For distinct natural numbers $a$, $b$, and $c$, $$\frac { 6 ^ { a } \cdot 15 ^ { b } } { 9 ^ { b } \cdot 10 ^ { c } }$$ is equal to an integer. Accordingly, which of the following orderings is correct? A) $a < b < c$ B) $b < a < c$ C) $b < c < a$ D) $c < a < b$ E) $c < b < a$
Let $p$ and $r$ be distinct prime numbers. The number $180 \cdot r$ is an integer multiple of the number $p$. Accordingly, the prime number $p$ definitely divides which of the following numbers? A) $12 \cdot r$ B) $18 \cdot r$ C) $20 \cdot r$ D) $30 \cdot r$ E) $45 \cdot r$
Let $x$ and $y$ be positive real numbers such that $$\begin{aligned}
& x ^ { 2 } + 3 y ^ { 2 } = 8 \\
& 2 x ^ { 2 } + y ^ { 2 } = 6
\end{aligned}$$ What is the product $x \cdot y$? A) 2 B) 4 C) 6 D) 8 E) 10
Let $m$ and $n$ be positive integers such that $$\begin{aligned}
& \gcd ( m , n ) + \text{lcm} ( m , n ) = 289 \\
& m + n \neq 289
\end{aligned}$$ What is the sum $m + n$? A) 41 B) 43 C) 45 D) 47 E) 49
Let $a$, $b$, $c$, and $d$ be real numbers such that $$\begin{aligned}
& a x ^ { 2 } + b x + 12 \geq 0 \\
& c x ^ { 2 } + d x + 24 \leq 0
\end{aligned}$$ To find the solution set of this system of inequalities, the following table is constructed and the solution set is found to be $[ - 2 , - 1 ] \cup [ 4,6 ]$. What is the sum $a + b + c + d$? A) 15 B) 16 C) 17 D) 18 E) 19
Let $a$, $b$, and $c$ be real numbers. In the rectangular coordinate plane, the graphs of the functions $f(x) + a$, $b \cdot f(x)$, and $f(c \cdot x)$ are given in the figure. What are the signs of the numbers $a$, $b$, and $c$ respectively? A) $-, +, -$ B) $+, -, +$ C) $-, +, -$ D) $-, -, +$ E) $-, -, -$
In the rectangular coordinate plane, the graph of the function $f(x)$ defined on the closed interval $[0,5]$ is given in the figure. If the function $(f \circ f \circ f)(x)$ attains its maximum value at the point $x = a$, in which of the following open intervals is the number $a$? A) $( 0,1 )$ B) $( 1,2 )$ C) $( 2,3 )$ D) $( 3,4 )$ E) $( 4,5 )$
For the equation $x^2 - 2x + c = 0$, the discriminant is also a root of this equation. What is the product of the possible values of the real number $c$? A) 1 B) 2 C) 4 D) $\frac{1}{2}$ E) $\frac{1}{4}$
A polynomial $P(x)$ with real coefficients and of degree four satisfies the inequality $$P(x) \geq x$$ for every real number $x$. $$\begin{aligned}
& P(1) = 1 \\
& P(2) = 4 \\
& P(3) = 3
\end{aligned}$$ according to, $\mathbf{P(4)}$ is equal to what?
Let $a$ and $b$ be digits. Given the sets $$\begin{aligned}
& A = \{ 5,6,7,8,9 \} \\
& B = \{ 1,4,5,7 \} \\
& C = \{ a , b \}
\end{aligned}$$ If the number of elements in the Cartesian product $(A \cup C) \times (B \cup C)$ is 28, what is the sum $a + b$? A) 5 B) 6 C) 8 D) 9 E) 11
For an arithmetic sequence $(a_n)$: $$\begin{gathered}
a _ { 2 } = 2 a _ { 1 } + 1 \\
a _ { 6 } + a _ { 22 } = 34
\end{gathered}$$ equalities are given. Accordingly, what is $a _ { 7 }$? A) $61^3$ B) 7 C) 8 D) 9 E) 10
Two different digits are randomly selected from the set $A = \{ 1,2,3,4,5,6,7 \}$. Given that the product of the selected digits is an even number, what is the probability that the sum of these digits is also an even number? A) $\frac { 1 } { 2 }$ B) $\frac { 1 } { 3 }$ C) $\frac { 1 } { 4 }$ D) $\frac { 1 } { 5 }$ E) $\frac { 1 } { 6 }$
Let $n$ be a natural number. Given that the arithmetic mean of all coefficients in the expansion of $$\left( x ^ { 3 } - \frac { 2 } { x ^ { 2 } } \right) ^ { n }$$ is 0.2, what is the coefficient of the $x ^ { 2 }$ term in this expansion? A) 12 B) 16 C) 24 D) 32 E) 40
On the set of real numbers greater than 1, a function $f$ is defined as $$f ( x ) = 3 \ln \left( x ^ { 2 } - 1 \right) + 2 \ln \left( x ^ { 3 } - 1 \right) - 5 \ln ( x - 1 )$$ Accordingly, $$\lim _ { x \rightarrow 1 ^ { + } } e ^ { f ( x ) }$$ what is the value of this limit? A) 30 B) 36 C) 60 D) 64 E) 72
Let $a$ and $b$ be real numbers. A function $f$ that is continuous on the set of real numbers is defined as $$f ( x ) = \begin{cases} x ^ { 2 } - 4 & , x \leq a \\ 5 x - 8 & , a < x \leq b \\ 7 & , x > b \end{cases}$$ Accordingly, what is the sum $a + b$? A) 4 B) 5 C) 6 D) 7 E) 8
Let $a$ and $b$ be real numbers. A function $f$ is defined on the set of positive real numbers as $$f ( x ) = a x ^ { a } + b x ^ { b }$$ $$\begin{aligned}
& f ( 1 ) = 6 \\
& f ^ { \prime } ( 1 ) = 20
\end{aligned}$$ Given that, what is $f''(1)$? A) 44 B) 46 C) 48 D) 50 E) 52
In the rectangular coordinate plane, the graph of $f ^ { \prime }$, the derivative of function $f$, is given on the closed interval $[ 0,10 ]$. The areas of the regions between this graph and the x-axis are shown as follows. $$f ( 0 ) = \frac { - 1 } { 2 }$$ Given that, how many different roots does the function $f$ have on the interval $[ 0 , 10 ]$? A) 1 B) 2 C) 3 D) 4 E) 5
Let $a$ and $b$ be real numbers. A function $f$ that is continuous on the set of real numbers is defined as $$f ( x ) = \begin{cases} 6 - \frac { 3 x ^ { 2 } } { 2 } , & x < 2 \\ a x - b & x \geq 2 \end{cases}$$ $$\int _ { 0 } ^ { 4 } f ( x ) d x = \int _ { 2 } ^ { 6 } f ( x ) d x$$ Given that, what is the sum $a + b$? A) 1 B) 2 C) 3 D) 4 E) 5
In the rectangular coordinate plane, $$\begin{aligned}
& f ( x ) = x ^ { 2 } - 2 x \\
& g ( x ) = - x ^ { 2 } + 4 x
\end{aligned}$$ The shaded region between the graphs of these functions and the x-axis is given below. Accordingly, what is the area of the shaded region in square units? A) $\frac { 17 } { 3 }$ B) $\frac { 19 } { 3 }$ C) $\frac { 20 } { 3 }$ D) $\frac { 22 } { 3 }$ E) $\frac { 23 } { 3 }$
Let $a \in \left( \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$. Given that $$\begin{aligned}
& x = \tan a \\
& y = \tan ( 2 a ) \\
& z = \tan ( 3 a )
\end{aligned}$$ Which of the following is the correct ordering of these numbers? A) $x < y < z$ B) $x < z < y$ C) $y < x < z$ D) $z < x < y$ E) $z < y < x$
Ali places the sharp end of a compass on a point on paper and, without changing the compass opening, draws a circle with a diameter of 21 cm. Given that the lengths of the compass legs are 7.5 and 12 cm, what is the measure of the angle between the compass legs in degrees? A) 30 B) 45 C) 60 D) 90 E) 120
In the figure, using the points $\mathrm { P } ( 0,1 )$ and $\mathrm { S } ( 1,0 )$ on the unit circle with center O and the positive directed angle $\theta$ that the line segment RO makes with the x-axis, new trigonometric functions are defined as follows: $$\begin{aligned}
& \text { kas } \theta = | \mathrm { RS } | \\
& \text { sas } \theta = | \mathrm { RP } |
\end{aligned}$$ Accordingly, $$\frac { \mathrm { kas } ^ { 2 } \theta } { 2 - \operatorname { sas } ^ { 2 } \theta }$$ For $\theta$ values where this expression is defined, which of the following is it equal to? A) $\sin ( 2 \theta )$ B) $\cos ^ { 2 } ( 2\theta )$ C) $\sec \theta$ D) $\tan \left( \frac { \theta } { 2 } \right)$
In the rectangular coordinate plane, one vertex of a triangle is at the origin, its centroid is at the point $( 0,6 )$, and its orthocenter is at the point $( 0,8 )$. Accordingly, what is the area of this triangle in square units? A) 18 B) 21 C) 24 D) 27 E) 30
In the rectangular coordinate plane, points A and B lie on the line $y = x + 2$, and the distance between them is 3 units. Given that the coordinates of the midpoint of segment [AB] are $( -1, 1 )$, in which regions of the analytic plane are points A and B located? A) Both in region II B) Both in region III C) One in region I, the other in region II D) One in region I, the other in region III E) One in region II, the other in region III
In the rectangular coordinate plane, two lines that intersect perpendicularly at point $A ( 3,4 )$ have slopes whose sum is $\frac { 3 } { 2 }$. If the points where these two lines intersect the x-axis are points B and C, what is the area of triangle ABC in square units? A) 24 B) 20 C) 16 D) 12 E) 8
In the rectangular coordinate plane, the symmetric point of $( 4,4 )$ with respect to a line passing through $( 1,0 )$ is $( a , 0 )$. Accordingly, what is the product of the values that $a$ can take? A) $-24$ B) $-16$ C) $-8$ D) $16$ E) $32$