Let $\mathbf { a }$ and $\mathbf { b }$ be real numbers such that $$\begin{aligned}
& a ^ { 2 } - a = b ^ { 2 } - b \\
& a \cdot b = - 1
\end{aligned}$$ Given this, what is the sum $a ^ { 2 } + b ^ { 2 }$? A) 6 B) 5 C) 4 D) 3 E) 2
For real numbers x and y $$2 ^ { x } = 6 ^ { x + y - 1 }$$ Given this, what is $3 ^ { \mathbf { x } }$ in terms of y? A) $3 ^ { 1 - y }$ B) $6 ^ { 1 - y }$ C) $6 ^ { y }$ D) $9 ^ { - y }$ E) $9 ^ { 1 + y }$
For real numbers $x , y$ and $z$ $$x + y < 0 < x < y + z$$ Given this, which of the following orderings is correct? A) $x < y < z$ B) $x < z < y$ C) $y < x < z$ D) $y < z < x$ E) $z < y < x$
$$\begin{aligned}
& a = \frac { x } { x - y } \\
& b = \frac { y } { x + y }
\end{aligned}$$ Given this, what is the value of the expression $\frac { a + b - 1 } { a \cdot b }$? A) $- 2$ B) $- 1$ C) $0$ D) $1$ E) $2$
For real numbers $\mathbf { x }$ and $\mathbf { y }$ $$\begin{aligned}
& y - x = 1 \\
& y - | x - y | = 2
\end{aligned}$$ Given this, what is the sum $\mathbf { x } + \mathbf { y }$? A) 5 B) 6 C) 7 D) 8 E) 9
For integers $\mathrm { x } , \mathrm { y }$ and z $$2 x = 3 y = 5 z$$ Given this, which of the possible values of the sum $x + y + z$ is closest to 100? A) 93 B) 96 C) 98 D) 103 E) 105
I. $f ( x ) = 2 x$ II. $f ( x ) = 2 ^ { x }$ III. $f ( x ) = x ^ { 2 }$ Which of these functions satisfy the equation $f ( a + b ) = f ( a ) \cdot f ( b )$ for every real number a and b? A) Only I B) Only II C) I and II D) I and III E) II and III
If Ahmet's salary is increased by half of Deniz's salary, then the sum of their salaries becomes 2 times Ahmet's initial salary. If Ahmet's salary is A TL and Deniz's salary is D TL, what is the relationship between $A$ and $D$? A) $5A = 8D$ B) $5A = 6D$ C) $4A = 5D$ D) $3A = 4D$ E) $2A = 3D$
Stationery materials are to be distributed to students in a class. Enough materials are brought to the class so that each of the 36 students receives one pencil, one pencil sharpener, and one eraser. However, on the distribution day, since some students are absent, each student present in the class is given 3 pencils, 2 pencil sharpeners, and 1 eraser. If a total of 42 items of these materials remain after distribution, how many pencil sharpeners are left? A) 10 B) 11 C) 12 D) 13 E) 14
All 5 different marbles are to be distributed among 3 siblings of different ages. In how many different ways can this distribution be made such that the oldest sibling gets 1 marble and the other two each get at least one marble? A) 45 B) 50 C) 60 D) 70 E) 75
A bag contains 10 balls numbered from 1 to 10. Given that the sum of the numbers on two balls randomly drawn from the bag is 15, what is the probability that ball number 7 was drawn? A) $\frac { 2 } { 3 }$ B) $\frac { 2 } { 5 }$ C) $\frac { 2 } { 7 }$ D) $\frac { 1 } { 2 }$ E) $\frac { 1 } { 3 }$
OABC is a parallelogram $\mathrm { A } = ( 5,0 )$ $\mathrm { C } = ( 3,4 )$ According to the given information above, what is the sum of the diagonal lengths of parallelogram OABC in units? A) $5 \sqrt { 5 }$ B) $6 \sqrt { 5 }$ C) $7 \sqrt { 5 }$ D) $7 \sqrt { 3 }$ E) $8 \sqrt { 3 }$
In the right coordinate plane shown in the figure, lines d and e are perpendicular to each other. Accordingly, what is the abscissa of the point where line d intersects the x-axis? A) $\frac { 9 } { 2 }$ B) $\frac { 11 } { 2 }$ C) $\frac { 13 } { 3 }$ D) $\frac { 14 } { 3 }$ E) $\frac { 25 } { 6 }$