Na)

Two polynomial fuctions \(f(x)\) and \(g(x)\) satisfy
\(\displaystyle\lim_{x\to 0}\!\dfrac{f(x)+g(x)}{x}=3\:\) and \(\:\displaystyle\lim_{x\to 0}\!\dfrac{f(x)+3}{xg(x)}=2\).
For the function \(h(x)=f(x)g(x)\), what is the value of \(h'(0)\)? [4 points]
  1. \(27\)
  2. \(30\)
  3. \(33\)
  4. \(36\)
  5. \(39\)
For a real number \(a\) in \(\dfrac{1}{4}<a<1\), let \(\mathrm{A}\) and \(\mathrm{B}\) be points where the line \(y=1\) meets the two curves \(y=\log_a x\) and \(y=\log_{4a} x\) respectively, and let \(\mathrm{C}\) and \(\mathrm{D}\) be points where the line \(y=-1\) meets the two curves \(y=\log_a x\) and \(y=\log_{4a} x\) respectively.
Which option only contains every correct statement in the <List>? [3 points]
  1. The point externally dividing the line segment \(\mathrm{AB}\) in the ratio \(1:4\) has coordinates \((0,1)\).
  2. If the quadrilateral \(\mathrm{ABCD}\) is a rectangle,
    then \(a=\dfrac{1}{2}\).
  3. If \(\overline{\mathrm{AB}}<\overline{\mathrm{CD}}\), then \(\dfrac{1}{2}<a<1\).
  1. a
  2. c
  3. a, b
  4. b, c
  5. a, b, c

Mathematics (Type Na)

A random variable \(X\) follows a normal distribution with a mean of \(8\) and a standard deviation of \(3\), and a random variable \(Y\) follows a normal distribution with a mean of \(m\) and a standard deviation of \(\sigma\). Two random variables \(X\) and \(Y\) satisfy
\(\mathrm{P}(4\leq X\leq 8)+\mathrm{P}(Y\geq 8)=\dfrac{1}{2}\).
What is the value of \(\mathrm{P}\!\left(\!Y\leq 8+\dfrac{2\sigma}{3}\!\right)\) computed using 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. \(0.8351\)
  2. \(0.8413\)
  3. \(0.9332\)
  4. \(0.9772\)
  5. \(0.9938\)
For a real number \(a\,(a>1)\), define the function \(f(x)\) as
\(f(x)=(x+1)(x-1)(x-a)\).
What is the maximum value of \(a\) for which the function
\(\displaystyle g(x)=x^2\int_0^x f(t)dt-\int_0^x t^2f(t)dt\)
only has one local extremum? [4 points]
  1. \(\dfrac{9\sqrt{2}}{8}\)
  2. \(\dfrac{3\sqrt{6}}{4}\)
  3. \(\dfrac{3\sqrt{2}}{2}\)
  4. \(\sqrt{6}\)
  5. \(2\sqrt{2}\)

Mathematics (Type Na)

A sequence \(\{a_n\}\) satisfies \(0<a_1<1\), and the following for all positive integers \(n\).
  1. \(a_{2n}=a_2\times a_n+1\)
  2. \(a_{2n+1}=a_2\times a_n-2\)
If \(a_7=2\), what is the value of \(a_{25}\)? [4 points]
  1. \(78\)
  2. \(80\)
  3. \(82\)
  4. \(84\)
  5. \(86\)
Short Answer Questions
In the expansion of the polynomial \((3x+1)^8\), compute the coefficient of \(x\). [3 points]
A function \(f(x)\) satisfies \(f'(x)=3x^2+4x+5\) and \(f(0)=4\). compute \(f(1)\). [3 points]

Mathematics (Type Na)

Compute \(\log_3 72 - \log_3 8\). [3 points]
Compute the value of the positive number \(k\) for which the curve \(y=4x^3-12x+7\) and the line \(y=k\) meet on exactly \(2\) points. [3 points]
The function
\(f(x)=\begin{cases} -3x+a &\; (x\leq1)\\\\ \dfrac{x+b}{\sqrt{x+3}-2} &\; (x>1) \end{cases}\)
is continuous on the set of all real numbers.
Compute \(a+b\). (※ \(a\) and \(b\) are constants.) [4 points]

Mathematics (Type Na)

Compute the area of the region enclosed by the curve \(y=x^2-7x+10\) and the line \(y=-x+10\). [4 points]
Consider a triangle \(\mathrm{ABC}\) where \(\angle\mathrm{A}=\dfrac{\pi}{3}\) and \(\overline{\mathrm{AB}}:\overline{\mathrm{AC}}=3:1\). If the circumcircle of the triangle \(\mathrm{ABC}\) has a radius of \(7\), the length of the line segment \(\mathrm{AC}\) is equal to \(k\). Compute \(k^2\). [4 points]

Mathematics (Type Na)

There is a sack containing \(5\) balls marked with numbers \(3,3,4,4\) and \(4\) respectively. Let us perform the following trial and set a score using this sack and a die.
Randomly take out a ball from the sack.
If the number marked on the ball taken out is \(3\), throw the die \(3\) times and set the score as the sum of the three numbers it lands on.
If the number marked on the ball taken out is \(4\), throw the die \(4\) times and set the score as the sum of the four numbers it lands on.
After performing this trial once, the probability that the score set is \(10\), is equal to \(\dfrac{q}{p}\). Compute \(p+q\).
(※ \(p\) and \(q\) are positive integers that are coprime.) [4 points]
Consider a cubic function \(f(x)\) with a leading coefficient of \(1\), and a linear function \(g(x)\).
Define the function \(h(x)\) as
\(h(x)=\begin{cases} |f(x)-g(x)| &\; (x<1)\\\\ f(x)+g(x) &\; (x\geq 1). \end{cases}\)
Given that the function \(h(x)\) is differentiable on the set of all real numbers, and \(h(0)=0\) and \(h(2)=5\), compute \(h(4)\). [4 points]