2015 College Scholastic Ability Test

Mathematics (Type A)

Multiple Choice Questions

What is the value of \(5\times 8^{\,\begin{array}{c}1 \\\hline 3\end{array}}\)? [2 points]

\(10\)
\(15\)
\(20\)
\(25\)
\(30\)

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]

\(5\)
\(6\)
\(7\)
\(8\)
\(9\)

What is the value of \(\displaystyle\lim_{n\;\!\to\;\!\infty}\frac{4n^2+6}{n^2+3n}\)? [2 points]

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

In the adjacency matrix of the following graph, what is the number of elements with a value of \(0\)? [3 points]

\(9\)
\(11\)
\(13\)
\(15\)
\(17\)

Mathematics (Type A)

A geometric progression \(\{a_n\}\) with a positive common ratio satisfies \(a_1=3\) and \(a_5=48\). What is the value of \(a_3\)? [3 points]

\(18\)
\(16\)
\(14\)
\(12\)
\(10\)

If \(\displaystyle\int_{0}^{1}(2x+a)dx=4\), what is the value of the constant \(a\)? [3 points]

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

In the expansion of the polynomial \((x+a)^6\), the coefficient of \(x^4\) is \(60\). What is the value of the positive number \(a\)? [3 points]

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

Mathematics (Type A)

Figure below is the graph of the function \(y=f(x)\).

What is the value of \(\displaystyle\lim_{x\;\!\to\;\!\,-0}f(x)+ \lim_{x\;\!\to\;\!1+0}\!f(x)\)? [3 points]

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

Let \(S_n\) be the sum of the first \(n\) terms of a sequence \(a_n\). Given that \(S_n=\dfrac{n}{n+1}\), what is the value of \(a_4\)? [3 points]

\(\dfrac{1}{22}\)
\(\dfrac{1}{20}\)
\(\dfrac{1}{18}\)
\(\dfrac{1}{16}\)
\(\dfrac{1}{14}\)

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 \(E_B=100E_A\), what is the value of \(P_A-P_B\)? [3 points]

\(30\)
\(25\)
\(20\)
\(15\)
\(10\)

Mathematics (Type A)

A geometric progression \(\{a_n\}\) satisfies \(a_1=3\) and \(a_2=1\). What is the value of \(\displaystyle\sum_{n\;\!=\;\!1}^{\infty} \left(a_n\right)^{2}\)? [3 points]

\(\dfrac{81}{8}\)
\(\dfrac{83}{8}\)
\(\dfrac{85}{8}\)
\(\dfrac{87}{8}\)
\(\dfrac{89}{8}\)

A research facility planted tomato seedlings and examined the length of tomato stems after \(3\) weeks. It was established that the length of a tomato stem follows a normal distribution with a mean of \(30\text{cm}\) and a standard deviation of \(2\text{cm}\). Suppose we randomly select a tomato seedling grown for \(3\) weeks

in this facility. What is the probability that the length of its stem is between \(27\text{cm}\) and \(32\text{cm}\), computed with the standard normal table to the right? [3 points]

\(z\)	\(\mathrm{P}(0\!\leq\! Z \!\leq\!z)\)
\(1.0\)	\(0.3413\)
\(1.5\)	\(0.4332\)
\(2.0\)	\(0.4772\)
\(2.5\)	\(0.4938\)

\(0.6826\)
\(0.7745\)
\(0.8185\)
\(0.9104\)
\(0.9270\)

Mathematics (Type A)

[13~14] For the function \(\boldsymbol{f(x)=x(x+1)(x-4)}\), answer the questions 13 and 14.

For the matrix \(A=\begin{pmatrix} 2&1 \\ 0&3 \end{pmatrix}\), what is the sum of all constants \(a\) that satisfy \(A \begin{pmatrix} 0 \\ f(a) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}\)? [3 points]

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

If the line \(y=5x+k\) and the graph of the function \(y=f(x)\) meet at two distinct points, what is the value of the positive number \(k\)? [4 points]

\(5\)
\(\dfrac{11}{2}\)
\(6\)
\(\dfrac{13}{2}\)
\(7\)

Mathematics (Type A)

What is the sum of all positive integers \(x\) that satisfy \(\left(\!\dfrac{1}{5}\!\right)^{\!1-2x}\leq 5^{x+4}\)? [4 points]

\(11\)
\(12\)
\(13\)
\(14\)
\(15\)

Two events \(A\) and \(B\) satisfy

\(\mathrm{P}(A)=\dfrac{1}{3}\:\) and \(\:\mathrm{P}(A\cap B)=\dfrac{1}{8}\).

What is the value of \(\mathrm{P}(B^C \,|\, A)\)?
(※ \(B^C\) is the complement of \(B\).) [4 points]

\(\dfrac{11}{24}\)
\(\dfrac{1}{2}\)
\(\dfrac{13}{24}\)
\(\dfrac{7}{12}\)
\(\dfrac{5}{8}\)

Mathematics (Type A)

An arithmetic progression \(\{a_n\}\) satisfies \(\displaystyle\sum_{k\;\!=\;\!1}^{n}a_{2k-1}=3n^2+n\). What is the value of \(a_8\)? [4 points]

\(16\)
\(19\)
\(22\)
\(25\)
\(28\)

What is the number of all \(4\)-tuples \((x,y,z,w)\) where \(x, y, z\) and \(w\) are nonnegative integers that satisfy the following system of equations? [4 points]

\(\begin{cases} x+y+z+3w=14\\\\ x+y+z+w=10 \end{cases}\)

\(40\)
\(45\)
\(50\)
\(55\)
\(60\)

Mathematics (Type A)

Two square matrices \(A\) and \(B\) of order \(2\) satisfy

\(A^2-AB=3E\:\) and \(\:A^2B-B^2A=A+B\).

What is the list of correct statements in the <List>? (※ \(E\) is the identity matrix.) [4 points]

\(A\) is invertible.
\(AB = BA\)
\((A+2B)^2=24E\)

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

A function \(f(x)\) satisfies \(f(x+3)=f(x)\) for all real numbers \(x\), and

\(f(x)=\begin{cases} x &\; (0\leq x< 1) \\ 1 &\; (1 \leq x< 2) \\ -x+3 &\; (2 \leq x< 3). \end{cases}\)

Given that \(\displaystyle\int_{-a}^{a}f(x)dx=13\), what is the value of the constant \(a\)? [4 points]

\(10\)
\(12\)
\(14\)
\(16\)
\(18\)

Mathematics (Type A)

Among all cubic functions \(f(x)\) that satisfy the following, what is the minimum value of \(f(2)\)? [4 points]

\(f(x)\) has a leading coefficient of \(1\).
\(f(0)=f'(0)\)
\(f(x)\geq f'(x)\) for all real numbers \(x\geq -1\).

\(28\)
\(33\)
\(38\)
\(43\)
\(48\)

Short Answer Questions

Compute \(\displaystyle\lim_{x\;\!\to\;\!0} \dfrac{x(x+7)}{x}\). [3 points]

Compute the value of the constant \(a\) for which the function

\(f(x)= \begin{cases} 2x+10 &\; (x < 1) \\\\ x+a &\; (x \geq 1) \end{cases}\)

is continuous on the set of all real numbers. [3 points]

Mathematics (Type A)

Two sequences \(\{a_n\}\) and \(\{b_n\}\) satisfy

\(\displaystyle\sum_{n\;\!=\;\!1}^{\infty}a_n=4\:\) and \(\:\displaystyle\sum_{n\;\!=\;\!1}^{\infty}b_n=10\).

Compute \(\displaystyle\sum_{n\;\!=\;\!1}^{\infty}(a_n+5b_n)\). [3 points]

A random variable \(X\) following the binomial distribution \(\mathrm{B}\!\left(\!n, \dfrac{1}{3}\!\right)\) satisfies \(\mathrm{V}(3X)=40\).
Compute \(n\). [3 points]

A polynomial function \(f(x)\) has a derivative \(f'(x)=6x^2+4\). Given that the graph of the function \(y=f(x)\) passes through the point \((0,6)\), compute \(f(1)\). [4 points]

Mathematics (Type A)

An absolutely continuous random variable \(X\)
can take all values in the interval \([0, 3]\), and the graph of its probability density function is as below.

Given that \(\mathrm{P}(0\leq X\leq 2)=\dfrac{q}{p}\), compute \(p+q\).
(※ \(k\) is a constant. \(p\) and \(q\) are positive integers that are coprime.) [4 points]

For positive integers \(k\), let

\(a_k=\displaystyle\lim_{n\;\!\to\;\!\infty} \dfrac{\left(\!\dfrac{6}{k}\!\right)^{\!n+1}} {\left(\!\dfrac{6}{k}\!\right)^{\!n}+1}\).

Compute \(\displaystyle\sum_{k\;\!=\;\!1}^{10}ka_k\). [4 points]

Mathematics (Type A)

Two polynomial functions \(f(x)\) and \(g(x)\) satisfy

\(g(x)=(x^3+2)f(x)\)

for all real numbers \(x\). Given that \(g(x)\) has a local minimum value of \(24\) at \(x=1\), compute \(f(1)-f'(1)\). [4 points]

For positive integers \(n\), let \(f(n)\) be the number of triangles \(\mathrm{OAB}\) on the \(xy\)-plane that satisfy the following. Compute \(f(1)+f(2)+f(3)\).
(※ \(\mathrm{O}\) is the origin.) [4 points]

Point \(\mathrm{A}\) has coordinates \((-2,3^n)\).
Let \((a,b)\) be the coordinates of point \(\mathrm{B}\). Then \(a\) and \(b\) are positive integers satisfying \(b\leq \log_2 a\).
The area of the triangle \(\mathrm{OAB}\) is \(50\) or less.