For integers $x$ and $y$ $$9 ^ { x } - 3 ^ { 2 x - 2 } = 2 ^ { y } \cdot 3 ^ { 6 }$$ the equality is satisfied. Accordingly, what is the sum $x + y$? A) 3 B) 4 C) 5 D) 6 E) 7
For real numbers $a$, $b$, and $c$ $$a - b < 0 < c < c - b$$ the inequality is given. Accordingly, I. $a \cdot b \cdot c > 0$ II. $( a + c ) \cdot b > 0$ III. $b - a + c > 0$ which of these statements are always true? A) Only I B) Only II C) I and II D) I and III E) II and III
For integers $x$ and $y$, $$| x - 3 | + | 2 x + y | + | 2 x + y - 1 | = 1$$ the equality is satisfied. Accordingly, what is the sum of the values that $y$ can take? A) $- 12$ B) $- 11$ C) $- 10$ D) $- 9$ E) $- 8$
The three-digit natural number ABA divided by the two-digit natural number A1 gives a quotient of 13 and a remainder of 19. Accordingly, what is the sum $A + B$? A) 8 B) 9 C) 10 D) 11 E) 12
Regarding sets $A$, $B$, and $C$ $$\begin{aligned}
& \{ ( 1,2 ) , ( 2,3 ) , ( 3,4 ) \} \subseteq A \times B \\
& \{ ( 1,2 ) , ( 3,4 ) , ( 4,2 ) , ( 4,4 ) \} \subseteq A \times C
\end{aligned}$$ it is known that. Accordingly, I. The set $A \cap B$ has at least 3 elements. II. The set $A \cap C$ has at least 3 elements. III. The set $B \cap C$ has at least 3 elements. which of these statements are always true? A) Only I B) Only II C) Only III D) I and II E) I and III
If the number of elements of a set whose all elements are positive integers is one more than the value of the smallest element of this set, this set is called a wide set. Let $A$, $B$, and $C$ be wide sets,
$A \cup B \cup C = \{ 1,2,3,4,5,6,7,8,9 \}$
$A \cap B = \{ 3 \}$
$1 \in A$
$6 \in B$
it is known that. Accordingly, which of the following is set $C$? A) $\{ 1,2 \}$ B) $\{ 3,4,8,9 \}$ C) $\{ 3,5,7,8 \}$ D) $\{ 4,5,6,7,8 \}$ E) $\{ 4,5,7,8,9 \}$
In the set of complex numbers $$\frac { i \cdot ( 2 - i ) \cdot ( 2 - 4 i ) } { ( 1 - i ) \cdot ( 1 + i ) }$$ what is the result of the operation? A) 2 B) 5 C) 10 D) $2 i$ E) $5 i$
Let $\bar{z}$ be the conjugate of the complex number $z$, $$\frac { 6 + 2 i } { z } = \bar { z } + i$$ the sum of the complex numbers $z$ that satisfy the equality is what? A) $1 + 3 i$ B) $2 + i$ C) $3 + 2 i$ D) $4 + i$
Let $a$ and $b$ be non-zero integers. A function $f$ is defined on the set of real numbers as $$f ( x ) = a x + b$$ $$( f \circ f ) ( x ) = f ( x + 2 ) + f ( x )$$ According to this, what is the value of $f(3)$? A) 7 B) 8 C) 9 D) 10 E) 11
In the rectangular coordinate plane, the graph of a function $f$ defined on the closed interval $[-5,5]$ is given in the figure. For distinct numbers $a, b, c$ and $d$ in the domain of this function $$\begin{aligned}
& f(a) = f(b) = 1 \\
& f(c) = f(d) = 3
\end{aligned}$$ the equalities are satisfied. Accordingly, regarding the ordering of $a, b, c$ and $d$ numbers I. $a < b < c < d$ II. $c < a < b < d$ III. $c < d < a < b$ Which of the following inequalities can be true? A) Only I B) Only II C) I and II D) II and III E) I, II and III
$P(x)$ and $Q(x)$ are non-constant polynomials, and $R(x)$ is a first-degree polynomial, where $$P(x) = Q(x) \cdot R(x)$$ the equality is satisfied. Accordingly, I. The constant terms of polynomials $P(x)$ and $R(x)$ are the same. II. If the graph of $P(x)$ is a parabola, then the graph of $Q(x)$ is a line. III. Every root of polynomial $Q(x)$ is also a root of polynomial $R(x)$. Which of the following statements are always true? A) Only II B) Only III C) I and II D) I and III E) II and III
For third-degree real-coefficient polynomials $P(x)$ and $R(x)$ whose highest degree terms have coefficient 1, the numbers 2 and 6 are common roots. When the polynomial $P(x) - R(x)$ is divided by $x - 1$, the remainder is 10. Accordingly, what is the value of $P(0) - R(0)$? A) 24 B) 27 C) 30 D) 33 E) 36
Where $m$ and $n$ are integers, $$\left(x^2 + 2y\right)^7$$ In the expansion of this expression, if one of the terms is $mx^ny^2$, what is the sum $m + n$? A) 56 B) 64 C) 72 D) 86 E) 94
Where $a$ and $b$ are positive real numbers, $$2ax^2 - 5bx + 8b = a$$ the roots of the equation are $a$ and $b$. Accordingly, what is the sum $a + b$? A) 5 B) 6 C) 10 D) 12 E) 15
Where $a, b$ and $c$ are real numbers, $$y = ax^2 + bx + c$$ the parabola intersects the line $y = 1$ at points B and C, and intersects the line $y = 6$ at only point A. The locations of points A, B and C in the rectangular coordinate plane are shown in the figure below. Accordingly, what are the signs of the numbers $a$, $b$ and $c$ respectively? A) +, -, - B) +, +, - C) -, +, + D) -, -, + E) -, -, -
When a stick is divided into 4 equal parts, the length of each part is $\log_5(x)$ units, and when divided into 10 equal parts, the length of each part is $\log_5\left(\frac{x^2}{25}\right)$ units. Accordingly, what is the length of the stick in units? A) 5 B) 8 C) 10 D) 12 E) 15
Where $n$ is an integer and $1 < n < 100$, $$\log_2\left(\log_3 n\right)$$ the value of this expression equals a positive integer. Accordingly, what is the sum of the values that $n$ can take? A) 36 B) 45 C) 63 D) 72 E) 90
Where $a$ and $b$ are positive real numbers different from 1, $$\log_a 2 < 0 < \log_2 b$$ the inequality is satisfied. Accordingly, I. $a + b > 1$ II. $b - a > a$ III. $a \cdot b > 1$ Which of the following statements are always true? A) Only I B) Only II C) I and II D) I and III E) II and III
The sequence $(a_n)$ of real numbers satisfies for every positive integer $n$ $$a_{n+1} = a_n + \frac{(-1)^n \cdot a_n}{2}$$ the equality. If $a_5 = 18$, what is $a_1$? A) 4 B) 8 C) 16 D) 32 E) 64
For a geometric sequence $(a_n)$ with all positive terms and common ratio $r$ $$\begin{aligned}
& a_1 + \frac{1}{2} + r \\
& a_7^2 = a_5 + 12 \cdot a_3
\end{aligned}$$ the equalities are given.
Between October 5, 2020 Monday and October 18, 2020 Sunday, including these two days, two meetings will be held on two different days within these 14 days. If an arrangement is to be made such that at least one of the meetings is on a weekday, in how many different ways can this arrangement be made? A) 70 B) 75 C) 80 D) 85 E) 90
An exam consisting of a total of 8 questions, with 4 questions each in the verbal and quantitative sections, has the following statement in its booklet: ``To pass the exam, you must answer at least 5 questions correctly in total, with at least 2 questions from each of the verbal and quantitative sections.'' Sevcan, who read this statement incompletely, randomly selected 5 out of 8 questions on the exam and answered each question she selected correctly. Accordingly, what is the probability that Sevcan passes the exam? A) $\frac{3}{4}$ B) $\frac{4}{5}$ C) $\frac{5}{6}$ D) $\frac{6}{7}$ E) $\frac{7}{8}$
Where $0 < x < \frac{\pi}{2}$, $$\frac{1 + \tan x}{\cot x} \cdot \frac{\sin x - \cos x}{\sin x} = 2$$ if this holds, what is the value of $\sin x$? A) $\frac{1}{3}$ B) $\frac{3}{5}$ C) $\frac{\sqrt{2}}{2}$ D) $\frac{\sqrt{3}}{2}$ E) $\frac{\sqrt{5}}{3}$
In the figure, the line segments $[OA]$ and $[OD]$ intersect perpendicularly. Accordingly, the ratio of the area of triangle OAB to the area of triangle OCD in terms of $\alpha$ is which of the following? A) $\tan \alpha$ B) $\cot \alpha$ C) $\tan^2 \alpha$ D) $\cot^2 \alpha$ E) $\sec^2 \alpha$
In the rectangular coordinate plane, a square $ABCD$ with two vertices at $A(0, a)$ and $B(0, b)$ is given. The vertex $C$ of square $ABCD$ lies on the line $y = \frac{x}{3}$. If $a + b = 15$, what is the sum of the coordinates of point $D$? A) 14 B) 18 C) 21 D) 24 E) 27
In the rectangular coordinate plane, it is known that a line $d$ passes through point $A(-4, 1)$ and is perpendicular to the line $2x - y = 5$. If the point where line $d$ intersects the x-axis is $(a, 0)$ and the point where it intersects the y-axis is $(0, b)$, what is the sum $a + b$? A) -3 B) -1 C) 0 D) 1 E) 3
In the rectangular coordinate plane, points $A(2, 7)$ and $B(-1, 4)$ are translated 3 units in the positive direction along the x-axis to obtain points $D$ and $C$ respectively. Accordingly, what is the area of the quadrilateral with vertices at points A, B, C, and D in square units? A) 9 B) 10 C) 11 D) 12 E) 13