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Q4 Permutations & Arrangements Factorial and Combinatorial Expression Simplification View

$\frac { 6 ! + 7 ! } { ( 4 ! ) ^ { 2 } }$\ What is the result of this operation?\ A) 10\ B) 12\ C) 15\ D) 18\ E) 20

Q5 Solving quadratics and applications Geometric or real-world application leading to a quadratic equation View

The numbers $\frac { x } { y }$, $x - y$, and $x$ are three consecutive even integers arranged from smallest to largest.\ Accordingly, what is the sum $\mathrm{x} + \mathrm{y}$?\ A) 8\ B) 10\ C) 12\ D) 14\ E) 16

Q7 Number Theory GCD, LCM, and Coprimality View

Let $a$ and $b$ be distinct positive integers such that LCM(a,b) equals a prime number.
Accordingly,\ I. $a$ and $b$ are coprime numbers.\ II. The sum $a + b$ is an odd number.\ III. The product $\mathrm{a} \cdot \mathrm{b}$ is an odd number.
Which of the following statements are always true?\ A) Only I\ B) Only II\ C) Only III\ D) I and II\ E) II and III

Q8 Polynomial Division & Manipulation View

$\frac { x z - y z + x y - y ^ { 2 } } { x ^ { 2 } - x y + x z - y z }$\ Which of the following is the simplified form of this expression?\ A) $\frac { z - y } { x - z }$\ B) $\frac { y + z } { x + z }$\ C) $\frac { x + z } { y + z }$\ D) $\frac { x } { x + y }$\ E) $\frac { y - z } { x + y }$

Q9 Simultaneous equations Evaluating an algebraic expression given a constraint View

For positive real numbers $\mathrm{a}$, $\mathrm{b}$, and $c$ $$\begin{aligned}& \frac { a + c } { b + 2 } = \frac { c } { b } \\& \frac { a } { b } = c\end{aligned}$$ the following equalities are given.\ Accordingly, what is b?\ A) $\sqrt { 2 }$\ B) $\sqrt { 3 }$\ C) $\sqrt { 6 }$\ D) 2\ E) 3

Q10 Indices and Surds Solving Equations Involving Surds View

$\frac { 1 } { \sqrt { \mathrm{a} } } - \frac { 2 } { \sqrt { 9 \mathrm{a} } } = 1$\ Given this, what is a?\ A) $\frac { 1 } { 3 }$\ B) $\frac { 2 } { 3 }$\ C) $\frac { 1 } { 4 }$\ D) $\frac { 1 } { 9 }$\ E) $\frac { 4 } { 9 }$

Q11 Inequalities Algebraic identities and properties of modulus View

For nonzero real numbers $x$, $y$, and $z$ whose absolute values are distinct from each other, $$\begin{aligned}| x + y | & = | x | - | y | \\| y + z | & = | y | + | z |\end{aligned}$$ the following equalities are satisfied.
Given that $x > 0$,\ I. $\frac { x } { x + y } < 1$\ II. $\frac { y } { y + z } < 1$\ III. $\frac { z } { x + z } < 1$\ Which of the following statements are always true?\ A) Only I\ B) Only II\ C) Only III\ D) I and III\ E) II and III

Q12 Inequalities Properties of Integer Sequences and Digit Analysis View

Three-digit natural numbers $ADB$, $ADC$, $DAA$, $DAD$ $$\begin{aligned}& \mathrm{ADB} < \mathrm{DAA} \\& \mathrm{DAD} < \mathrm{ADC}\end{aligned}$$ satisfy the inequalities.\ Accordingly, which of the following orderings is correct?\ A) A $=$ D $<$ B $<$ C\ B) C $<$ A $=$ B $<$ D\ C) D $<$ A $=$ B $<$ C\ D) B $<$ A $=$ D $<$ C\ E) C $<$ A $=$ D $<$ B

Q13 Inequalities Ordering and Sign Analysis from Inequality Constraints View

For nonzero real numbers $x$ and $y$, given that $y < x$ and $x ^ { 2 } < y ^ { 2 }$,\ I. $x \cdot y > 0$\ II. $x + y < 0$\ III. $\frac { 1 } { x } - \frac { 1 } { y } > 0$\ Which of the following statements are always true?\ A) Only I\ B) Only II\ C) I and II\ D) I and III\ E) II and III

Q14 Combinations & Selection Subset Counting with Set-Theoretic Conditions View

Let A be a subset of the set $\{ 1,2,3,4,5,6,7 \}$. $$A \cap \{ 5,6,7 \}$$ The elements of the set are odd numbers.\ Accordingly, how many three-element sets A satisfy this condition?\ A) 12\ B) 14\ C) 16\ D) 18\ E) 20

Q15 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

Sets $A$, $B$, and $C$ are defined as $$\begin{aligned}& A = \{ ( x , x ) : x \in \mathbb { R } \} \\& B = \{ ( x , 3 - x ) : x \in \mathbb { R } \} \\& C = \{ ( x , x + 4 ) : x \in \mathbb { R } \}\end{aligned}$$ Given that $( p , q ) \in A \cap B$ and $( r , s ) \in B \cap C$, $$\frac { p - r } { q + s }$$ what is the value of this expression?\ A) $\frac { 1 } { 3 }$\ B) $\frac { 1 } { 4 }$\ C) $\frac { 3 } { 4 }$\ D) $\frac { 4 } { 5 }$\ E) $\frac { 2 } { 5 }$

Q16 Solving quadratics and applications Evaluate Composition from Algebraic Definitions View

Functions $f$ and $g$ are defined on the set of real numbers as $$\begin{aligned}& f ( x ) = \frac { x \cdot ( x - 2 ) } { 2 } \\& g ( x ) = \frac { x \cdot ( x - 1 ) \cdot ( x - 2 ) } { 3 }\end{aligned}$$ The sum of the $\mathbf{x}$ values satisfying the equality $$f ( 2 x ) = g ( x + 1 )$$ is what?\ A) 1\ B) 3\ C) 4\ D) 6\ E) 8

Q17 Composite & Inverse Functions Evaluating an algebraic expression given a constraint View

A function $f$ on the set of real numbers is defined for every real number $x$ where $n$ is an integer as $$f ( x ) = x - n , \quad x \in [ n , n + 1 )$$ Accordingly, $$f ( 1 ) + f \left( \frac { 7 } { 3 } \right) + f \left( \frac { 13 } { 6 } \right)$$ what is this sum?\ A) $\frac { 1 } { 2 }$\ B) $\frac { 2 } { 3 }$\ C) $\frac { 7 } { 6 }$\ D) 1\ E) 2

Q18 Curve Sketching Determine Domain or Range of a Composite Function View

A function $f$ on the set of real numbers is defined as $$f ( x ) = \frac { | x | } { 1 + | x | }$$ Accordingly, which of the following is the image set of the interval $[ - 2,1 )$ under the function $\mathbf{f}$?\ A) $[ 0,1 ]$\ B) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right]$\ C) $\left[ \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$\ D) $\left[ 0 , \frac { 1 } { 3 } \right]$\ E) $\left[ 0 , \frac { 2 } { 3 } \right]$

Q22 Discriminant and conditions for roots Determine coefficients or parameters from root conditions View

The sum of the roots of the equation $x ^ { 2 } - a x + 1 = 0$, which has two real roots, is a root of the equation $$x ^ { 2 } + 6 x + a = 0$$ Accordingly, what is a?\ A) - 3\ B) - 4\ C) - 5\ D) - 6\ E) - 7

Q23 Complex Numbers Arithmetic Powers of i or Complex Number Integer Powers View

$\frac { \left( 1 - i ^ { 2 } \right) \cdot \left( 1 - i ^ { 6 } \right) \cdot \left( 1 - i ^ { 10 } \right) } { ( 1 - i ) \cdot \left( 1 - i ^ { 3 } \right) \cdot \left( 1 - i ^ { 5 } \right) }$\ What is the result of this operation?\ A) 1\ B) 2\ C) $2 + 2 i$\ D) $2 + 2 i$\ E) $1 + 2 i$

Q24 Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View

$4 z - 3 \bar { z } = \frac { 1 - 18 i } { 2 - i }$\ Which of the following is the complex number $z$ that satisfies this equality?\ A) $- 2 + i$\ B) $- 3 + i$\ C) $4 + 2 i$\ D) $3 - 2 i$\ E) $4 - i$

Q25 Inequalities Integer Solutions of an Inequality View

$( x - 1 ) ^ { 2 } < | x - 1 | + 6$\ What is the sum of the integers $x$ that satisfy this inequality?\ A) 2\ B) 3\ C) 4\ D) 5\ E) 6

Q26 Inequalities Solve Polynomial/Rational Inequality for Solution Set View

$\frac { 6 x + 1 } { ( x + 1 ) ^ { 2 } } > 1$\ Which of the following is the set of all real numbers that satisfy this inequality?\ A) $( - 1,4 )$\ B) $( - 1,6 )$\ C) $( 0,4 )$\ D) $( 0 , \infty )$\ E) $( 2 , \infty )$

Q27 Polynomial Division & Manipulation Polynomial Construction from Root/Value Conditions View

A third-degree polynomial $P ( x )$ with real coefficients has roots $- 3$, $- 1$, and $2$.\ Given that $P ( 0 ) = 12$, what is the coefficient of the $x ^ { 2 }$ term?\ A) - 4\ B) - 3\ C) - 2\ D) 1\ E) 2

Q28 Polynomial Division & Manipulation Divisibility and Factor Determination View

Let $a$ and $b$ be integers such that $$\begin{aligned}& P ( x ) = x ^ { 3 } - a x ^ { 2 } - ( b + 2 ) x + 4 b \\& Q ( x ) = x ^ { 2 } - 2 a x + b\end{aligned}$$ For the polynomials

$\mathrm{P} ( - 4 ) = 0$
$\mathrm{Q} ( - 4 ) \neq 0$

it is known that.\ If the roots of polynomial $\mathbf{Q} ( \mathbf{x} )$ are also roots of polynomial $\mathbf{P} ( \mathbf{x} )$, what is the difference $b - a$?\ A) 8\ B) 9\ C) 11\ D) 13\ E) 14

Q29 Polynomial Division & Manipulation Divisibility and Factor Determination View

How many second-degree polynomials have coefficients from the set $\{ 0,1,2 , \ldots , 9 \}$ and have one root equal to $\frac { - 2 } { 3 }$?\ A) 5\ B) 7\ C) 8\ D) 10\ E) 11

Q32 Laws of Logarithms Solve Exponential Equation for Unknown Variable View

$x ^ { \ln 4 } - 6 \cdot 2 ^ { \ln x } + 8 = 0$\ What is the product of the $x$ values that satisfy this equation?\ A) $e ^ { 6 }$\ B) $e ^ { 4 }$\ C) $e ^ { 3 }$\ D) $\frac { e ^ { 2 } } { 2 }$\ E) $\frac { e ^ { 3 } } { 3 }$

Q33 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

$\frac { \log _ { 3 } \sqrt { 27 } + \log _ { 27 } \sqrt { 3 } } { \log _ { 3 } \sqrt { 27 } - \log _ { 27 } \sqrt { 3 } }$\ What is the result of this operation?\ A) $\frac { 3 } { 2 }$\ B) $\frac { 4 } { 3 }$\ C) $\frac { 5 } { 4 }$\ D) $\frac { 6 } { 5 }$\ E) $\frac { 7 } { 6 }$

Q34 Laws of Logarithms Solve a Logarithmic Equation View

$\ln x + \ln y = 9$ $$\ln x - \ln y = 3$$ Given this, what is the value of $\log _ { y } x$?\ A) 1\ B) 2\ C) 3\ D) 4\ E) 5

Q35 Arithmetic Sequences and Series Find Specific Term from Given Conditions View

Let $\left( a _ { n } \right)$ be an arithmetic sequence such that $$\begin{aligned}& a _ { 10 } + a _ { 7 } = 6 \\& a _ { 9 } - a _ { 6 } = 1\end{aligned}$$ the following equalities are given.\ Accordingly, what is $a _ { 1 }$?\ A) $\frac { 7 } { 3 }$\ B) $\frac { 5 } { 2 }$\ C) $\frac { 4 } { 3 }$\ D) $\frac { 5 } { 6 }$\ E) $\frac { 1 } { 2 }$

Q37 Chain Rule Limit Evaluation Involving Composition or Substitution View

$\lim _ { x \rightarrow \pi } \frac { x ^ { 2 } \cdot \sin ( \pi - x ) + \pi ^ { 2 } \cdot \sin ( x - \pi ) } { ( x - \pi ) ^ { 2 } }$\ What is the value of this limit?\ A) $- 2 \pi$\ B) $- \pi$\ C) $\pi$\ D) $2 \pi$\ E) $3 \pi$

Q41 Applied differentiation Geometric or applied optimisation problem View

A crystal in the shape of a cube with one edge of length $x$ units has a production cost of 5 TL per unit cube based on volume, and a selling price of 20 TL per unit square based on surface area.
Accordingly, for what value of x in units will the profit from selling this crystal be maximum?\ A) 16\ B) 18\ C) 20\ D) 22\ E) 24

Q42 Tangents, normals and gradients Determine unknown parameters from tangent conditions View

Let a and b be real numbers, and $$f ( x ) = a \cdot \ln x + b \cdot x ^ { 2 } + 3$$ The equation of the tangent line drawn to the graph of the function at the point $(1, f(1))$ is given as $y - 2x + 1 = 0$.
Accordingly, what is the product $\mathbf{a} \cdot \mathbf{b}$?\ A) $- 18$\ B) $- 16$\ C) $- 12$\ D) $- 8$\ E) $- 6$

Q43 Differential equations Asymptote Determination View

Let a be a real number, and $$f ( x ) = \ln ( 2 x + 8 )$$ The vertical asymptote of the function $$g ( x ) = \frac { \sin x } { x ^ { 2 } + a x }$$ is also a vertical asymptote of the function.\ Accordingly, what is a?\ A) 0\ B) 1\ C) 2\ D) 3\ E) 4

Q44 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

$\int_{\pi/6}^{?} 2 \tan ( 2 x ) \, d x$\ What is the value of the integral?\ A) $\ln 2$\ B) $\ln 3$\ C) $\ln 4$\ D) $\ln 5$\ E) $\ln 6$

Q45 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View

$\int \sqrt { 1 + e^{x^{2}} } \, d x$\ In the integral, if the substitution $u = \sqrt { 1 + e ^ { x } }$ is made, which of the following integrals is obtained?\ A) $\int \frac { 2 u } { u ^ { 2 } + 1 } d u$\ B) $\int \frac { u ^ { 2 } } { u ^ { 2 } + 1 } d u$\ C) $\int \frac { 1 } { u ^ { 2 } - 1 } d u$\ D) $\int \frac { u } { u ^ { 2 } - 1 } d u$\ E) $\int \frac { 2 u ^ { 2 } } { u ^ { 2 } - 1 } d u$

Q46 Integration by Parts Definite Integral Evaluation by Parts View

$\int _ { 1/2 } ^ { e } x \ln ( 2 x ) \, d x$\ What is the value of the integral?\ A) $\frac { e ^ { 2 } } { 2 }$\ B) $\frac { e ^ { 2 } - 1 } { 4 }$\ C) $\frac { e ^ { 2 } + 1 } { 16 }$\ D) 1\ E) 2

Q47 Integration with Partial Fractions View

$\int _ { 4 } ^ { 5 } \frac { x + 1 } { x ^ { 2 } - 5 x + 6 } d x$\ What is the value of the integral?\ A) $5 \ln 3 - \ln 2$\ B) $5 \ln 2 - 2 \ln 3$\ C) $3 \ln 2 + 2 \ln 3$\ D) $2 \ln 2 + 3 \ln 3$\ E) $7 \ln 2 - 3 \ln 3$

Q48 Areas by integration View

The function $f ( x ) = x ^ { 2 }$ is defined on the set of real numbers. For real numbers in the interval $[-3, 3]$, the graph of $y = f(x)$ is given in the coordinate plane divided into unit squares as shown in the figure.
In the unit squares divided by this graph; the regions below the graph are colored blue, and the regions above are colored yellow as shown in the figure.
Accordingly, what is the ratio of the sum of the areas of the blue regions to the sum of the areas of the yellow regions?\ A) $\frac { 2 } { 3 }$\ B) $\frac { 3 } { 4 }$\ C) $\frac { 4 } { 5 }$\ D) $\frac { 5 } { 6 }$\ E) $\frac { 6 } { 7 }$

Q49 Reciprocal Trig & Identities View

Given that $0 < x < \frac { \pi } { 2 }$, $$\frac { \sec ( x ) - 1 } { 2 } = \frac { 3 } { \sec ( x ) + 1 }$$ the equality holds.\ Accordingly, what is the value of $\tan ( x )$?\ A) $\sqrt { 2 }$\ B) $\sqrt { 3 }$\ C) $\sqrt { 5 }$\ D) $\sqrt { 6 }$\ E) $\sqrt { 7 }$

Q50 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View

Given that $x \in [ 0, 2 \pi )$, $$\cos ( 5 x ) = \cos ( 3 x ) \cdot \cos ( 2 x )$$ How many different solutions does the equation have?\ A) 3\ B) 6\ C) 8\ D) 11\ E) 12

Q51 Addition & Double Angle Formulae Trigonometric Identity Proof or Derivation View

For every real number $x$, the number $A$ is defined as $$\sum _ { k = 2 } ^ { 4 } \cos ( 2 k x ) = A$$ Accordingly, $$\sum _ { k = 2 } ^ { 4 } \cos ^ { 2 } ( k x )$$ What is the equivalent of the expression in terms of A?\ A) $A + 2$\ B) $A + 4$\ C) $\frac { \mathrm { A } + 1 } { 2 }$\ D) $\frac { A + 2 } { 2 }$\ E) $\frac { A + 3 } { 2 }$

Q57 Circles Compute area of a triangle or related figure View

ABC and BDE are equilateral triangles\ $[ \mathrm { BD } ] \perp [ \mathrm { AC } ]$\ $[ \mathrm { BF } ] \perp [ \mathrm { DE } ]$\ $[ \mathrm { FH } ] \perp [ \mathrm { BE } ]$\ $| \mathrm { AB } | = 16$ units
Accordingly, what is the area of triangle BFH in square units?\ A) $12 \sqrt { 3 }$\ B) $15 \sqrt { 3 }$\ C) $18 \sqrt { 3 }$\ D) $20 \sqrt { 3 }$\ E) $24 \sqrt { 3 }$

Q58 Circles Compute area of a triangle or related figure View

$ABC$ right triangle\ $[ \mathrm { AC } ] \perp [ \mathrm { BC } ]$\ $[AB]$ // $[DE]$\ $[BC]$ // $[FH]$\ $| \mathrm { AD } | = | \mathrm { DH } | = | \mathrm { HC } |$\ $| \mathrm { GE } | = 4$ units\ $| \mathrm { GF } | = 2$ units
Accordingly, what is the area of triangle ABC in square units?\ A) $9 \sqrt { 3 }$\ B) $12 \sqrt { 3 }$\ C) $15 \sqrt { 3 }$\ D) $18 \sqrt { 3 }$\ E) $20 \sqrt { 3 }$

Q59 Circles Area Computation in Coordinate Geometry View

Below are given squares $\mathrm { ABCD }$, $\mathrm { BLPR }$, and KLMN with side lengths of 3, 2, and 1 units respectively.
In the figure, points $\mathrm { A }$, $\mathrm { B }$, $\mathrm { K }$, and L are collinear.\ Accordingly, what is the area of triangle DNP in square units?\ A) 3\ B) 4\ C) 5\ D) 6\ E) 8

Q60 Straight Lines & Coordinate Geometry Section Ratio and Division of Segments View

The square ABCD given above is divided into four rectangles of equal area.
Accordingly, what is the ratio $\frac { | AE | } { | AD | }$?\ A) $\frac { 2 } { 3 }$\ B) $\frac { 3 } { 4 }$\ C) $\frac { 3 } { 5 }$\ D) $\frac { 5 } { 8 }$\ E) $\frac { 9 } { 16 }$

Q61 Circles Compute area of a triangle or related figure View

ABCD right trapezoid, ABD equilateral triangle\ $[AB]$ // $[DC]$\ $| \mathrm { BF } | = 4 | \mathrm { DF } |$\ $| \mathrm { AB } | = 8$ units
Accordingly, what is the area of right trapezoid ABCE in square units?\ A) $10 \sqrt { 3 }$\ B) $12 \sqrt { 3 }$\ C) $16 \sqrt { 3 }$\ D) $18 \sqrt { 3 }$\ E) $20 \sqrt { 3 }$

Q62 Circles Compute area of a triangle or related figure View

ABCD kite\ $[ \mathrm { AC } ] \perp [ \mathrm { BD } ]$\ $| \mathrm { AB } | = | \mathrm { BC } |$\ $| \mathrm { AD } | = | \mathrm { DC } |$\ $| \mathrm { BE } | = 4 | \mathrm { ED } |$\ $| \mathrm { AC } | = 16$ units
The area of kite ABCD in the figure is 160 square units.\ Accordingly, what is the perimeter of kite ABCD in units?\ A) $20 \sqrt { 5 }$\ B) $24 \sqrt { 5 }$\ C) $28 \sqrt { 5 }$

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