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Q1 Indices and Surds Number-Theoretic Reasoning with Indices View

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) $c < b < a$

Q2 Number Theory Prime Number Properties and Identification View

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) $45 \cdot r$

Q3 Simultaneous equations View

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

Q4 Number Theory GCD, LCM, and Coprimality View

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

Q5 Inequalities Simultaneous/Compound Quadratic Inequalities View

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

Q6 Function Transformations View

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) $-, -, -$

Q7 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

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) $( 4,5 )$

Q8 Proof Computation of a Limit, Value, or Explicit Formula View

Regarding a two-digit natural number $AB$, the following propositions are given: p: The number $AB$ is even. q: The number $A^{AB}$ is prime. r: $A + B = 11$
If the proposition $(p \Rightarrow q) \wedge (q' \wedge r)$ is true, what is the product $A \cdot B$?
A) 18
B) 20
C) 24
D) 28
E) 30

Q9 Discriminant and conditions for roots Root relationships and Vieta's formulas View

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) $\frac{1}{4}$

Q10 Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View

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?

Q11 Combinations & Selection Basic Combination Computation View

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

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

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

Q13 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

On a calculator, when an operation is performed, the machine displays the result as a whole number if the result is an integer, or displays the integer part along with the first two decimal places after the decimal point if the result is a decimal number.
When Nevzat performs the operation $\ln ( 9{,}6 )$ on this calculator, he sees the value 2.26 on the screen, and when he performs the operation $\ln ( 0{,}3 )$, he sees the value $-1{,}20$ on the screen.
When Nevzat performs the operation $\ln ( 0{,}5 )$ on this calculator, what value does he see on the screen?
A) $-0{,}61$
B) $-0{,}65$
C) $-0{,}69$
D) $-0{,}73$
E) $-0{,}77$

Q14 Combinations & Selection Pigeonhole Principle Application View

A project team of 100 people has a certain number of projects, and everyone in the team will be assigned to some of these projects. It is desired that everyone in the team works on an equal number of projects, but no two people work on exactly the same projects. This condition cannot be satisfied if everyone works on 3 projects, but it can be satisfied if everyone works on 4 projects.
Accordingly, how many projects does the team have?
A) 6
B) 7
C) 8
D) 9
E) 10

Q15 Probability Definitions Conditional Probability and Bayes' Theorem View

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) $\frac { 1 } { 6 }$

Q16 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View

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

Q17 Laws of Logarithms Analyze a Logarithmic Function (Limits, Monotonicity, Zeros, Extrema) View

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

Q18 Composite & Inverse Functions Find or Apply an Inverse Function Formula View

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

Q19 Differentiating Transcendental Functions Higher-order or nth derivative computation View

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

Q21 Applied differentiation Applied modeling with differentiation View

The amount of fuel consumed in 1 hour by a rocket moving at a constant speed of $V$ kilometers per hour is calculated by the function
$$f ( V ) = \frac { V ^ { 3 } } { 20 } - 7 \cdot V ^ { 2 } + 265 \cdot V$$
in units.
Accordingly, what is the minimum amount of fuel in units that this rocket must consume to travel 100 kilometers at a constant speed?
A) 1000
B) 2000
C) 3000
D) 4000
E) 5000

Q22 Stationary points and optimisation Determine parameters from given extremum conditions View

Let $a$ and $b$ be real numbers. It is known that the polynomial
$$f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 1$$
is

increasing on the interval $( - \infty , 1 )$,
decreasing on the interval $( 1,5 )$,
increasing on the interval $( 5 , \infty )$.

Accordingly, what is $f ( 2 )$?
A) 0
B) 3
C) 6
D) 9
E) 12

Q23 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View

$$\int \frac { ( 3 \sqrt { x } + 2 ) ^ { 5 } } { \sqrt { x } } d x$$
Which of the following is this integral equal to? (c is an arbitrary constant.)
A) $\frac { 1 } { 18 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
B) $\frac { 1 } { 9 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
C) $\frac { 2 } { 9 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
D) $\frac { 1 } { 3 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$
E) $\frac { 2 } { 3 } \cdot ( 3 \sqrt { x } + 2 ) ^ { 6 } + c$

Q24 Indefinite & Definite Integrals Recovering Function Values from Derivative Information View

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

Q25 Indefinite & Definite Integrals Finding a Function from an Integral Equation View

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

Q26 Areas by integration View

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) $\frac { 23 } { 3 }$

Q27 Standard trigonometric equations Evaluate trigonometric expression given a constraint View

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) $z < y < x$

Q28 Sine and Cosine Rules Find an angle using the cosine rule View

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

Q29 Reciprocal Trig & Identities View

$$\frac { \cos ^ { 2 } \left( 80 ^ { \circ } \right) + 5 \sin ^ { 2 } \left( 80 ^ { \circ } \right) - 3 } { \cos \left( 50 ^ { \circ } \right) }$$
Which of the following is this expression equal to?
A) $\cot \left( 50 ^ { \circ } \right)$
B) $\sec \left( 20 ^ { \circ } \right)$
C) $\sec \left( 40 ^ { \circ } \right)$
D) $\operatorname { cosec } \left( 20 ^ { \circ } \right)$
E) $\operatorname { cosec } \left( 40 ^ { \circ } \right)$

Q30 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View

Let $0 \leq x \leq \frac { 3 \pi } { 2 }$. Given that
$$| \sin x | = \cos \left( 50 ^ { \circ } \right)$$
what is the sum of the $x$ values that satisfy this equation?
A) $\frac { 13 \pi } { 18 }$
B) $\frac { 11 \pi } { 90 }$
C) $\frac { 3 \pi } { 2 }$
D) $\frac { 31 \pi } { 18 }$
E) $\frac { 20 \pi } { 9 }$

Q31 Reciprocal Trig & Identities View

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) $\tan \left( \frac { \theta } { 2 } \right)$

Q32 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View

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

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

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

Q34 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View

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

Q35 Radians, Arc Length and Sector Area View

Ayşe and Ferhat enter a store to buy pizza. From a whole pizza divided into 13 circular slices in this store; the 2 slices that Ayşe buys are identical to each other, while the 11 slices that Ferhat buys are also identical to each other.
Later, they combine three of these slices to obtain a semicircular pizza.
Accordingly, what is the measure of the central angle of one of the larger slices in degrees?
A) 90
B) 81
C) 75
D) 72
E) 60

Q38 Straight Lines & Coordinate Geometry Reflection and Image in a Line View

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) $32$

Q39 Circles Circle-Line Intersection and Point Conditions View

In the rectangular coordinate plane, the line $y = m x$
$$x ^ { 2 } - 26 x + y ^ { 2 } + 144 = 0$$
intersects the circle at two different points.
Accordingly, which of the following is the interval showing all possible values of $m$?
A) $\left( - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right)$
B) $\left( - \frac { 3 } { 8 } , \frac { 3 } { 8 } \right)$
C) $\left( - \frac { 4 } { 9 } , \frac { 4 } { 9 } \right)$
D) $\left( - \frac { 5 } { 12 } , \frac { 5 } { 12 } \right)$
E) $\left( - \frac { 7 } { 24 } , \frac { 7 } { 24 } \right)$

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