In building construction, tubes of different sizes are used for water network installation. These measurements are known by their diameter, often measured in inches. Some of these tubes, with measurements in inches, are tubes of $\frac{1}{2}, \frac{3}{8}$ and $\frac{5}{4}$. Placing the values of these measurements in increasing order, we find
(A) $\frac{1}{2}, \frac{3}{8}, \frac{5}{4}$
(B) $\frac{1}{2}, \frac{5}{4}, \frac{3}{8}$
(C) $\frac{3}{8}, \frac{1}{2}, \frac{5}{4}$
(D) $\frac{3}{8}, \frac{5}{4}, \frac{1}{2}$
(E) $\frac{5}{4}, \frac{1}{2}, \frac{3}{8}$

On a map with scale 1 : 250 000, the distance between cities A and B is 13 cm. On another map, with scale 1 : 300 000, the distance between cities A and C is 10 cm. On a third map, with scale 1 : 500 000, the distance between cities A and D is 9 cm. The actual distances between city A and cities B, C, and D are, respectively, equal to $X$, $Y$, and $Z$ (in the same unit of length).
The distances $X$, $Y$, and $Z$, in increasing order, are given in
(A) $X, Y, Z$.
(B) $Y, X, Z$.
(C) $Y, Z, X$.
(D) $Z, X, Y$.
(E) $Z, Y, X$.

If $2 ^ { x } - 2 ^ { y } < 3 ^ { - x } - 3 ^ { - y }$ , then
A． $\ln ( y - x + 1 ) > 0$
B． $\ln ( y - x + 1 ) < 0$
C． $\ln | x - y | > 0$
D． $\ln | x - y | < 0$

A region is defined by the inequalities $x + y > 6$ and $x - y > - 4$
Consider the three statements:
$1 x > 1$
$2 y > 5$
$3 ( x + y ) ( x - y ) > - 24$
Which of the above statements is/are true for every point in the region?

In this question $x$ and $y$ are non-zero real numbers.
Which one of the following is sufficient to conclude that $x < y$ ?

$a , b$ and $c$ are real numbers with $a < b < c < 0$
Which of the following statements must be true?
I $a c < a b < a ^ { 2 }$
II $b ( c + a ) > 0$
III $\frac { c } { b } > \frac { a } { b }$

The real numbers $a , b , c$ and $d$ satisfy both
$$0 < a + b < c + d$$
and
$$0 < a + c < b + d$$
Which of the following inequalities must be true? I $a < d$ II $b < c$ III $a + b + c + d > 0$
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III

A region $R$ in the ( $x , y$ )-plane is defined by the simultaneous inequalities
$$\begin{array} { r } y - x < 3 \\ y - x ^ { 2 } < 1 \end{array}$$
Which of the following statements is/are true for every point in $R$ ?
I $- 1 < x < 2$ II $\quad ( y - x ) \left( y - x ^ { 2 } \right) < 3$ III $y < 5$
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III

For real numbers $x , y$ and $z$
$$\begin{aligned} & y > 0 \\ & x - y > z \end{aligned}$$
Given this, which of the following is always true?
A) $x > z$
B) $x > y$
C) $z > y$
D) $x > 0$
E) $z > 0$

For given positive real numbers $a$, $c$ and negative real number $b$, $$a^{2}b > abc + c^{2}$$ Given that the inequality is satisfied, which of the following is necessarily true?
A) $a = |b|$
B) $a = c$
C) $c > |b|$
D) $a < c$
E) $c < a$

$$\sqrt { 2 } < x < \sqrt { 3 }$$
Given this, which of the following can x be?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 7 } { 4 }$
E) $\frac { 6 } { 5 }$

For integers x and y, $x + 2y = 11$. Given that,
I. x is an odd number. II. x is greater than y. III. Both x and y are positive.
Which of the following statements are always true?
A) Only I B) Only III C) I and II D) I and III E) II and III

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

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) $z < y < x$

For positive real numbers $x$ and $y$
$$\frac { x } { 8 } = \frac { y } { 12 } = \frac { 9 } { y - x }$$
Given this, what is the sum $x + y$?
A) 10
B) 15
C) 20
D) 25
E) 30

Given that $| a | = 2 , | b | = 5$ and $| c | = 6$,
$$\begin{aligned} & \mathrm { c } < \mathrm { a } < \mathrm { b } \\ & \mathrm { a } \cdot \mathrm {~b} \cdot \mathrm { c } > 0 \end{aligned}$$
What is the sum $a + b + c$?
A) - 9
B) - 3
C) - 1
D) 1
E) 3

Let $a , b , c$ be real numbers and $a \cdot b \cdot c > 0$ such that
$$\begin{aligned} & a \cdot b = - 2 | a | \\ & \frac { b } { c } = 3 | b | \end{aligned}$$
Given that $\mathbf { a } + \mathbf { b } + \mathbf { c } = \mathbf { 0 }$, what is a?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 9 } { 2 }$
D) $\frac { 7 } { 3 }$
E) $\frac { 8 } { 3 }$

For real numbers $\mathbf { a }$ and $\mathbf { b }$
$$b ^ { 2 } < a \cdot b < b - a$$
Given that, which of the following orderings is correct?
A) $a < 0 < b$ B) $b < 0 < a$ C) $0 < a < b$ D) $\mathrm { b } < \mathrm { a } < 0$ E) $a < b < 0$

For integers a and b
$$16 ^ { a } \cdot 9 ^ { a } = 6 ^ { b } \cdot 8 ^ { 2 }$$
Given this equality, what is the sum $\mathbf { a } + \mathbf { b }$?
A) 6
B) 9
C) 12
D) 15
E) 20

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

Let $\mathrm { a }$, $\mathrm { b }$ and $c$ be non-zero real numbers,
$$\begin{aligned} & \mathrm { p } : \mathrm { a } + \mathrm { b } = 0 \\ & \mathrm { q } : \mathrm { a } + \mathrm { c } < 0 \\ & \mathrm { r } : \mathrm { c } < 0 \end{aligned}$$
the propositions are given.
$$( p \wedge q ) \Rightarrow r$$
Given that the proposition is false; what are the signs of $\mathbf { a }$, $\mathbf { b }$ and $\mathbf { c }$ respectively?
A) $+$, $-$, $+$ B) $+$, $-$, $-$ C) $-$, $-$, $+$ D) $+$, $+$, $-$

For distinct real numbers $a , b$ and $c$,
$$\begin{aligned} & a + b = | a | \\ & b + c = | b | \end{aligned}$$
equalities are given. Accordingly; what is the correct ordering of the numbers $\mathbf { a , b }$ and c?
A) a < b < c
B) a $<$ c $<$ b
C) b $<$ a $<$ c
D) b $<$ c $<$ a
E) c $<$ a $<$ b

For real numbers $a$, $b$, and $c$
$$a - b < 0 < c < c - b$$
the inequality is given.
Accordingly, I. $a \cdot b \cdot c > 0$ II. $( a + c ) \cdot b > 0$ III. $b - a + c > 0$ which of these statements are always true?
A) Only I
B) Only II
C) I and II
D) I and III
E) II and III

On the number line given below, the distance of K to 1 is equal to the distance of L to 2.
Accordingly, which of the following could be the value of the product $K \cdot L$?
A) A
B) B
C) C
D) D
E) E

Bilge will choose two of the soup, salad, and fruit options given as one portion each at lunch based on the required calorie amount. Regarding the choices she can make, Bilge has calculated that the required calorie amount is
- exceeded when she chooses soup and fruit, - not exceeded when she chooses fruit and salad, - exactly met when she chooses salad and soup.
If the calories of one portion of soup, fruit, and salad are Ç, M, and S respectively, which of the following is the correct ordering of these values?
A) Ç $<$ M $\leq$ S B) Ç $\leq$ S $<$ M C) S $\leq$ Ç $<$ M D) S $<$ M $\leq$ Ç E) M $\leq$ S $<$ Ç