1994 College Scholastic Ability Test No.2

Mathematics·Studies (I)

If \(\sin\theta + \cos\theta= \dfrac{1}{3}\), what is the value of \(\dfrac{1}{\cos\theta}\left(\!\tan\theta+\dfrac{1}{\tan^2\theta}\!\right)\)?

\(\dfrac{45}{16}\)
\(\dfrac{43}{16}\)
\(\dfrac{41}{16}\)
\(\dfrac{39}{16}\)
\(\dfrac{37}{16}\)

If two distinct real numbers \(\alpha\) and \(\beta\) satisfy \(\alpha+\beta=1\), what is the value of \(\displaystyle\lim_{x\to\infty}\frac{\sqrt{x+\alpha^2}-\sqrt{x+\beta^2}}{\sqrt{4x+\alpha}-\sqrt{4x+\beta}}\)?

\(1\)
\(\dfrac{1}{2}\)
\(2\)
\(\dfrac{1}{4}\)
\(4\)

Figure to the right is the graph of \(y=f(x)\). Let a function \(g(x)\) be

\(g(x)=\displaystyle\int_x^{x+1}f(t)dt.\)

What is the minimum value of \(g(x)\)?

\(g(1)\)
\(g(2)\)
\(g\!\left(\!\dfrac{5}{2}\!\right)\)
\(g\!\left(\!\dfrac{7}{2}\!\right)\)
\(g(4)\)

Consider an arithmetic progression with an initial term of \(m\) and a common difference of \(1\). If the sum of the first \(n\) terms of this sequence is \(50\), what is the value of \(m+n\)? (※ \(m\) is a positive integer \(\leq 10\).)

\(13\)
\(14\)
\(15\)
\(16\)
\(17\)

Mathematics·Studies (I)

Given that the infinite geometric series \(\displaystyle\sum_{n=1}^\infty r^n\) converges, which option below cannot be said to always converge?

\(\displaystyle\sum_{n=1}^\infty(r^n+r^{2n})\)
\(\displaystyle\sum_{n=1}^\infty(r^n-2r^{2n})\)
\(\displaystyle\sum_{n=1}^\infty\frac{r^n+(-r)^n}{2}\)
\(\displaystyle\sum_{n=1}^\infty\left(\!\frac{r-1}{2}\!\right)^{\!n}\)
\(\displaystyle\sum_{n=1}^\infty\left(\!\frac{r}{2}-1\!\right)^{\!n}\)

The line \(y=3x+2\) translated \(k\) units horizontally, is a tangent line to the parabola \(y^2=4x\). What is the value of \(k\)?

\(\dfrac{5}{9}\)
\(\dfrac{4}{9}\)
\(\dfrac{2}{9}\)
\(\dfrac{2}{3}\)
\(\dfrac{1}{3}\)

Set \(A\) is a subset of the set of all real numbers and satisfies the following condition.

If \(x\in A\), then \(\dfrac{1}{2}x\in A\).

Which option below is always correct?

If \(\sqrt{2}\in A\), then \(0 \notin A\).
If \(A\) is a finite set, then \(2 \notin A\).
If \(A\) is an infinite set, then \(0 \in A\).
If \(x\in A\) and \(y \in A\), then \(x+y\in A\).
If \(x\in A\) and \(y \in A\), then \(xy\in A\).

For positive integers \(a\) and \(b\), let \(a◇b\) be the remainder of the division of \(a\) by \(b\). For example,
\(1993◇5=3\). Which option below is incorrect?

\(2^{4n}◇5=1\) for all positive integers \(n\).
\(2^{n}◇5\ne0\) for all positive integers \(n\).
\(2^{m+n}◇5=\{2^m(2^n◇5)\}◇5\)
for all positive integers \(m\) and \(n\).
\(2^{m+n}◇5=\{(2^m◇5)(2^n◇5)\}◇5\)
for all positive integers \(m\) and \(n\).
\((2^m+2^n)◇5=(2^m◇5)+(2^n◇5)\)
for all positive integers \(m\) and \(n\).

Mathematics·Studies (I)

For a square matrix \(A\) of order \(2\), Which option contains every correct statement in the <List>?
(※ \(E\) is the identity matrix of size \(2\).)

If \(A^3=A^5=E\), then \(A=E\).
If \(A^3+A^2+A+E=O\), then \(A\) is invertible.
If there are distinct positive integers \(k, m\) and \(n\) such that \(A^k=A^m=A^n=E\), then \(A=E\).

a, b
a, c
b, c
a
c

Consider a cross section of the cube shown in the figure and a plane. How many types of shapes can this cross section be among the shapes in the <List>?

A triangle
A rectangle that is not a square
A rhombus that is not a square
A pentagon
A hexagon

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

Let us list the numbers that appear on the figure

to the right in ascending order to create the following sequence.

\(1, 2, 3, 11, 12, 13,\) \(21, 22, 23, 31, 32, 33,\) \(111, 112, 113, 121, \cdots\)

What is the \(200\)th term
of this sequence?

\(13323\)
\(13332\)
\(21111\)
\(21113\)
\(21122\)

Mathematics·Studies (I)

\(a\) and \(b\) are two distinct integers, and a polynomial \(f(x)\) has the following properties \((A)\) and \((B)\).

\((A)\:\) All coefficients of \(f(x)\) are integers.

\((B)\:\) \(f(a)f(b)=-(a-b)^2\)

The following is a process proving that \(\dfrac{f(a)}{a-b}\) is an integer, using the properties above and the theorem \((C)\) below.

\((C)\:\) For integers \(m\) and \(n\), if a solution to the quadratic equation \(x^2+mx+n=0\) is a rational number, then that solution is an integer.

(Proof) \(a^n-b^n\) is divisible by \(a-b\) for all positive integers \(n\), therefore \(f(a)-f(b)\) is divisible by \(a-b\) by the property \((A)\)

ⓐ

.

Therefore, \(\dfrac{f(a)-f(b)}{a-b}\) is an integer.
A quadratic equation with two solutions \(\dfrac{f(a)}{a-b}\) and \(\dfrac{-f(b)}{a-b}\), is \(x^2-\left(\!\dfrac{f(a)-f(b)}{a-b}\!\right)+1=0\), using the relation between solutions and coefficients, and the property \((B)\)

ⓑ

.

\(\dfrac{f(a)}{a-b}\) is a rational number by the property \((A)\)

ⓒ

,

and \(\dfrac{f(a)-f(b)}{a-b}\) is an integer,
so \(\dfrac{f(a)}{a-b}\) is an integer by theorem \((C)\)

ⓓ

Among the underlined places in the proof above, which place wrongly used either \((A), (B)\) or \((C)\)?

ⓐ
ⓑ
ⓒ
ⓓ
None.

What is the number of pairs \((a,b)\) where \(a\) and \(b\) are positive integers that satisfy the inequality \(\big|\log_2a-\log_2 10\big|+\log_2b\leq 1\)?

\(15\)
\(17\)
\(19\)
\(21\)
\(23\)

It is given that the element in the product of matrices

\(\begin{pmatrix}x&y\end{pmatrix} \begin{pmatrix}a&b\\b&a\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}\)

is nonnegative for all real numbers \(x\) and \(y\).
What is the minimum value of \(a^2+(b-2)^2\)?

\(1\)
\(\dfrac{1}{2}\)
\(2\)
\(\dfrac{1}{4}\)
\(4\)

Mathematics·Studies (I)

For a real number \(x\), let \(f(x)\) be the number of real numbers \(t\) that satisfy \(t^2=x^3-x\). Which option below is an appropriate graph of \(y=f(x)\)?

For positive numbers \(a, b\) and \(c\), which option below is a necessary and sufficient condition for the following system of inequalities?

\(\begin{cases} ax^2-bx+c<0\\ cx^2-bx+a<0 \end{cases}\)

\(a+c<\dfrac{b}{2}\)
\(a+c<b\)
\(a+c<2b\)
\(a+c<1\)
\(a+c<2\)

For the function \(f(x)=4x^2-4x+1\:(0\leq x\leq1)\), the graphs of \(y=f(x)\) and \(y=f(f(x))\) are as the image below.

What is the number of elements in the set \(\{x\,|\,f(f(f(x)))=x,\:0\leq x\leq1\}\)?

\(16\)
\(12\)
\(8\)
\(6\)
\(5\)

Mathematics·Studies (I)

A person with cancer is diagnosed as having cancer with a probability of \(98\%\), and a person without cancer is diagnosed as not having cancer with a probability of \(92\%\), by some doctor. Suppose this doctor examined \(400\) people who actually had cancer and \(600\) people who actually did not have cancer, to diagnose whether they had cancer or not. If one of these \(1000\) people is chosen at random, what is the probability that the person has been diagnosed as having cancer?

\(39.2\%\)
\(40.0\%\)
\(40.8\%\)
\(44.0\%\)
\(44.8\%\)

A \(1\) meter tall fence surrounds a circular land with a radius of \(5\) meters. Suppose a light source is at a position \(6\) meters vertically apart from a position on the ground, \(2\) meters apart from the center \(\mathrm{O}\) of the circle. What is the area of the shadow of the fence cast by this light source?

\(11\pi\text{m}^2\)
\(14\pi\text{m}^2\)
\(17\pi\text{m}^2\)
\(20\pi\text{m}^2\)
\(24\pi\text{m}^2\)

Suppose an express train starts to move, and for the time that it travels the first \(3\)km, its speed at time \(t\) minutes is \(v(t)=\dfrac{3}{4}t^2+\dfrac{1}{2}t\) (km/minutes), and after that its speed is constant. What is the distance that this train travels for the first \(5\) minutes?

\(17\)km
\(16\)km
\(15\)km
\(14\)km
\(13\)km