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Q3 Indices and Surds Solving Exponential or Index Equations View

For integers $x$ and $y$
$$2^{3x-1} - 8^{x-1} = 3^{y+3} \cdot 4^{x+1}$$
Given this equality.
Accordingly, what is the product $\mathbf{x} \cdot \mathbf{y}$?
A) $-10$ B) $-6$ C) $-2$ D) 4 E) 8

Q5 Inequalities Ordering and Sign Analysis from Inequality Constraints View

Let $x$ and $y$ be real numbers,
$$x^{2} \cdot y^{2} < x \cdot y < x^{2} \cdot y$$
Given this inequality.
Accordingly,
I. $x < 1$ II. $y < 1$ III. $x \cdot y < 1$
Which of these statements are true?
A) Only I B) Only II C) I and III D) II and III E) I, II and III

Q11 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

Let $a$ and $b$ be real numbers. For functions $f$ and $g$ defined on the set of real numbers
$$\begin{aligned} & f(x) = x^{2} + ax + b \\ & g(x) = ax + 2 \\ & (f + g)(3) = 4 \\ & (f - g)(5) = 6 \end{aligned}$$
These equalities are satisfied.
Accordingly, what is the difference $\mathrm{a} - \mathrm{b}$?
A) 17 B) $\frac{52}{3}$ C) 18 D) $\frac{56}{3}$ E) 19

Q12 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View

Let $a$ be a positive real number,
$$\left(x + \frac{a - 7}{x}\right)^{13}$$
In the expansion of this expression, the coefficient of the $x^{11}$ term is $\frac{234}{a}$.
Accordingly, what is $a$?
A) 9 B) 12 C) 13 D) 15 E) 18

Q14 Probability Definitions Finite Equally-Likely Probability Computation View

Veysel has four gift vouchers of 200, 400, 600 and 800 TL from a clothing store. Veysel randomly gives one of these four gift vouchers to each of his two daughters, Yasemin and Zehra, who want to shop at this clothing store. On different days, Yasemin likes a dress for 300 TL and Zehra likes a dress for 500 TL.
Accordingly, what is the probability that both girls can buy the dresses they like with only the gift vouchers they received from their father?
A) $\frac{1}{2}$ B) $\frac{1}{3}$ C) $\frac{1}{4}$ D) $\frac{1}{5}$ E) $\frac{1}{6}$

Q15 Simultaneous equations View

Let $x$ and $y$ be real numbers,
$$\begin{aligned} & x^{2} + 8xy = 60 \\ & y^{2} - 3xy = -15 \end{aligned}$$
Given that this holds, what is the product $\mathrm{x} \cdot \mathrm{y}$?
A) 3 B) 4 C) 5 D) 6 E) 7

Q16 Inequalities Simultaneous/Compound Quadratic Inequalities View

Let $a$ and $b$ be positive integers,
$$\begin{aligned} & (x - a)(x + 2a) < 0 \\ & (x - b)(x + 2b) > 0 \end{aligned}$$
Given this system of inequalities.
If $a + b = 8$ and the solution set of this system of inequalities contains 16 integers, what is the product $a \cdot b$?
A) 7 B) 10 C) 12 D) 15 E) 16

Q17 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

Functions $f$ and $g$ are defined on the set of real numbers as
$$\begin{aligned} & f(x) = \frac{3x + 4}{2} \\ & g(x) = \frac{2x - 4}{3} \end{aligned}$$
If $(\mathbf{f} \circ \mathbf{g})(\mathbf{a}) = \mathbf{f}(\mathbf{a}) = \mathbf{b}$, what is the product $\mathbf{a} \cdot \mathbf{b}$?
A) $-20$ B) $-12$ C) $-8$ D) 4 E) 16

Q18 Factor & Remainder Theorem Polynomial Construction from Root/Value Conditions View

A third-degree polynomial $\mathrm{P}(\mathrm{x})$ with real coefficients and leading coefficient 3 is known to have exactly 2 different real roots.
If $\mathbf{P}(1) = \mathbf{P}(2) = \mathbf{0}$, then the value $\mathbf{P}(3)$ is
I. 6 II. 12 III. 18
Which of these numbers can it equal?
A) Only I B) Only II C) Only III D) I and II E) II and III

Q19 Roots of polynomials Factorization and Root Analysis View

Let $a$ and $b$ be integers,
$$P(x) = x^{3} + ax^{2} + bx - 2$$
It is known that the polynomial has exactly one real root.
If $\mathbf{P}(1) = 0$, what is the smallest value that the integer $a$ can take?
A) $-6$ B) $-5$ C) $-4$ D) $-3$ E) $-2$

Q20 Solving quadratics and applications Geometric or real-world application leading to a quadratic equation View

Uncle Ahmet has a rectangular field with side lengths $x + 20$ and $2x + 30$ meters. He grows sunflowers in a square-shaped part with side length $x$ meters as shown in the figure.
If the area of the remaining part of the field is 1400 square meters, what is the perimeter of the entire field in meters?
A) 148 B) 154 C) 160 D) 166 E) 172

Q21 Curve Sketching Identifying the Correct Graph of a Function View

For real numbers $\mathrm{a}$, $\mathrm{b}$ and $\mathrm{c}$ with $\mathrm{a} \cdot \mathrm{b} \cdot \mathrm{c} > 0$, a function $f$ is defined on the set of real numbers as
$$f(x) = ax^{2} + bx + c$$
Accordingly, the graph of function $f$ can be which of the graphs shown (I, II, III)?
A) Only I B) Only II C) I and III D) II and III E) I, II and III

Q23 Laws of Logarithms Logarithmic Function Graph Intersection or Geometric Analysis View

In the Cartesian coordinate plane, the graphs of functions $f(x) = \log_{2} x$ and $g(x) = \log_{\frac{1}{4}} x$ are given in the figure. Points $A$ and $D$ are on the graph of function $f$, and points $B$ and $C$ are on the graph of function $g$. In the figure, the line segment $[AB]$ passing through the point $(4, 0)$ and the line segment $[CD]$ are both perpendicular to the x-axis, and the area of triangle $ABC$ is 6 square units.
Accordingly, what is the area of triangle $ACD$ in square units?
A) 12 B) 11 C) 10 D) 9 E) 8

Q24 Laws of Logarithms Solve a Logarithmic Equation View

Let $x$ be a positive real number,
$$\log_{4}(x + 5) + \log_{4}(x + 4) - \log_{4}(x + 3) = \log_{2} 3$$
What is the value of $x$ that satisfies this equality?
A) $\sqrt{6}$ B) $\sqrt{7}$ C) $2\sqrt{2}$ D) $2\sqrt{5}$ E) $3\sqrt{2}$

Q25 Sequences and series, recurrence and convergence Direct term computation from recurrence View

The sequence $(a_n)$ of real numbers satisfies for every positive integer $n$
$$a_{n} + (-1)^{n} \cdot a_{n+1} = 2^{n}$$
If $a_{1} = 0$, what is the sum $a_{3} + a_{4} + a_{5} + a_{6}$?
A) 6 B) 8 C) 12 D) 16 E) 20

Q26 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View

A geometric sequence $(b_n)$ with first two terms $b_{1} = \frac{4}{3}$ and $b_{2} = 2$ and an arithmetic sequence $(a_n)$ whose common difference equals the common ratio of this geometric sequence are given.
If $b_{7} = a_{11}$, what is $a_{1}$?
A) $\frac{1}{4}$ B) $\frac{1}{8}$ C) $\frac{3}{8}$ D) $\frac{3}{16}$ E) $\frac{5}{16}$

Q27 Trig Proofs Ordering or Comparing Trigonometric Expressions View

Let $a \in \left(\frac{3\pi}{4}, \pi\right)$,
$$\begin{aligned} & x = \sin(2a) \cdot \tan(a) \\ & y = \cos(2a) \cdot \cot(2a) \\ & z = \sin(a) \cdot \cot(2a) \end{aligned}$$
Given these equalities.
Accordingly, what are the signs of $\mathbf{x}$, $y$ and $\mathbf{z}$ respectively?
A) $+, +, -$ B) $+, -, -$ C) $-, -, -$ D) $-, +, +$ E) $-, -, +$

Q28 Trig Proofs Trigonometric Identity Simplification View

$$\frac{1}{1 + \cot x} - \frac{\sin x}{\sin x - \cos x}$$
What is the simplified form of this expression?
A) $\sec(2x)$ B) $\sec^{2}(2x)$ C) $\tan(2x)$ D) $2 \cdot \sec x$ E) $2 \cdot \tan x$

Q29 Trig Proofs Trigonometric Equation Constraint Deduction View

In Figure 1, a parallelogram with two sides of lengths 4 units and 8 units and the angle between these sides measuring $x$ degrees is given. In Figure 2, a parallelogram with two sides of lengths 4 units and 6 units and the angle between these sides measuring $2x$ degrees is given.
If the area of the parallelogram in Figure 1 is 24 square units, what is the area of the parallelogram in Figure 2 in square units?
A) $6\sqrt{7}$ B) $7\sqrt{7}$ C) $8\sqrt{7}$ D) $9\sqrt{7}$ E) $10\sqrt{7}$

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

Let $\pi < x < 2\pi$,
$$\frac{2\cos^{2} x + 2\sin x}{5\sin(2x)} = \tan x$$
What is the sum of the real numbers $x$ that satisfy this equation?
A) $2\pi$ B) $3\pi$ C) $4\pi$ D) $5\pi$ E) $6\pi$

Q31 Reciprocal Trig & Identities Trigonometric Identity Simplification View

In the figure, a semicircle with center $O$ and diameter $AD$, a rectangle $ABCD$, and a triangle $OEF$ are given. Points $C$, $F$, $E$, $B$ are collinear; points $E$ and $F$ are on the circle.
Accordingly, what is the ratio of the area of rectangle $ABCD$ to the area of triangle $OEF$ in terms of $x$?
A) $\tan\frac{x}{2}$ B) $2 \cdot \sec x$ C) $2 \cdot \operatorname{cosec}\frac{x}{2}$ D) $2 \cdot \tan x$ E) $\cot x$

Q38 Straight Lines & Coordinate Geometry Slope and Angle Between Lines View

In the rectangular coordinate plane, the line $2x + y = 12$ and a line d intersect at point $\mathrm{A}(4,4)$. These two lines divide every circle centered at point $\mathrm{A}(4,4)$ into four equal areas.
Accordingly, which of the following is the equation of line d?
A) $-2x + y = -4$ B) $x - 3y = -8$ C) $3x + y = 16$ D) $x + 2y = 12$ E) $x - 2y = -4$

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