Positive real numbers $a$ and $b$ satisfy the equality $$a ^ { 2 } - 2 a b - 3 b ^ { 2 } = 0$$ Accordingly, what is the value of the expression $\frac { a + b } { a - b }$? A) 2 B) 3 C) 4 D) 5 E) 6
For integers a and b $$16 ^ { a } \cdot 9 ^ { a } = 6 ^ { b } \cdot 8 ^ { 2 }$$ Given this equality, what is the sum $\mathbf { a } + \mathbf { b }$? A) 6 B) 9 C) 12 D) 15 E) 20
Real numbers $x$ and $y$ satisfy the equality $$\| x | + | y | | = | x + y |$$ Accordingly, which of the following inequalities is always true? A) $x \cdot y \geq 0$ B) $x \cdot y \leq 0$ C) $x + y \geq 0$ D) $x + y \leq 0$ E) $x - y \leq 0$
$$\mathrm { A } = \left\{ \mathrm { n } ( - 1 ) ^ { \mathrm { n } } : \mathrm { n } = 1,2,3 , \ldots , \mathrm { k } \right\}$$ The difference between the largest and smallest elements of the set is 25. Accordingly, how many positive elements does set A have? A) 4 B) 6 C) 8 D) 10 E) 12
Integers a and b satisfy the inequality $$1 < a < b - a < 5$$ Accordingly, what is the sum of the values that b can take? A) 11 B) 14 C) 15 D) 16 E) 18
The smaller of two numbers is 3 less than the arithmetic mean of these two numbers, and the larger is 4 more than the geometric mean of these two numbers. Accordingly, what is the sum of these two numbers? A) 7 B) 9 C) 10 D) 12 E) 14
Functions $f$ and $g$ defined on the set of real numbers satisfy the equalities $$\begin{aligned}
& ( f + g ) ( x ) = x ^ { 2 } \\
& ( f - g ) ( 2 x ) = x
\end{aligned}$$ Accordingly, what is the product $f ( 4 ) \cdot g ( 4 )$? A) 45 B) 51 C) 54 D) 60 E) 63
Function f is defined for every $\mathrm { x } \in ( 0,3 ]$ as $$f ( x ) = 2 x + 1$$ and satisfies the equality $$f ( x ) = f ( x + 3 )$$ for every real number x. Accordingly, what is the sum $\mathbf { f } ( \mathbf { 6 } ) + \mathbf { f } ( \mathbf { 7 } ) + \mathbf { f } ( \mathbf { 8 } )$? A) 8 B) 12 C) 15 D) 18 E) 21
Let N be the set of natural numbers. The sets $$\begin{aligned}
& C = \{ 2 n : n \in \mathbb { N } \} \\
& K = \left\{ n ^ { 2 } : n \in \mathbb { N } \right\}
\end{aligned}$$ are given. Accordingly, which of the following is an element of the Cartesian product set $$( \mathrm { K } \backslash \mathrm { C } ) \times ( \mathrm { C } \backslash \mathrm { K } )$$ ? A) $( 3,2 )$ B) $( 9,4 )$ C) $( 15,1 )$ D) $( 16,12 )$ E) $( 25,8 )$
A table consisting of two rows and 7 cells is given in the figure. Patterns are created by painting 4 cells of this table black. How many different patterns are there such that each row has at least one painted cell? A) 26 B) 28 C) 30 D) 32 E) 34
In the figure, 3 of the 6 edges of a regular tetrahedron are randomly painted. Accordingly, what is the probability that all three painted edges are on the same face? A) $\frac { 1 } { 2 }$ B) $\frac { 1 } { 3 }$ C) $\frac { 1 } { 4 }$ D) $\frac { 1 } { 5 }$ E) $\frac { 1 } { 6 }$
$$P ( x ) = ( x + 1 ) ^ { 2 } \left( x ^ { 2 } + 1 \right) ^ { 4 }$$ What is the coefficient of the $x ^ { 4 }$ term in the polynomial? A) 8 B) 10 C) 12 D) 14 E) 16
$$P ( x ) = x ^ { 3 } - m x + 1$$ The remainder when $P ( x - 1 )$ is divided by $x + 1$ equals the remainder when $P ( x + 1 )$ is divided by $x - 1$. Accordingly, what is m? A) 2 B) 4 C) 6 D) - 1 E) - 8
A third-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with real coefficients and leading coefficient 1 satisfies the equalities $$P ( 1 ) = P ( 3 ) = P ( 5 ) = 7$$ Accordingly, what is the value of $\mathbf { P } ( \mathbf { 0 } )$? A) - 1 B) - 4 C) - 8 D) 4 E) 8
Let a be a real number. One root of the equation $$a x ^ { 2 } - 18 x + 18 = 0$$ is 2 times the other. Accordingly, what is a? A) 2 B) 3 C) 4 D) 5 E) 6
$\cos x = \frac { \sqrt { 5 } } { 3 }$ Accordingly, I. $\sin \mathrm { x }$ II. $\sin 2 x$ III. $\cos 2 x$ Which of the following values equals a rational number? A) Only I B) Only III C) I and II D) I and III E) II and III
Let z be a complex number satisfying the equality $$i \cdot z + 1 = 2 ( 1 - \bar { z } )$$ What is the real part of the complex number z? A) $\frac { 1 } { 6 }$ B) $\frac { 1 } { 4 }$ C) $\frac { 1 } { 2 }$ D) $\frac { 2 } { 3 }$ E) $\frac { 5 } { 6 }$
Below, line segments $[ A B ]$ and $[ C D ]$ are given in the complex number plane. For each complex number z taken on these line segments, the number $\mathrm { w } = \mathrm { z } \cdot \overline { \mathrm { z } }$ is defined. Accordingly, in which of the following are the minimum and maximum values that w can take given respectively? A) 5 and 20 B) 5 and 25 C) 5 and 30 D) 10 and 20 E) 10 and 25
Let t be a real number. The equalities $$\begin{aligned}
& x = e ^ { 2 \cos t } \\
& y = e ^ { 3 \sin t }
\end{aligned}$$ are given. Accordingly, which of the following gives the relationship between $x$ and y that is satisfied for every real number t? A) $\ln ^ { 2 } x + \ln ^ { 2 } y = 4$ B) $\ln ^ { 2 } x + \ln ^ { 2 } y = 9$ C) $9 \ln ^ { 2 } x + 2 \ln ^ { 2 } y = 27$ D) $\ln ^ { 2 } x + 4 \ln ^ { 2 } y = 28$ E) $9 \ln ^ { 2 } x + 4 \ln ^ { 2 } y = 36$
$$\left( \sum _ { k = 1 } ^ { 9 } k \right) \cdot \left( \sum _ { n = 1 } ^ { 8 } \frac { 1 } { n ( n + 1 ) } \right)$$ What is the result of this operation? A) 27 B) 30 C) 32 D) 36 E) 40
In the first quadrant of the rectangular coordinate plane; a square $A _ { 1 }$ is drawn with two sides on the coordinate axes and one vertex on the line $\mathrm { d } : \mathrm { y } = 4 - \mathrm { x }$. Then, a square $A _ { 2 }$ adjacent to the square $A _ { 1 }$ with one side on the x-axis and one vertex on line d is drawn. Continuing in a similar manner, a sequence of squares $\mathrm { A } _ { 1 } , \mathrm {~A} _ { 2 } , \mathrm {~A} _ { 3 } , \ldots$ is obtained as shown in the figure. Accordingly, what is the sum of the areas of all the squares $\mathbf { A } _ { \mathbf { n } }$ obtained in square units? A) $\frac { 9 } { 2 }$ B) $\frac { 11 } { 2 }$ C) $\frac { 14 } { 3 }$ D) $\frac { 16 } { 3 }$ E) $\frac { 20 } { 3 }$
The inverse of matrix A is $A ^ { - 1 } = \left[ \begin{array} { l l } 1 & 0 \\ 2 & 1 \end{array} \right]$. Given that $$A \cdot \left[ \begin{array} { l } 1 \\ a \end{array} \right] = \left[ \begin{array} { l } b \\ 4 \end{array} \right]$$ what is the sum $\mathrm { a } + \mathrm { b }$? A) 5 B) 7 C) 8 D) 9 E) 11
$$A = \left[ \begin{array} { r r } 1 & 0 \\ - 1 & 3 \end{array} \right], \quad B = \left[ \begin{array} { r r } - 1 & 1 \\ 0 & m \end{array} \right]$$ The matrices satisfy the equality $$\operatorname { det } ( A + B ) = \operatorname { det } ( A ) + \operatorname { det } ( B )$$ Accordingly, what is m? A) - 3 B) - 1 C) 0 D) 2 E) 4
$$3 x - y = 2$$ $$5 x + 2 y = 3$$ The matrix representation of the linear equation system is $$A \cdot \left[ \begin{array} { l } x \\ y \end{array} \right] = \left[ \begin{array} { l } 2 \\ 3 \end{array} \right]$$ Given that $$A \cdot \left[ \begin{array} { l } 1 \\ 2 \end{array} \right] = \left[ \begin{array} { l } a \\ b \end{array} \right]$$ what is the sum $\mathbf { a + b }$? A) 4 B) 6 C) 8 D) 10 E) 12
$$f ( x ) = \left\{ \begin{array} { c c } \frac { a x } { x + 2 b } \cdot \cot x & , x \neq 0 \\ 2 & , x = 0 \end{array} \right.$$ The function is continuous at the point $x = 0$. Accordingly, what is the ratio $\frac { a } { b }$? A) 1 B) 2 C) 4 D) $\frac { 1 } { 3 }$ E) $\frac { 1 } { 6 }$
$$f ( x ) = \left| \frac { 2 x - 1 } { x - 1 } \right|$$ The graph of the function intersects its horizontal asymptote at the point (a, b). Accordingly, what is the sum $a + b$? A) $\frac { 5 } { 2 }$ B) $\frac { 7 } { 2 }$ C) $\frac { 8 } { 3 }$ D) $\frac { 9 } { 4 }$
Let $f ( x ) = e ^ { x }$. The function $g$ is defined as $$g ( x ) = ( f \circ f ) ( x )$$ Accordingly, what is the value of the derivative of the $\mathbf { g }$ function at the point $\mathbf { x } = \boldsymbol { \ln } \mathbf { 2 }$, that is, $\mathbf { g } ^ { \prime } ( \ln 2 )$? A) e B) $\ln 2$ C) $2 \ln 2$ D) $e ^ { 2 }$ E) $2 e ^ { 2 }$
Let a and b be real numbers. In the rectangular coordinate plane, the parabola $$y = a x ^ { 2 } + b x$$ passes through the point $( 1,2 )$, and the tangent line to the parabola at this point intersects the y-axis at the point $( 0,1 )$. Accordingly, what is the product $a \cdot b$? A) - 3 B) - 2 C) - 1 D) 2 E) 4
In the rectangular coordinate plane $$y ^ { 2 } + \sin \left( x ^ { 2 } - 1 \right) = 4$$ What is the slope of the tangent line to the curve given by this equation at the point $\mathbf { P } ( - \mathbf { 1 } , - \mathbf { 2 } )$? A) - 1 B) $\frac { 1 } { 2 }$ C) 2 D) $\frac { - 1 } { 2 }$
Let $f$ be a function defined on the set of real numbers, and let the derivative of $f$ be denoted by $f ^ { \prime }$. The graph of the function $f ^ { \prime }$ is the parabolic curve shown in the figure. Accordingly, regarding the function f: I. $f ( 0 ) < 0$ II. It is decreasing on the interval (-a, a). III. $f ( a )$ is a local minimum value. Which of the following statements are definitely true? A) Only II B) Only III C) I and II D) II and III E) I, II and III
In the rectangular coordinate plane, rectangles are drawn such that two vertices lie on the x-axis and the other two vertices lie on the parabola $y = 27 - x ^ { 2 }$, and the rectangles lie between this parabola and the x-axis. Accordingly, what is the perimeter of the rectangle with the largest area? A) 40 B) 42 C) 44 D) 46 E) 48
The graph of a one-to-one and onto function f defined on the interval [2, 6] is given in the figure. Given that the area of the shaded region is 13 square units, $$\int _ { 2 } ^ { 6 } f ^ { - 1 } ( x ) d x$$ What is the value of the integral? A) 18 B) 19 C) 20 D) 21 E) 22
The graph of a function f defined on the interval [-1, 7] is given in the rectangular coordinate plane divided into unit squares as shown in the figure. Accordingly, what is the value of the integral $\int _ { - 1 } ^ { 7 } f ( x ) d x$? A) 2 B) 4 C) 6 D) 8 E) 10
Let k be a positive real number. The area of the bounded region between the line $\mathrm { y } = \mathrm { kx }$ and the parabola $y = x ^ { 2 }$ is $\frac { 9 } { 16 }$ square units. Accordingly, what is the value of $\mathbf { k }$? A) $\frac { 3 } { 2 }$ B) $\frac { 4 } { 3 }$ C) $\frac { 7 } { 4 }$ D) $\frac { 7 } { 6 }$ E) $\frac { 8 } { 5 }$
In the rectangular coordinate plane, the region between the lines $y = - x + 5$, $y = x + 3$ and the coordinate axes is shown below. What is the volume of the solid of revolution obtained by rotating this region $360 ^ { \circ }$ about the y-axis? A) $37 \pi$ B) $38 \pi$ C) $40 \pi$ D) $41 \pi$ E) $42 \pi$