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Q1 Complex Numbers Arithmetic Complex Division/Multiplication Simplification View

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

Q2 Number Theory Prime Number Properties and Identification View

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

Q3 Solving quadratics and applications Counting solutions or configurations satisfying a quadratic system View

Let $n$ and $k$ be positive integers. The value of $n _ { k }$ is defined as
- If $n$ is divisible by $k$, then $n _ { k } = \frac { n } { k }$ - If $n$ is not divisible by $k$, then $n _ { k } = 0$
Example: $$\begin{aligned} & 10 _ { 2 } = 5 \\ & 10 _ { 3 } = 0 \end{aligned}$$
Accordingly,
$$n _ { 2 } + n _ { 3 } = 10$$
what is the sum of the $n$ numbers that satisfy the equality?
A) 24 B) 28 C) 32 D) 36 E) 40

Q4 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View

On a circular playground, five players named Ali, Büşra, Cem, Deniz, and Ekin are playing with a ball in positions shown in the figure. In each turn of this game; the player holding the ball passes it to the third player after them in the direction of the arrow.
Initially, the ball is in Ali's hands and the game starts when Ali passes the ball to Deniz. Deniz received the ball on the 1st turn, Büşra on the 2nd turn, and the game continued in this way.
Accordingly, who received the ball on the 99th turn?
A) Ali B) Büşra C) Cem D) Deniz E) Ekin

Q5 Inequalities Ordering and Sign Analysis from Inequality Constraints View

Let $\mathrm { a }$, $\mathrm { b }$ and $c$ be non-zero real numbers,
$$\begin{aligned} & \mathrm { p } : \mathrm { a } + \mathrm { b } = 0 \\ & \mathrm { q } : \mathrm { a } + \mathrm { c } < 0 \\ & \mathrm { r } : \mathrm { c } < 0 \end{aligned}$$
the propositions are given.
$$( p \wedge q ) \Rightarrow r$$
Given that the proposition is false; what are the signs of $\mathbf { a }$, $\mathbf { b }$ and $\mathbf { c }$ respectively?
A) $+$, $-$, $+$ B) $+$, $-$, $-$ C) $-$, $-$, $+$ D) $+$, $+$, $-$

Q6 Proof True/False Justification View

Let $a$ and $b$ be integers. The notation $\mathrm { a } \mid \mathrm { b }$ means that $a$ divides $b$ exactly.
A student wants to prove that the proposition "If integers $a$, $b$ and $c$ satisfy the conditions $a \mid c$ and $b \mid c$, then $(a + b) \mid c$ also holds." is false by using the counterexample method.
Accordingly, which of the following could be the example given by the student?

Q7 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View

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

Q8 Inequalities Simultaneous/Compound Quadratic Inequalities View

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) $( - 2 , - 1 ) \cup ( 1,2 )$

Q9 Roots of polynomials Determine coefficients or parameters from root conditions View

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

Q10 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

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

Q11 Laws of Logarithms Solve a Logarithmic Equation View

The arithmetic mean of $\log _ { 4 } \mathrm { x }$ and $\log _ { 8 } \frac { 1 } { \mathrm { x } }$ is $\frac { 1 } { 2 }$.
Accordingly, what is the value of $\log _ { 16 } \mathbf { x }$?
A) $\frac { 1 } { 2 }$ B) $\frac { 3 } { 2 }$ C) $\frac { 5 } { 2 }$ D) $\frac { 1 } { 4 }$ E) $\frac { 5 } { 4 }$

Q12 Arithmetic Sequences and Series Find Specific Term from Given Conditions View

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

Q13 Completing the square and sketching Graph translation and resulting quadratic equation View

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

Q14 Discriminant and conditions for roots Root relationships and Vieta's formulas View

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

Q15 Combinations & Selection Counting Arrangements with Run or Pattern Constraints View

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

Q16 Probability Definitions Finite Equally-Likely Probability Computation View

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) $\frac { 1 } { 9 }$

Q18 Differential equations Qualitative Analysis of DE Solutions View

$$f ( x ) = \left\{ \begin{array} { l l l } 10 - x ^ { 2 } & , & x < 0 \\ a x + b & , & 0 \leq x \leq 3 \\ ( 1 - x ) ^ { 2 } & , & x > 3 \end{array} \right.$$
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

Q19 Stationary points and optimisation Analyze function behavior from graph or table of derivative View

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) $\mathrm { f } ( 2 ) < \mathrm { f } ( 1 ) < \mathrm { f } ( 0 )$

Q20 Differentiating Transcendental Functions Higher-order or nth derivative computation View

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

Q21 Tangents, normals and gradients Determine unknown parameters from tangent conditions View

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

Q22 Applied differentiation Applied modeling with differentiation View

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

Q23 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

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

Q24 Integration by Parts Integration by Parts within Function Analysis View

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

Q25 Areas by integration View

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

Q26 Areas by integration View

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) $\frac { 5 } { 6 }$

Q27 Trig Proofs Trigonometric Identity Simplification View

$$\frac { \cot \left( 34 ^ { \circ } \right) \cdot \sin \left( 44 ^ { \circ } \right) } { \sin \left( 22 ^ { \circ } \right) \cdot \sin \left( 56 ^ { \circ } \right) }$$
What is the equivalent of this expression?
A) $2 \cot \left( 22 ^ { \circ } \right)$ B) $2 \cos \left( 56 ^ { \circ } \right)$ C) $4 \sin \left( 44 ^ { \circ } \right)$ D) $4 \cos \left( 34 ^ { \circ } \right)$ E) $4 \tan \left( 56 ^ { \circ } \right)$

Q28 Quadratic trigonometric equations View

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) $2 \pi$

Q29 Trig Proofs Trigonometric Identity Simplification View

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) $\sec x$

Q30 Sine and Cosine Rules Heights and distances / angle of elevation problem View

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?

Q31 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View

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) $18 \sqrt { 2 }$

Q32 Circles Area and Geometric Measurement Involving Circles View

A square frame made by assembling four wires of equal length and fixed to the wall with nails at its corners as shown in Figure 1 covers an area of 100 square units on the wall.
As a result of the nails on corners A and B coming loose, one side slides down to form a rhombus shape as shown in Figure 2. In this frame, the height of corners A and B from the ground has decreased by 6 units each, while the position of the other two corners has not changed.
Accordingly, by how many square units has the area covered by the frame on the wall decreased?
A) 18 B) 20 C) 26 D) 30 E) 32

Q33 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View

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

Q34 Circles Optimization on a Circle View

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) $2 \sqrt { 10 }$

Q35 Linear transformations View

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

Q36 Conic sections Equation Determination from Geometric Conditions View

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) $3 - \sqrt { 2 }$

Q37 Vectors Introduction & 2D Dot Product Computation View

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) $\frac { 3 } { 10 }$

Q38 Vectors Introduction & 2D Magnitude of Vector Expression View

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

Q39 Vectors Introduction & 2D Magnitude of Vector Expression View

Three cubes, each with edge length 1 unit, are glued together such that at least one face of each cube completely overlaps with a face of another cube.
Accordingly, which of the following cannot be the distance between two selected vertices of the solid obtained in this way, in units?
A) $\sqrt { 7 }$ B) $\sqrt { 8 }$ C) $\sqrt { 9 }$ D) $\sqrt { 10 }$ E) $\sqrt { 11 }$

Q40 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View

In space, a plane E is given with points A and B on it, and a point P at a distance of 4 units from this plane.
The perpendicular projections of line segments PA and PB onto plane E, together with line segment AB, form an equilateral triangle with side length 2 units.
Accordingly, what is the product $| \mathbf { P A } | \cdot | \mathbf { P B } |$?
A) 8 B) 12 C) 16 D) 18 E) 20

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