turkey-yks

Q1 Indices and Surds Solving Exponential or Index Equations View

$$\left( 1 - 3 ^ { - 1 } + a ^ { - 1 } \right) ^ { - 3 } = 8$$
Given this, what is a?
A) $- 6$
B) $- 4$
C) $\frac { - 2 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 1 } { 6 }$

Q2 Completing the square and sketching Solving Equations Involving Surds View

$$\sqrt { x } - \sqrt { y } = \sqrt { x + y - 1 }$$
Given this, what is the product $x \cdot y$?
A) $\frac { 1 } { 3 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 2 } { 5 }$
E) $\frac { 4 } { 5 }$

Q3 Simultaneous equations Ratio and Proportion Problems View

$x , y$ are positive real numbers and
$$\frac { 2 y } { x + \frac { 1 } { y } } - \frac { 3 x } { y + \frac { 1 } { x } } = \frac { 5 x ^ { 2 } } { x \cdot y + 1 }$$
Given this, what is the ratio $\frac { x } { y }$?
A) $\frac { 2 } { 5 }$
B) $\frac { 1 } { 5 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 1 } { 3 }$
E) $\frac { 1 } { 2 }$

Q4 Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View

$$4 ^ { x } \cdot 6 ^ { x } \cdot 9 ^ { x } = 36$$
Given this, what is x?
A) $\frac { 2 } { 3 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 3 } { 8 }$
E) $\frac { 4 } { 9 }$

Q5 Inequalities Ordering and Sign Analysis from Inequality Constraints View

Given that $x < 0 < y$, I. $y - x ^ { - 1 }$ II. $x ^ { 2 } + y ^ { - 1 }$ III. $( x \cdot y ) ^ { - 1 }$ Which of these expressions have negative values?
A) Only I
B) Only II
C) Only III
D) I and III
E) II and III

Q6 Proof Prime Number Properties and Identification View

a, b are positive integers, p is a prime number and
$$a ^ { 3 } - b ^ { 3 } = p$$
Given this, which of the following is the equivalent of $a ^ { 2 } + b ^ { 2 }$ in terms of $p$?
A) $\frac { p + 1 } { 2 }$
B) $\frac { p + 3 } { 2 }$
C) $\frac { p + 2 } { 3 }$
D) $\frac { 2 p - 1 } { 2 }$
E) $\frac { 2 p + 1 } { 3 }$

Q7 Partial Fractions View

$\mathbf { a } , \mathbf { b } , \mathbf { c }$ are non-zero real numbers and $\mathbf { a } + \mathbf { b } + \mathbf { c } = \mathbf { a b }$. Given this,
$$\frac { a b + a c + b c + c ^ { 2 } } { a b c }$$
Which of the following is this expression equal to?
A) $\frac { a + 1 } { a }$
B) $\frac { b + 1 } { b }$
C) $\frac { c + 1 } { c }$
D) $\frac { b } { a }$
E) $\frac { b } { c }$

Q8 Inequalities Integer Solutions of an Inequality View

a, b are real numbers and
$$\begin{aligned} & 0 < a < 3 a ^ { 2 } \\ & b - 1 = 6 a \end{aligned}$$
Given this, what is the smallest integer value that b can take?
A) 3
B) 4
C) 5
D) 6
E) 7

Q9 Arithmetic Sequences and Series Factorial and Combinatorial Expression Simplification View

$$( n + 2 ) ! - ( n + 1 ) ! - n ! = 2 ^ { 3 } \cdot 3 \cdot 5 ^ { 2 } \cdot 7$$
Given this, what is $n$?
A) 5
B) 6
C) 7
D) 8
E) 9

Q10 Number Theory Divisibility and Divisor Analysis View

Let n be a positive integer. If every prime number p that divides n also divides $p ^ { 2 }$ into n, then n is called a powerful number.
Which of the following is NOT a powerful number?
A) 27
B) 64
C) 72
D) 99
E) 108

Q13 Number Theory Congruence Reasoning and Parity Arguments View

n is an integer greater than 1 and
$$\begin{aligned} & 73 \equiv 3 ( \bmod n ) \\ & 107 \equiv 2 ( \bmod n ) \end{aligned}$$
Given this, what is the sum of the possible values of n?
A) 39
B) 41
C) 47
D) 51
E) 54

Q14 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

$$f ( x ) = - 3 x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$$
Given this, what is the product $x ^ { 3 } \cdot f \left( \frac { 1 } { x } \right)$ equal to?
A) $x ^ { 3 } - 2 x ^ { 2 } + 5 x - 3$
B) $x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$
C) $3 x ^ { 3 } - 5 x ^ { 2 } + 2 x - 1$
D) $3 x ^ { 3 } - 2 x ^ { 2 } + 5 x + 1$
E) $5 x ^ { 3 } - x ^ { 2 } + 2 x - 3$

Q15 Composite & Inverse Functions Find or Apply an Inverse Function Formula View

$f : [ 1 , \infty ) \rightarrow [ 1 , \infty )$ is a function and
$$f \left( e ^ { x } \right) = \sqrt { x } + 1$$
Given this, what is the value of $f ^ { - 1 } ( 2 )$?
A) 1
B) $e - 1$
C) e
D) $e ^ { 2 }$
E) $\ln 2$

Q17 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View

$$\mathrm { P } ( \mathrm { x } ) = ( \mathrm { x } - 1 ) ^ { 4 } + ( \mathrm { x } - 1 ) ^ { 5 }$$
In this polynomial, what is the coefficient of the $x ^ { 3 }$ term?
A) 4
B) 6
C) 9
D) 10
E) 11

Q18 Factor & Remainder Theorem Remainder by Quadratic or Higher Divisor View

$$P ( x ) = x ^ { 11 } - 2 x ^ { 10 } + x - 2$$
What is the remainder when this polynomial is divided by $x ^ { 2 } - 5 x + 6$?
A) $3 ^ { 10 } + 1$
B) $3 ^ { 10 } - 1$
C) $3 ^ { 11 } + 1$
D) $3 ^ { 11 } - 1$

Q19 Polynomial Division & Manipulation View

A second-degree polynomial $P ( x )$ with leading coefficient 3 satisfies
$$P ( 1 ) - P ( 0 ) = 2$$
Given this, what is the value of $\mathbf { P } ( \mathbf { 2 } ) - \mathbf { P } ( \mathbf { 1 } )$?
A) 4
B) 5
C) 6
D) 7
E) 8

Q20 Discriminant and conditions for roots Root relationships and Vieta's formulas View

Let k be a positive real number. If the roots of the equation
$$2 x ^ { 2 } + k x - 1 = 0$$
have a difference of 2, what is k?
A) 1
B) 2
C) $\sqrt { 2 }$
D) $2 \sqrt { 2 }$
E) $\sqrt { 3 }$

Q21 Completing the square and sketching Two quadratic functions: intersection, tangency, or equality conditions View

The parabolas $f ( x )$ and $g ( x )$ whose graphs are shown above intersect each other at their vertices.
Given this, what is the value of $\mathbf { g } ( \mathbf { 0 } )$?
A) 3
B) 4
C) 5
D) 6
E) 7

Q22 Probability Definitions Optimization of Probability or Strategy View

A bag contains nine balls numbered from 1 to 9. Ayşe will choose a number from 1 to 9 and then draw a ball randomly from the bag. Ayşe wins the game if the sum of the number on the ball and the number she chose is at most 9 and their product is at least 9.
Which number should Ayşe choose so that her probability of winning the game is highest?
A) 2
B) 3
C) 4
D) 5
E) 6

Q23 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View

Given that $0 < \mathrm { x } < \pi$,
$$\sin ^ { 4 } x = \cos ^ { 4 } x$$
What is the sum of the $\mathbf { x }$ values that satisfy this equality?
A) $\frac { 3 \pi } { 2 }$
B) $\frac { 4 \pi } { 3 }$
C) $\frac { 5 \pi } { 4 }$
D) $\pi$
E) $2 \pi$

Q24 Reciprocal Trig & Identities View

$$\frac { \cot x } { \tan x + \cot x } = 4 \sin x - 3$$
Given this, what is the value of $\sin x$?
A) $3 - 2 \sqrt { 2 }$
B) $1 - \sqrt { 3 }$
C) $- 1 + \sqrt { 2 }$
D) $- 1 + \sqrt { 3 }$
E) $- 2 + 2 \sqrt { 2 }$

Q25 Addition & Double Angle Formulae Trigonometric Equation Solving via Identities View

Given that $\alpha , \beta \in \left[ 0 , \frac { \pi } { 2 } \right]$,
$$\sin ( \alpha - \beta ) = \sin \alpha \cdot \cos \beta$$
Which of the following is true?
A) $\alpha = 0$ or $\beta = \frac { \pi } { 2 }$
B) $\alpha = 0$ or $\beta = \frac { \pi } { 4 }$
C) $\alpha = \frac { \pi } { 2 }$ or $\beta = 0$
D) $\alpha = \frac { \pi } { 2 }$ or $\beta = \frac { \pi } { 2 }$
E) $\alpha = \frac { \pi } { 4 }$ or $\beta = 0$

Q26 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View

$z$ is a complex number, $\operatorname { Im } ( z ) \neq 0$ and $z ^ { 3 } = - 1$. Given this,
$$( z - 1 ) ^ { 10 }$$
Which of the following is this expression equal to?
A) $z + 1$
B) $z - 1$
C) $z$
D) $- z$
E) $- z - 1$

Q27 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View

$$\frac { | z | ^ { 2 } + z } { \bar { z } } = z + i$$
Which of the following is the set of complex numbers z that satisfy this equality? (R is the set of real numbers.)
A) $\{ a + a i \mid a \in R , a \neq 0 \}$
B) $\{ a - a i \mid a \in R , a \neq 0 \}$
C) $\{ a + 2 a i \mid a \in R , a \neq 0 \}$
D) $\{ a - 2 a i \mid a \in R , a \neq 0 \}$
E) $\{ 2 a - a i \mid a \in R , a \neq 0 \}$

Q28 Complex Numbers Arithmetic Trigonometric/Polar Form and De Moivre's Theorem View

$$\frac { 1 } { z } = \left( \cos \frac { \pi } { 4 } + i \sin \frac { \pi } { 4 } \right)$$
Which of the following is the complex number z that satisfies this equation?
A) $\sqrt { 2 } ( 1 + i )$
B) $\sqrt { 2 } ( 1 - \mathrm { i } )$
C) $\frac { \sqrt { 2 } } { 2 } ( 1 + i )$
D) $\frac { \sqrt { 2 } } { 2 } ( 1 - \mathrm { i } )$
E) $\frac { 1 + i } { 2 }$

Q29 Laws of Logarithms Solve Exponential Equation for Unknown Variable View

$$\log _ { 8 } \left( \log _ { 9 } ( \sqrt { x + 1 } ) \right) = \frac { - 2 } { 3 }$$
Given this, what is x?
A) 2
B) 3
C) 5
D) 7
E) 8

Q30 Laws of Logarithms Find or Apply an Inverse Function Formula View

$$\begin{aligned} & f ( x ) = - \log _ { 2 } x \\ & g ( x ) = \log _ { 10 } x \end{aligned}$$
Given this, what is the value of a that satisfies the equality $\left( \right.$ gof $\left. ^ { - 1 } \right) ( a ) = \ln 2$?
A) $\ln 2$
B) $\frac { \ln 2 } { \ln 10 }$
C) $\frac { \ln 10 } { \ln 2 }$
D) $\ln \left( \frac { 1 } { 10 } \right)$
E) $\ln \left( \frac { 1 } { 2 } \right)$

Q31 Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View

$$9 ^ { x + 1 } + 3 ^ { x + 1 } - 6 = 0$$
Given this, which of the following is x?
A) $\frac { \ln 3 } { \ln 2 }$
B) $\frac { 1 + \ln 3 } { \ln 2 }$
C) $\frac { 2 + \ln 3 } { \ln 2 }$
D) $\frac { 3 + \ln 2 } { \ln 3 }$
E) $\frac { \ln 2 - \ln 3 } { \ln 3 }$

Q32 Sequences and series, recurrence and convergence Direct term computation from recurrence View

Let $a _ { 1 } , a _ { 2 }$ be real numbers. The sequence $\left( a _ { n } \right)$ satisfies the relation
$$a _ { n + 2 } = a _ { n + 1 } + a _ { n } \quad ( n = 1,2 , \cdots )$$
Given that $a _ { 8 } = 6$, what is the sum $a _ { 6 } + a _ { 9 }$?
A) 9
B) 10
C) 12
D) 15
E) 16

Q33 Number Theory Properties of Integer Sequences and Digit Analysis View

For a positive integer n, the greatest odd divisor of n is denoted by $\overline{n}$. The terms of the sequence $( a _ { n } )$ are defined for $n = 1,2 , \ldots$ as
$$a _ { n } = \begin{cases} n + 1 , & \text{if } n \equiv 1 ( \bmod 4 ) \\ n - 1 , & \text{if } n \equiv 3 ( \bmod 4 ) \end{cases}$$
Given this, what is the difference $a _ { 18 } - a _ { 12 }$?
A) 2
B) 4
C) 6
D) 8
E) 10

Q34 Matrices Determinant and Rank Computation View

$$A = \left[ \begin{array} { l l } 3 & 2 \\ 0 & 1 \end{array} \right]$$
Given this, what is the value of the determinant $\left| A - A ^ { \top } \right|$?
A) 3
B) 4
C) 5
D) 6
E) 9

Q35 Matrices Compute eigenvalues of a given matrix View

Let m be a positive real number and $u = \left[ \begin{array} { l l } x & y \end{array} \right]$. Given that
$$\mathrm { u } \cdot \left[ \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right] = \mathrm { u } \cdot \left[ \begin{array} { c c } \mathrm { m } & 0 \\ 0 & \mathrm {~m} \end{array} \right]$$
the matrix equation has infinitely many solutions for u, what is m?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) 3
E) 4

Q36 Matrices Matrix Algebra and Product Properties View

Let A be a $3 \times 3$ matrix. Given that
$$\begin{aligned} & { \left[ \begin{array} { l l l } 2 & 1 & 3 \end{array} \right] \cdot A = \left[ \begin{array} { l l l } 0 & 2 & 2 \end{array} \right] } \\ & { \left[ \begin{array} { l l l } 1 & 4 & 0 \end{array} \right] \cdot A = \left[ \begin{array} { l l l } 3 & 1 & 5 \end{array} \right] } \end{aligned}$$
What is the product $\left[ \begin{array} { l l l } 5 & 6 & 6 \end{array} \right] \cdot A$ equal to?
A) $\left[ \begin{array} { l l l } 2 & 1 & 3 \end{array} \right]$
B) $\left[ \begin{array} { l l l } 3 & 3 & 7 \end{array} \right]$
C) $\left[ \begin{array} { l l l } 3 & 5 & 9 \end{array} \right]$
D) $\left[ \begin{array} { l l l } 6 & 2 & 10 \end{array} \right]$
E) $\left[ \begin{array} { l l l } 6 & 4 & 12 \end{array} \right]$

Q39 Exponential Functions Modular Arithmetic Computation View

$$\lim _ { x \rightarrow \infty } \frac { e ^ { - 3 x } + e ^ { 2 x } } { \ln x + 3 e ^ { 2 x } }$$
What is the value of this limit?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 1 } { 3 }$
D) 0
E) 1

Q40 Indefinite & Definite Integrals Recovering Function Values from Derivative Information View

Below, the graph of the derivative of a function f is given. Given that $f ( 0 ) = 1$, what is the value of $f ( 2 )$?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 4 } { 3 }$
D) $\frac { - 1 } { 2 }$
E) $\frac { - 1 } { 3 }$

Q41 Differentiating Transcendental Functions Higher-order or nth derivative computation View

$$f ( x ) = e ^ { 2 x } - e ^ { - 2 x }$$
What is the value of the 15th order derivative of the function at the point $x = \ln 2$, that is $\mathbf { f } ^ { \mathbf { ( 1 5 ) } } ( \mathbf { \ln } \mathbf { 2 } )$?
A) $17 \cdot 2 ^ { 13 }$
B) $15 \cdot 2 ^ { 13 }$
C) $9 \cdot 2 ^ { 13 }$
D) $15 \cdot 2 ^ { 12 }$
E) $7 \cdot 2 ^ { 12 }$

Q42 Implicit equations and differentiation Horizontal tangent point on implicit curve (single-step) View

In the analytic plane
$$x y ^ { 2 } - x ^ { 3 } y - 6 = 0$$
Given that the tangent line passing through the point $\mathbf { P } \left( \mathbf { x } _ { 0 } , \mathbf { y } _ { 0 } \right)$ on the curve given by the equation is parallel to the x-axis, what is $\mathrm { x } _ { 0 }$?
A) $- 3$
B) $- 2$
C) $\frac { - 3 } { 2 }$

Q43 Tangents, normals and gradients Determine unknown parameters from tangent conditions View

Given that the function $f$ has derivative $f ^ { \prime } ( x ) = 3 x ^ { 2 }$ and the tangent line at the point $x = a ( a > 0 )$ is the line $y - 12 x + 14 = 0$, what is the value of $f ( 1 )$?
A) $- 2$
B) 0
C) 1
D) 3
E) 5

Q44 Stationary points and optimisation Geometric or applied optimisation problem View

A tour company charges 140 TL per person for a tour it will organize. If the number of registered participants exceeds 80, a refund of 50 kuruş will be made to all participants for each person above 80. The capacity is limited to 200 people.
For example, if 100 people participate in the tour, everyone receives a 10 TL refund and the per-person fee is 130 TL.
Accordingly, how many people should participate in the tour so that the company's revenue from participants is maximum?
A) 160
B) 165
C) 175
D) 180
E) 185

Q45 Integration by Substitution Definite Integral Evaluation via Substitution or Standard Forms View

$$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin 2 x \cdot \cot x \, d x$$
What is the value of this integral?
A) $\frac { \pi + 1 } { 2 }$
B) $\frac { \pi + 1 } { 3 }$
C) $\frac { \pi + 2 } { 4 }$
D) $\frac { \pi - 1 } { 6 }$
E) $\frac { \pi - 2 } { 6 }$

Q46 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

For a continuous function f defined on the set of real numbers,
$$\int _ { 1 } ^ { 3 } f ( x ) d x = 5$$
is known. Accordingly,
$$\int _ { 0 } ^ { 1 } ( 4 + f ( 2 x + 1 ) ) d x$$
What is the value of this integral?
A) 1
B) 2
C) 3
D) $\frac { 5 } { 2 }$
E) $\frac { 13 } { 2 }$

Q47 Areas Between Curves Select Correct Integral Expression View

The function $f$ is one-to-one, and the shaded region between the lines $y = x$ and $x = 1$ and the curve $y = f ( x )$ in the first quadrant is given below.
Which of the following is the expression of the area of the shaded region in terms of $\mathbf { f } ^ { - \mathbf { 1 } } ( \mathbf { x } )$?
A) $\int _ { 0 } ^ { 2 } f ^ { - 1 } ( x ) d x$
B) $\int _ { 0 } ^ { 2 } \left( 2 - f ^ { - 1 } ( x ) \right) d x$
C) $\int _ { 0 } ^ { 1 } \left( x - f ^ { - 1 } ( x ) \right) d x$
D) $\int _ { 0 } ^ { 1 } \left( 2 - f ^ { - 1 } ( x ) \right) d x + \int _ { 1 } ^ { 2 } f ^ { - 1 } ( x ) d x$
E) $\int _ { 0 } ^ { 1 } \left( x - f ^ { - 1 } ( x ) \right) d x + \int _ { 1 } ^ { 2 } \left( 1 - f ^ { - 1 } ( x ) \right) d x$

Q48 Numerical integration Riemann Sum Computation from a Given Formula View

$$\begin{aligned} & f : [ 1,3 ] \rightarrow [ 2,10 ] \\ & f ( x ) = 1 + x ^ { 2 } \end{aligned}$$
The interval $[ 1,3 ]$ is divided into two subintervals of equal length, and the right endpoints of these subintervals are marked as $x _ { 1 }$ and $x _ { 2 }$. Then, two rectangles are drawn with each subinterval as the base and heights $f \left( x _ { 1 } \right), f \left( x _ { 2 } \right)$ respectively.
If the sum of the areas of these rectangles is A and the area of the region between the function f and the x-axis is B, what is the difference A - B in square units?
A) $\frac { 11 } { 2 }$
B) $\frac { 13 } { 3 }$
C) $\frac { 15 } { 4 }$
D) $\frac { 19 } { 6 }$
E) $\frac { 23 } { 6 }$

Q49 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Let $n$ be a natural number,
$$\begin{aligned} & f _ { n } : [ n , n + 1 ) \rightarrow \left[ 0 , \frac { 1 } { 2 ^ { n } } \right) \\ & f _ { n } ( x ) = \frac { ( x - n ) ^ { 2 } } { 2 ^ { n } } \end{aligned}$$
The regions between the functions defined in this form and the x-axis are given shaded in the figure below.
Accordingly, what is the sum of the areas of all shaded regions in square units?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 5 } { 6 }$
D) $\frac { 8 } { 9 }$
E) $\frac { 11 } { 12 }$

Q50 Volumes of Revolution Volume of Revolution about a Horizontal Axis (Evaluate) View

In the analytic plane; the region bounded by the x-axis, the line $x + y = 2$, and the curve $y = \sqrt { x }$ is rotated $360 ^ { \circ }$ around the x-axis.
What is the volume of the solid of revolution obtained in cubic units?
A) $\frac { \pi } { 2 }$
B) $\frac { 2 \pi } { 3 }$
C) $\frac { 3 \pi } { 4 }$
D) $\frac { 5 \pi } { 6 }$
E) $\frac { 7 \pi } { 6 }$

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