Let a be a real number. In complex numbers, $$\frac { 1 - a i } { a - i } = i$$ the equality is given. Accordingly, what is a? A) 4 B) 3 C) 2 D) 1 E) 0
Let $x$, $y$ and $z$ be distinct prime numbers, $$\begin{aligned}
& x ( z - y ) = 18 \\
& y ( z - x ) = 40
\end{aligned}$$ the equalities are given. Accordingly, what is the sum $\mathbf { x } + \mathbf { y } + \mathbf { z }$? A) 17 B) 19 C) 21 D) 23 E) 25
Let $a$ and $b$ be non-zero real numbers. A function $f$ defined on the set of real numbers $$\begin{aligned}
& f ( a x + b ) = x \\
& f ( a ) = \frac { b } { a }
\end{aligned}$$ satisfies the equalities. Accordingly, what is the value of $\mathrm { f } ( 0 )$? A) $\frac { - 1 } { 2 }$ B) $\frac { - 1 } { 3 }$ C) $\frac { - 2 } { 3 }$ D) 1 E) 2
In the Cartesian coordinate plane, the graphs of functions $f$, $g$ and $h$ whose domains consist of real numbers are given in the figure. Accordingly, for $x \in [ - 2,2 ]$, $$\begin{aligned}
& f ( x ) \cdot g ( x ) > 0 \\
& g ( x ) \cdot h ( x ) < 0
\end{aligned}$$ the solution set of the system of inequalities is which of the following? A) $( - 2 , - 1 )$ B) $( - 1,0 )$ C) $( 1,2 )$ D) $( - 2 , - 1 ) \cup ( 1,2 )$
A 4th degree polynomial $P ( x )$ with real coefficients and leading coefficient 1 satisfies $$P ( x ) = P ( - x )$$ for every real number $x$. $$P ( 2 ) = P ( 3 ) = 0$$ Given that, what is $\mathbf { P ( 1 ) }$? A) 12 B) 18 C) 24 D) 30 E) 36
On a ruler-like scale with integers from 1 to 50 written on it, the distance of each integer $n$ from 1 is $\log n$ units. When two identical rulers with this property are placed one below the other as shown in the figure, the number 42 on the upper ruler aligns with the number 28 on the lower ruler, and the number 33 on the upper ruler aligns with the number $x$ on the lower ruler. Accordingly, what is $x$? A) 18 B) 19 C) 20 D) 21 E) 22
For an arithmetic sequence $(a_n)$ with distinct terms and common difference $r$, $$\begin{aligned}
& a _ { 1 } = 3 \cdot r \\
& a _ { 6 } = a _ { 2 } \cdot a _ { 4 }
\end{aligned}$$ the equalities are given. Accordingly, what is $\mathbf { a } _ { \mathbf { 1 0 } }$? A) 10 B) 8 C) 6 D) 4 E) 2
Let $a$ and $b$ be positive real numbers. In the Cartesian coordinate plane, using the parabola $$p ( x ) = ( x - a ) ^ { 2 } - b$$ that passes through the origin, three parabolas defined as $$\begin{aligned}
& p ( x + a ) + b \\
& p ( x + a ) - b \\
& p ( x - a ) - b
\end{aligned}$$ have their vertices at the vertices of a triangle with an area of 16 square units. Accordingly, what is the sum $a + b$? A) 6 B) 9 C) 12 D) 15 E) 18
Let $m$ and $n$ be two non-zero and distinct real numbers, $$x ^ { 2 } + ( m + 1 ) x + n - m = 0$$ one of the roots of the equation is the number $m - n$. Accordingly, what is the ratio $\frac { \mathbf { n } } { \mathbf { m } }$? A) 2 B) 3 C) 4 D) 5 E) 6
If the arrangement of letters in a word from left to right is the same as from right to left, this word is called a palindrome word. For example; NEDEN is a palindrome word. Engin will create a 5-letter palindrome word using each of 3 distinct vowels and 4 distinct consonants as many times as he wants. In this word, two vowels should not be adjacent and two consonants should not be adjacent either. Accordingly, how many different palindrome words can Engin create that satisfy these conditions? A) 72 B) 84 C) 96 D) 108 E) 120
There is an ant at each of the vertices $K$ and $L$ of a regular tetrahedron. Each of these ants starts walking along one of the edges emanating from their respective corners, chosen at random, and stops when reaching the other end of that edge. Accordingly, what is the probability that the ants meet? A) $\frac { 1 } { 3 }$ B) $\frac { 2 } { 3 }$ C) $\frac { 1 } { 4 }$ D) $\frac { 3 } { 4 }$ E) $\frac { 1 } { 9 }$
$$f ( x ) = \left\{ \begin{array} { l l l }
10 - x ^ { 2 } & , & x < 0 \\
a x + b & , & 0 \leq x \leq 3 \\
( 1 - x ) ^ { 2 } & , & x > 3
\end{array} \right.$$ The function is continuous on the set of real numbers. Accordingly, what is the sum $\mathbf { a } + \mathbf { b }$? A) 16 B) 15 C) 12 D) 9 E) 8
The graph of the derivative function $f ^ { \prime }$ of a function f defined on the set of real numbers is given in the following Cartesian coordinate plane. Accordingly; what is the correct ordering of the values $\mathbf { f } ( \mathbf { 0 } )$, $\mathbf { f } ( \mathbf { 1 } )$ and $\mathbf { f } ( \mathbf { 2 } )$? A) $\mathrm { f } ( 0 ) < \mathrm { f } ( 1 ) < \mathrm { f } ( 2 )$ B) $\mathrm { f } ( 0 ) < \mathrm { f } ( 2 ) < \mathrm { f } ( 1 )$ C) $f ( 1 ) < f ( 2 ) < f ( 0 )$ D) $\mathrm { f } ( 2 ) < \mathrm { f } ( 0 ) < \mathrm { f } ( 1 )$ E) $\mathrm { f } ( 2 ) < \mathrm { f } ( 1 ) < \mathrm { f } ( 0 )$
What is the value of the 16th order derivative $f ^ { ( 16 ) } ( x )$ of the function $f ( x ) = e ^ { x } \cdot \cos x$ at the point $x = 0$? A) 32 B) 64 C) 128 D) 256 E) 512
Let a, b and c be real numbers. The equation of the tangent line to the curve $$y = \frac { a } { x + a }$$ at point $P ( a , b )$ is given in the form $$y = \frac { - x } { 8 } + c$$ Accordingly, what is the sum $a + b + c$? A) $\frac { 7 } { 4 }$ B) $\frac { 11 } { 4 }$ C) $\frac { 13 } { 4 }$ D) 2 E) 3
An internet company can serve at most 1000 customers and can reach this number when it sets the monthly internet fee at 40 TL. The company has observed that after each 5 TL increase in the monthly internet fee, the number of customers decreases by 50. At what monthly internet fee should this company set its rate to maximize the total revenue from the monthly internet fee? A) 55 B) 60 C) 65 D) 70 E) 75
For an increasing and continuous function f defined on the set of real numbers, $$\begin{aligned}
& f ( 0 ) = 2 \\
& f ( 1 ) = 3 \\
& f ( 2 ) = 4
\end{aligned}$$ equalities are given. Accordingly, the value of the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ could be which of the following? A) 4 B) 4.5 C) 6 D) 7.5 E) 8
In a mathematics class where the topic of integration by parts is being taught, Teacher Ebru writes on the board $$\int u d v = u v - \int v d u$$ rule. Then, in solving a problem, Mehmet chooses functions $f ( x )$ and $g ( x )$ in place of u and v respectively and applies this rule to obtain $$\int f ( x ) g ^ { \prime } ( x ) d x = \frac { f ( x ) } { x } - \int \frac { 2 } { x ^ { 2 } } d x$$ equality. Given that $\mathbf { f } ( \mathbf { 1 } ) = \mathbf { 2 }$, what is the value of $\mathbf { f } ( \mathbf { e } )$? A) 2 B) 4 C) 6 D) 8 E) 10
In the Cartesian coordinate plane, the graphs of functions $f$, $g$ and $h$ are shown below. The areas of the shaded regions A1, A2 and A3 shown in the figure are 1, 3 and 9 square units, respectively. Accordingly, $$\int _ { a } ^ { c } ( h ( x ) - g ( x ) ) d x + \int _ { b } ^ { d } ( f ( x ) - h ( x ) ) d x$$ what is the value of the integral? A) 5 B) 8 C) 12 D) 13 E) 17
Let a and b be positive real numbers. In the Cartesian coordinate plane, the region between the curve $$y = a x ^ { 2 } + b$$ and the lines $x = 0$, $x = 2$ and $y = 0$ is divided by the line passing through points $(2,0)$ and $(0, b)$ into two regions whose areas have a ratio of 3. Accordingly, what is the ratio $\frac { \mathbf { a } } { \mathbf { b } }$? A) $\frac { 1 } { 2 }$ B) $\frac { 2 } { 3 }$ C) $\frac { 3 } { 4 }$ D) $\frac { 4 } { 5 }$ E) $\frac { 5 } { 6 }$
For $0 < x < \pi$, $$\frac { \sin x \cdot \cos x } { \sin x + \cos x } = \frac { \sin x - \cos x } { 2 }$$ What is the sum of the $\mathbf { x }$ values that satisfy the equality? A) $\frac { \pi } { 2 }$ B) $\frac { 5 \pi } { 4 }$ C) $\frac { 7 \pi } { 4 }$ D) $\pi$ E) $2 \pi$
Below are shown a semicircle with center O and radius 1 unit, and right triangles OAB and ODC. Points A and C lie on both the triangle OAB and the semicircle. Accordingly, $$\frac { | \mathrm { AB } | + | \mathrm { BC } | } { | \mathrm { CD } | + | \mathrm { DA } | }$$ What is the equivalent of this ratio in terms of x? A) $\sin x$ B) $\tan x$ C) $\cot x$ D) $\csc x$ E) $\sec x$
Captain Temel will take the tourists on his boat from island A to island B in the morning, from island B to island C at noon, and from island C to island A in the evening. The points where the boat will dock at the islands are marked as the vertices of a triangle ABC where side AB equals side BC, as shown in the figure. Since Captain Temel knows he will travel in the dark on the return journey, as he travels from A to B and from B to C, he notes on a piece of paper the angle between the compass needle pointing north and the path he follows. Accordingly, how should Captain Temel set his compass to go from C to A?
In the Cartesian coordinate plane; a triangle with one vertex at the origin and the other vertices on the lines $y = x$ and $y = - x$ has its medians intersecting at point $(2,4)$. Accordingly, what is the area of this triangle in square units? A) 18 B) 24 C) 27 D) $9 \sqrt { 2 }$ E) $18 \sqrt { 2 }$
The square shown in the figure in the Cartesian coordinate plane is divided into two regions of equal area by a line with slope $\frac { - 1 } { 4 }$. If this line intersects the x-axis at point $(a, 0)$, what is a? A) 12 B) 14 C) 16 D) 18 E) 20
In the Cartesian coordinate plane, two circles with one centered at $(12,0)$ and the other centered at $(0,9)$ intersect only at point $(4,6)$. What is the distance between the points on these circles that are closest to the origin? A) $\sqrt { 5 }$ B) $\sqrt { 10 }$ C) $\sqrt { 13 }$ D) $2 \sqrt { 5 }$ E) $2 \sqrt { 10 }$
In the Cartesian coordinate plane, the square $ABCD$ with vertex coordinates $$\mathrm { A } ( - 1 , - 1 ) , \mathrm { B } ( 1 , - 1 ) , \mathrm { C } ( 1,1 ) , \mathrm { D } ( - 1,1 )$$ is given below. To this square, the following transformations are applied in order: - Rotation counterclockwise by $45 ^ { \circ }$ about the origin, - Reflection with respect to the y-axis, - Rotation clockwise by $45 ^ { \circ }$ about the origin. In the final state, which of the following are the vertex points of this square whose coordinates remain unchanged? A) A and B B) A and C C) A and D D) B and C E) C and D
In the Cartesian coordinate plane, an ellipse with center at the origin and foci at points E and F is given below. The vertical line drawn from point F intersects the ellipse at points, and the point with positive y-coordinate is denoted by K. The equation of the line passing through points K and E is $\mathrm { y } = \mathrm { x } + 1$. Accordingly, what is the value of a? A) $\sqrt { 2 } + 1$ B) $\sqrt { 3 } + 2$ C) $\sqrt { 5 } + 1$ D) $3 - \sqrt { 2 }$
In the analytic plane, the sides of a regular pentagon are named as vectors $\vec { a } , \vec { b } , \vec { c } , \vec { d }$ and $\vec { e }$ as shown in the figure. Accordingly, what is the probability that the dot product of two vectors randomly selected from these five vectors is positive? A) $\frac { 1 } { 2 }$ B) $\frac { 1 } { 5 }$ C) $\frac { 2 } { 5 }$ D) $\frac { 1 } { 10 }$ E) $\frac { 3 } { 10 }$
In the Cartesian coordinate plane, vectors $\overrightarrow { \mathrm { u } _ { 1 } } = ( 3,4 )$ and $\overrightarrow { \mathrm { u } _ { 2 } } = ( 8 , - 6 )$ are given. For a vector $\vec { V }$ taken in this plane, the perpendicular projection vector onto the $\overrightarrow { u _ { 1 } }$ vector is 3 units, and the perpendicular projection vector onto the $\overrightarrow { u _ { 2 } }$ vector is 1 unit in length. Accordingly, what is the length of the $\vec { v }$ vector in units? A) $\sqrt { 5 }$ B) $\sqrt { 10 }$ C) $5 \sqrt { 5 }$ D) 5 E) 10